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1. Shard Calculation

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Calculated Shard: 11 (from laksa131)

2. Crawled Status Check

Query:

curl -X POST \
  'http://laksa011.int.ahrefs:8124/' \
  -H 'Content-Type: text/plain' \
  -H 'X-ClickHouse-Database: crawler3' \
  -H 'Authorization: Basic YXBpOg==' \
  -d 'SELECT getAhrefsURLFromUnparsed(src_unparsed) AS found_url, ifNull(toUnixTimestamp(download_stamp), 0) AS crawl_time, ifNull(toUnixTimestamp(props_url_first_seen), 0) AS first_indexed_time, download_http_code AS http_code, src_unparsed AS src_unparsed, src_root_hash AS src_root_hash, history_drop_reason AS history_drop_reason, meta_title AS meta_title, meta_descriptions AS meta_descriptions, attrs_boilerpipe_text AS attrs_boilerpipe_text, attrs_markdown AS attrs_markdown, attrs_readable_markdown AS attrs_readable_markdown, meta_canonical AS meta_canonical FROM crawler3.page_info_local FINAL PREWHERE (src_root_hash, src_unparsed) IN ((getAhrefsRootHashFromUnparsed(getAhrefsUnparsedNoserviceFromURL(\'https://www.scribbr.com/statistics/coefficient-of-determination/\')), getAhrefsUnparsedNoserviceFromURL(\'https://www.scribbr.com/statistics/coefficient-of-determination/\'))) FORMAT JSONEachRow'

Response:

{"found_url":"https:\/\/www.scribbr.com\/statistics\/coefficient-of-determination\/","crawl_time":1775454106,"first_indexed_time":1650895560,"http_code":200,"src_unparsed":"com,scribbr!www,\/statistics\/coefficient-of-determination\/ s443","src_root_hash":"9175914091137272011","history_drop_reason":null,"meta_title":"Coefficient of Determination (R²) | Calculation & Interpretation","meta_descriptions":["The coefficient of determination (R²) is a number between 0 and 1 that measures how well a statistical model predicts an outcome."],"attrs_boilerpipe_text":"Published on\n    April 22, 2022\n    by\nShaun Turney\n.\n                \n        Revised on\n        June 22, 2023.\nThe coefficient of determination is a number between 0 and 1 that measures how well a statistical\nmodel\npredicts an outcome.\nInterpreting the coefficient of determination\nCoefficient of determination (\nR\n2\n)\nInterpretation\n0\nThe model\ndoes not\npredict the outcome.\nBetween 0 and 1\nThe model\npartially\npredicts the outcome.\n1\nThe model\nperfectly\npredicts the outcome.\nThe coefficient of determination is often written as\nR\n2\n, which is pronounced as “r squared.” For\nsimple linear regressions\n, a lowercase\nr\nis usually used instead (\nr\n2\n).\nTable of contents\nWhat is the coefficient of determination?\nCalculating the coefficient of determination\nInterpreting the coefficient of determination\nReporting the coefficient of determination\nPractice questions\nOther interesting articles\nFrequently asked questions about the coefficient of determination\nWhat is the coefficient of determination?\nThe coefficient of determination (\nR\n²) measures how well a statistical\nmodel\npredicts an outcome. The outcome is represented by the model’s\ndependent variable\n.\nThe lowest possible value of\nR\n² is 0 and the highest possible value is 1. Put simply, the better a model is at making predictions, the closer its\nR\n² will be to 1.\nExample: Coefficient of determination\nImagine that you perform a simple linear regression that predicts students’ exam scores (dependent variable) from their time spent studying (\nindependent variable\n).\nIf the\nR\n2\nis 0, the linear regression model doesn’t allow you to predict exam scores any better than simply estimating that everyone has an\naverage\nexam score.\nIf the\nR\n2\nis between 0 and 1, the model allows you to partially predict exam scores. The model’s estimates are not perfect, but they’re better than simply using the average exam score.\nIf the\nR\n2\nis 1, the model allows you to perfectly predict anyone’s exam score.\nMore technically,\nR\n2\nis a measure of goodness of fit. It is the proportion of\nvariance\nin the dependent variable that is explained by the model.\nGraphing your linear regression data usually gives you a good clue as to whether its\nR\n2\nis high or low. For example, the graphs below show two sets of simulated data:\nThe observations are shown as dots.\nThe model’s predictions (the line of best fit) are shown as a black line.\nThe distance between the observations and their predicted values (the residuals) are shown as purple lines.\nYou can see in the first dataset that when the\nR\n2\nis high, the observations are\nclose to the model’s predictions\n. In other words, most points are close to the line of best fit:\nNote:\nThe coefficient of determination is always positive, even when the\ncorrelation\nis negative.\nIn contrast, you can see in the second dataset that when the\nR\n2\nis low, the observations are\nfar from the model’s predictions\n. In other words, when the\nR\n2\nis low, many points are far from the line of best fit:\nReceive feedback on language, structure, and formatting\nProfessional editors proofread and edit your paper by focusing on:\nAcademic style\nVague sentences\nGrammar\nStyle consistency\nSee an example\nCalculating the coefficient of determination\nYou can choose between two formulas to calculate the coefficient of determination (\nR\n²) of a simple linear regression. The first formula is specific to\nsimple linear regressions\n, and the second formula can be used to calculate the\nR\n² of many types of statistical models.\nFormula 1: Using the correlation coefficient\nFormula 1:\n \n \nWhere\nr\n=\nPearson correlation coefficient\nExample: Calculating\nR\n² using the correlation coefficient\nYou are studying the relationship between heart rate and age in children, and you find that the two variables have a negative Pearson correlation:\n \n \nThis value can be used to calculate the coefficient of determination (\nR\n²) using Formula 1:\n \n \n \n \n \n \nFormula 2: Using the regression outputs\nFormula 2:\n \n \nWhere:\nRSS = sum of squared residuals\nTSS = total sum of squares\nExample: Calculating\nR\n² using regression outputs\nAs part of performing a simple linear regression that predicts students’ exam scores (dependent variable) from their study time (independent variable), you calculate that:\n \n \n \n \nThese values can be used to calculate the coefficient of determination (\nR\n²) using Formula 2:\n \n \n \n \n \n \n \n \nInterpreting the coefficient of determination\nYou can interpret the coefficient of determination (\nR\n²) as the proportion of\nvariance\nin the\ndependent variable\nthat is predicted by the statistical\nmodel\n.\nAnother way of thinking of it is that the\nR\n² is the proportion of variance that is shared between the independent and dependent variables.\nYou can also say that the\nR\n² is the proportion of variance “explained” or “accounted for” by the model. The proportion that remains (1 −\nR\n²) is the variance that is not predicted by the model.\nIf you prefer, you can write the\nR\n² as a percentage instead of a proportion. Simply multiply the proportion by 100.\nR\n² as an effect size\nLastly, you can also interpret the\nR\n² as an\neffect size\n: a measure of the strength of the relationship between the dependent and independent variables. Psychologist and statistician Jacob Cohen (1988) suggested the following rules of thumb for\nsimple linear regressions\n:\nR²\nas an effect size\nMinimum coefficient of determination (\nR\n²) value\nEffect size interpretation\n.01\nSmall\n.09\nMedium\n.25\nLarge\nBe careful: the\nR\n² on its own can’t tell you anything about\ncausation\n.\nExample: Interpreting\nR\n²\nA simple linear regression that predicts students’ exam scores (dependent variable) from their study time (independent variable) has an\nR\n² of .71. From this\nR\n² value, we know that:\n71% of the variance in students’ exam scores is predicted by their study time\n29% of the variance in student’s exam scores is unexplained by the model\nThe students’ study time has a large effect on their exam scores\nStudying longer may or may not\ncause\nan improvement in the students’ scores. Although this causal relationship is very plausible, the\nR\n² alone can’t tell us why there’s a relationship between students’ study time and exam scores.\nFor example, students might find studying less frustrating when they understand the course material well, so they study longer.\nReporting the coefficient of determination\nIf you decide to include a coefficient of determination (\nR\n²) in your\nresearch paper\n,\ndissertation\nor\nthesis\n, you should report it in your\nresults section\n. You can follow these rules if you want to\nreport statistics in APA Style\n:\nYou should use “\nr\n²” for statistical models with one independent variable (such as simple linear regressions). Use “\nR\n²” for statistical models with multiple independent variables.\nYou don’t need to provide a reference or formula since the coefficient of determination is a commonly used statistic.\nYou should italicize\nr\n² and\nR\n² when reporting their values (but don’t italicize the ²).\nYou shouldn’t include a leading zero (a zero before the decimal point) since the coefficient of determination can’t be greater than one.\nYou should provide two significant digits after the decimal point.\nVery often, the coefficient of determination is provided alongside related statistical results, such as the\nF\nvalue\n,\ndegrees of freedom\n, and\np\nvalue\n.\nExample: Reporting\nr\n² in APA Style\nStudents’ exam scores were predicted by their study time,\nr\n² = .71,\nF\n(1,32) = 7.33,\np\n= .003.\nPractice questions\nOther interesting articles\nIf you want to know more about\nstatistics\n,\nmethodology\n, or\nresearch bias\n, make sure to check out some of our other articles with explanations and examples.\nStatistics\nChi square test of independence\nStatistical power\nDescriptive statistics\nDegrees of freedom\nPearson correlation\nNull hypothesis\nMethodology\nDouble-blind study\nCase-control study\nResearch ethics\nData collection\nHypothesis testing\nStructured interviews\nFrequently asked questions about the coefficient of determination\nCite this Scribbr article\nIf you want to cite this source, you can copy and paste the citation or click the “Cite this Scribbr article” button to automatically add the citation to our free Citation Generator.\nTurney, S. \n                    (2023, June 22).\nCoefficient of Determination (R²) | Calculation & Interpretation.\nScribbr. \n        Retrieved April 2, 2026, \n        from https:\/\/www.scribbr.com\/statistics\/coefficient-of-determination\/\nYou have already voted. 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Coefficient of Determination (R²) \\| Calculation & Interpretation\n\n### Statistics\n- [Statistical analysis step by step](https:\/\/www.scribbr.com\/category\/statistics\/)\n- [Data collection](https:\/\/www.scribbr.com\/methodology\/data-collection\/ \"Data Collection | Definition, Methods & Examples\")\n  - [Data collection guide](https:\/\/www.scribbr.com\/methodology\/data-collection\/ \"Data Collection | Definition, Methods & Examples\")\n  - [Experimental design](https:\/\/www.scribbr.com\/methodology\/experimental-design\/ \"Guide to Experimental Design | Overview, Steps, & Examples\")\n  - [Populations and samples](https:\/\/www.scribbr.com\/methodology\/population-vs-sample\/ \"Population vs. Sample | Definitions, Differences & Examples\")\n    - [Populations and samples](https:\/\/www.scribbr.com\/methodology\/population-vs-sample\/ \"Population vs. Sample | Definitions, Differences & Examples\")\n    - [Sampling methods](https:\/\/www.scribbr.com\/methodology\/sampling-methods\/ \"Sampling Methods | Types, Techniques & Examples\")\n  - [Types of variables](https:\/\/www.scribbr.com\/methodology\/types-of-variables\/ \"Types of Variables in Research & Statistics | Examples\")\n    - [Types of variables](https:\/\/www.scribbr.com\/methodology\/types-of-variables\/ \"Types of Variables in Research & Statistics | Examples\")\n    - [Levels of measurement](https:\/\/www.scribbr.com\/statistics\/coefficient-of-determination\/)\n    - [Nominal data](https:\/\/www.scribbr.com\/statistics\/nominal-data\/ \"Nominal Data | Definition, Examples, Data Collection & Analysis\")\n    - [Ordinal data](https:\/\/www.scribbr.com\/statistics\/ordinal-data\/ \"Ordinal Data | Definition, Examples, Data Collection & Analysis\")\n    - [Interval data](https:\/\/www.scribbr.com\/statistics\/interval-data\/ \"Interval Data and How to Analyze It | Definitions & Examples\")\n    - [Ratio data](https:\/\/www.scribbr.com\/statistics\/ratio-data\/ \"Ratio Scales | Definition, Examples, & Data Analysis\")\n  - [Data cleansing](https:\/\/www.scribbr.com\/methodology\/data-cleansing\/ \"What Is Data Cleansing? | Definition, Guide & Examples\")\n    - 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[Overview](https:\/\/www.scribbr.com\/statistics\/probability-distributions\/ \"Probability Distribution | Formula, Types, & Examples\")\n    - [Normal distribution](https:\/\/www.scribbr.com\/statistics\/normal-distribution\/ \"Normal Distribution | Examples, Formulas, & Uses\")\n    - [Standard normal distribution](https:\/\/www.scribbr.com\/statistics\/standard-normal-distribution\/ \"The Standard Normal Distribution | Calculator, Examples & Uses\")\n    - [Poisson distribution](https:\/\/www.scribbr.com\/statistics\/poisson-distribution\/ \"Poisson Distributions | Definition, Formula & Examples\")\n    - [Chi-square distribution](https:\/\/www.scribbr.com\/statistics\/chi-square-distributions\/ \"Chi-Square (Χ²) Distributions | Definition & Examples\")\n    - [Chi-square table](https:\/\/www.scribbr.com\/statistics\/chi-square-distribution-table\/ \"Chi-Square (Χ²) Table | Examples & Downloadable Table\")\n    - [*t* distribution](https:\/\/www.scribbr.com\/statistics\/t-distribution\/ \"T-distribution: What it is and how to use it\")\n    - 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[Statistical significance](https:\/\/www.scribbr.com\/statistics\/statistical-significance\/ \"An Easy Introduction to Statistical Significance (With Examples)\")\n    - [*p* value](https:\/\/www.scribbr.com\/statistics\/p-value\/ \"Understanding P values | Definition and Examples\")\n    - [Type I & Type II errors](https:\/\/www.scribbr.com\/statistics\/type-i-and-type-ii-errors\/ \"Type I & Type II Errors | Differences, Examples, Visualizations\")\n    - [Statistical power](https:\/\/www.scribbr.com\/statistics\/statistical-power\/ \"Statistical Power and Why It Matters | A Simple Introduction\")\n- [Statistical tests](https:\/\/www.scribbr.com\/statistics\/statistical-tests\/ \"Choosing the Right Statistical Test | Types & Examples\")\n  - [Choosing the right test](https:\/\/www.scribbr.com\/statistics\/statistical-tests\/ \"Choosing the Right Statistical Test | Types & Examples\")\n  - [Assumptions for hypothesis testing](https:\/\/www.scribbr.com\/statistics\/skewness\/ \"Skewness | Definition, Examples & Formula\")\n    - [Skewness](https:\/\/www.scribbr.com\/statistics\/skewness\/ \"Skewness | Definition, Examples & Formula\")\n    - [Kurtosis](https:\/\/www.scribbr.com\/statistics\/kurtosis\/ \"What Is Kurtosis? | Definition, Examples & Formula\")\n  - [Correlation](https:\/\/www.scribbr.com\/methodology\/correlation-vs-causation\/ \"Correlation vs. Causation | Difference, Designs & Examples\")\n    - [Correlation vs. causation](https:\/\/www.scribbr.com\/methodology\/correlation-vs-causation\/ \"Correlation vs. Causation | Difference, Designs & Examples\")\n    - [Correlation coefficient](https:\/\/www.scribbr.com\/statistics\/correlation-coefficient\/ \"Correlation Coefficient | Types, Formulas & Examples\")\n    - [Pearson correlation](https:\/\/www.scribbr.com\/statistics\/pearson-correlation-coefficient\/ \"Pearson Correlation Coefficient (r) | Guide & Examples\")\n  - [Regression analysis](https:\/\/www.scribbr.com\/statistics\/simple-linear-regression\/ \"Simple Linear Regression | An Easy Introduction & Examples\")\n    - [Simple linear regression](https:\/\/www.scribbr.com\/statistics\/simple-linear-regression\/ \"Simple Linear Regression | An Easy Introduction & Examples\")\n    - [Multiple linear regression](https:\/\/www.scribbr.com\/statistics\/multiple-linear-regression\/ \"Multiple Linear Regression | A Quick Guide (Examples)\")\n    - [Linear regression in R](https:\/\/www.scribbr.com\/statistics\/linear-regression-in-r\/ \"Linear Regression in R | A Step-by-Step Guide & Examples\")\n  - [*t* tests](https:\/\/www.scribbr.com\/statistics\/t-test\/ \"An Introduction to t Tests | Definitions, Formula and Examples\")\n  - [ANOVAs](https:\/\/www.scribbr.com\/statistics\/one-way-anova\/ \"One-way ANOVA | When and How to Use It (With Examples)\")\n    - [One-way ANOVA](https:\/\/www.scribbr.com\/statistics\/one-way-anova\/ \"One-way ANOVA | When and How to Use It (With Examples)\")\n    - [Two-way ANOVA](https:\/\/www.scribbr.com\/statistics\/two-way-anova\/ \"Two-Way ANOVA | Examples & When To Use It\")\n    - [ANOVA in R](https:\/\/www.scribbr.com\/statistics\/anova-in-r\/ \"ANOVA in R | A Complete Step-by-Step Guide with Examples\")\n  - [Chi-square](https:\/\/www.scribbr.com\/statistics\/chi-square-tests\/ \"Chi-Square (Χ²) Tests | Types, Formula & Examples\")\n    - [Overview chi-square tests](https:\/\/www.scribbr.com\/statistics\/chi-square-tests\/ \"Chi-Square (Χ²) Tests | Types, Formula & Examples\")\n    - [Chi-square goodness of fit test](https:\/\/www.scribbr.com\/statistics\/chi-square-goodness-of-fit\/ \"Chi-Square Goodness of Fit Test | Formula, Guide & Examples\")\n    - [Chi-square test of independence](https:\/\/www.scribbr.com\/statistics\/chi-square-test-of-independence\/ \"Chi-Square Test of Independence | Formula, Guide & Examples\")\n  - [Effect size](https:\/\/www.scribbr.com\/statistics\/effect-size\/ \"What is Effect Size and Why Does It Matter? (Examples)\")\n    - [Overview of effect sizes](https:\/\/www.scribbr.com\/statistics\/effect-size\/ \"What is Effect Size and Why Does It Matter? (Examples)\")\n    - [Coefficient of determination](https:\/\/www.scribbr.com\/statistics\/coefficient-of-determination\/ \"Coefficient of Determination (R²) | Calculation & Interpretation\")\n  - [Model selection](https:\/\/www.scribbr.com\/statistics\/akaike-information-criterion\/ \"Akaike Information Criterion | When & How to Use It (Example)\")\n    - [Akaike information criterion](https:\/\/www.scribbr.com\/statistics\/akaike-information-criterion\/ \"Akaike Information Criterion | When & How to Use It (Example)\")\n- [Reporting statistics in APA](https:\/\/www.scribbr.com\/apa-style\/numbers-and-statistics\/ \"Reporting Statistics in APA Style | Guidelines & Examples\")\n\n### Interesting topics\n- [Parts of speech](https:\/\/www.scribbr.com\/category\/parts-of-speech\/ \"View all articles about Parts of speech\")\n- [Working with sources](https:\/\/www.scribbr.com\/category\/working-with-sources\/ \"View all articles about Working with sources\")\n- 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services\n[![](data:image\/svg+xml,%3Csvg%20xmlns='http:\/\/www.w3.org\/2000\/svg'%20viewBox='0%200%2032%2032'%3E%3C\/svg%3E) ![](https:\/\/www.scribbr.com\/wp-content\/uploads\/2024\/09\/human-proofreading.svg) Proofreading & Editing Have a human editor polish your writing to ensure your arguments are judged on merit, not grammar errors. 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Try for free](https:\/\/www.scribbr.com\/paraphrasing-tool\/?scr_source=knowledgebase&scr_medium=sidebar&scr_campaign=minimalistic-sidebar)\n\n[![](data:image\/svg+xml,%3Csvg%20xmlns='http:\/\/www.w3.org\/2000\/svg'%20viewBox='0%200%2032%2032'%3E%3C\/svg%3E) ![](https:\/\/www.scribbr.com\/wp-content\/uploads\/2024\/09\/grammer-checker.svg) Grammar Checker Eliminate grammar errors and improve your writing with our free AI-powered grammar checker. Try for free](https:\/\/www.scribbr.com\/grammar-checker\/?scr_source=knowledgebase&scr_medium=sidebar&scr_campaign=minimalistic-sidebar)\n\n# Coefficient of Determination (R²) \\| Calculation & Interpretation\nPublished on April 22, 2022 by  [Shaun Turney](https:\/\/www.scribbr.com\/author\/shaunt\/ \"All articles by Shaun Turney\"). Revised on June 22, 2023.\n\nThe coefficient of determination is a number between 0 and 1 that measures how well a statistical [model](https:\/\/www.scribbr.com\/frequently-asked-questions\/what-is-a-model\/) predicts an outcome.\n\n| Coefficient of determination (*R*2) | Interpretation |\n|---|---|\n| 0 | The model **does not** predict the outcome. |\n| Between 0 and 1 | The model **partially** predicts the outcome. |\n| 1 | The model **perfectly** predicts the outcome. |\n\nThe coefficient of determination is often written as *R*2, which is pronounced as “r squared.” For [simple linear regressions](https:\/\/www.scribbr.com\/statistics\/simple-linear-regression\/), a lowercase *r* is usually used instead (*r*2).\n\n## Table of contents\n1. [What is the coefficient of determination?](https:\/\/www.scribbr.com\/statistics\/coefficient-of-determination\/#what)\n2. [Calculating the coefficient of determination](https:\/\/www.scribbr.com\/statistics\/coefficient-of-determination\/#calculating)\n3. [Interpreting the coefficient of determination](https:\/\/www.scribbr.com\/statistics\/coefficient-of-determination\/#interpreting)\n4. [Reporting the coefficient of determination](https:\/\/www.scribbr.com\/statistics\/coefficient-of-determination\/#reporting)\n5. [Practice questions](https:\/\/www.scribbr.com\/statistics\/coefficient-of-determination\/#quiz)\n6. [Other interesting articles](https:\/\/www.scribbr.com\/statistics\/coefficient-of-determination\/#other)\n7. [Frequently asked questions about the coefficient of determination](https:\/\/www.scribbr.com\/statistics\/coefficient-of-determination\/#frequently-asked-questions-about-the-coefficient-of-determination)\n## What is the coefficient of determination?\nThe coefficient of determination (*R*²) measures how well a statistical [model](https:\/\/www.scribbr.com\/frequently-asked-questions\/what-is-a-model\/) predicts an outcome. The outcome is represented by the model’s [dependent variable](https:\/\/www.scribbr.com\/methodology\/independent-and-dependent-variables\/#dependent).\n\nThe lowest possible value of *R*² is 0 and the highest possible value is 1. Put simply, the better a model is at making predictions, the closer its *R*² will be to 1.\n\nExample: Coefficient of determination\n\nImagine that you perform a simple linear regression that predicts students’ exam scores (dependent variable) from their time spent studying ([independent variable](https:\/\/www.scribbr.com\/methodology\/independent-and-dependent-variables\/#independent)).\n\n- If the *R*2 is 0, the linear regression model doesn’t allow you to predict exam scores any better than simply estimating that everyone has an [average](https:\/\/www.scribbr.com\/statistics\/mean\/) exam score.\n- If the *R*2 is between 0 and 1, the model allows you to partially predict exam scores. The model’s estimates are not perfect, but they’re better than simply using the average exam score.\n- If the *R*2 is 1, the model allows you to perfectly predict anyone’s exam score.\n\nMore technically, *R*2 is a measure of goodness of fit. It is the proportion of [variance](https:\/\/www.scribbr.com\/statistics\/variance\/) in the dependent variable that is explained by the model.\n\nGraphing your linear regression data usually gives you a good clue as to whether its *R*2 is high or low. For example, the graphs below show two sets of simulated data:\n\n- The observations are shown as dots.\n- The model’s predictions (the line of best fit) are shown as a black line.\n- The distance between the observations and their predicted values (the residuals) are shown as purple lines.\n\nYou can see in the first dataset that when the *R*2 is high, the observations are close to the model’s predictions. In other words, most points are close to the line of best fit:\n\n![Coefficient of determination (R2) = 0.9](data:image\/svg+xml,%3Csvg%20xmlns='http:\/\/www.w3.org\/2000\/svg'%20viewBox='0%200%20660%20480'%3E%3C\/svg%3E)![Coefficient of determination (R2) = 0.9](https:\/\/www.scribbr.com\/wp-content\/uploads\/2022\/04\/Coefficient-of-determination_0.9.webp)\n\n**Note:** The coefficient of determination is always positive, even when the [correlation](https:\/\/www.scribbr.com\/frequently-asked-questions\/what-is-a-correlation\/) is negative.\n\nIn contrast, you can see in the second dataset that when the *R*2 is low, the observations are far from the model’s predictions. In other words, when the *R*2 is low, many points are far from the line of best fit:\n\n![Coefficient of determination (R2) = 0.2](data:image\/svg+xml,%3Csvg%20xmlns='http:\/\/www.w3.org\/2000\/svg'%20viewBox='0%200%20660%20480'%3E%3C\/svg%3E)![Coefficient of determination (R2) = 0.2](https:\/\/www.scribbr.com\/wp-content\/uploads\/2022\/04\/Coefficient-of-determination_0.2.webp)\n\n### Receive feedback on language, structure, and formatting\nProfessional editors proofread and edit your paper by focusing on:\n\n- Academic style\n- Vague sentences\n- Grammar\n- Style consistency\n\n[See an example](https:\/\/www.scribbr.com\/proofreading-editing\/paper-editing-service\/?scr_source=knowledgebase&scr_medium=in-text&scr_campaign=proofreading&src_content=feedback+language+structure+layout)\n\n![](data:image\/svg+xml,%3Csvg%20xmlns='http:\/\/www.w3.org\/2000\/svg'%20viewBox='0%200%20830%20660'%3E%3C\/svg%3E)\n\n![](https:\/\/www.scribbr.com\/wp-content\/uploads\/2018\/01\/dissertation-proofreading-service.png)\n## Calculating the coefficient of determination\nYou can choose between two formulas to calculate the coefficient of determination (*R*²) of a simple linear regression. The first formula is specific to [simple linear regressions](https:\/\/www.scribbr.com\/statistics\/simple-linear-regression\/), and the second formula can be used to calculate the *R*² of many types of statistical models.\n\n### Formula 1: Using the correlation coefficient\nFormula 1:\n\n![\\\\begin{equation\\*}R^2=(r)^2\\\\end{equation\\*}](data:image\/svg+xml,%3Csvg%20xmlns='http:\/\/www.w3.org\/2000\/svg'%20viewBox='0%200%2078%2023'%3E%3C\/svg%3E)![\\\\begin{equation\\*}R^2=(r)^2\\\\end{equation\\*}](https:\/\/www.scribbr.com\/wp-content\/ql-cache\/quicklatex.com-8c4d82b2fe20f208da8c31b613e5fd89_l3.png)\n\nWhere *r* = [Pearson correlation coefficient](https:\/\/www.scribbr.com\/statistics\/pearson-correlation-coefficient\/)\n\nExample: Calculating *R*² using the correlation coefficient\n\nYou are studying the relationship between heart rate and age in children, and you find that the two variables have a negative Pearson correlation:\n\n![\\\\begin{equation\\*}r=-0.28\\\\end{equation\\*}](data:image\/svg+xml,%3Csvg%20xmlns='http:\/\/www.w3.org\/2000\/svg'%20viewBox='0%200%2082%2013'%3E%3C\/svg%3E)![\\\\begin{equation\\*}r=-0.28\\\\end{equation\\*}](https:\/\/www.scribbr.com\/wp-content\/ql-cache\/quicklatex.com-028980c08bb3f318f52836e30ee9cfa9_l3.png)\n\nThis value can be used to calculate the coefficient of determination (*R*²) using Formula 1:\n\n![\\\\begin{equation\\*}R^2=(r)^2\\\\end{equation\\*}](data:image\/svg+xml,%3Csvg%20xmlns='http:\/\/www.w3.org\/2000\/svg'%20viewBox='0%200%2078%2023'%3E%3C\/svg%3E)![\\\\begin{equation\\*}R^2=(r)^2\\\\end{equation\\*}](https:\/\/www.scribbr.com\/wp-content\/ql-cache\/quicklatex.com-8c4d82b2fe20f208da8c31b613e5fd89_l3.png)\n\n![\\\\begin{equation\\*}R^2=(-0.28)^2\\\\end{equation\\*}](data:image\/svg+xml,%3Csvg%20xmlns='http:\/\/www.w3.org\/2000\/svg'%20viewBox='0%200%20117%2023'%3E%3C\/svg%3E)![\\\\begin{equation\\*}R^2=(-0.28)^2\\\\end{equation\\*}](https:\/\/www.scribbr.com\/wp-content\/ql-cache\/quicklatex.com-335562cda79b94c68f75c3b43bf7a341_l3.png)\n\n![\\\\begin{equation\\*}R^2=0.08\\\\end{equation\\*}](data:image\/svg+xml,%3Csvg%20xmlns='http:\/\/www.w3.org\/2000\/svg'%20viewBox='0%200%2080%2018'%3E%3C\/svg%3E)![\\\\begin{equation\\*}R^2=0.08\\\\end{equation\\*}](https:\/\/www.scribbr.com\/wp-content\/ql-cache\/quicklatex.com-7c06a31c5618b67f96fec79ffb74e1e3_l3.png)\n### Formula 2: Using the regression outputs\nFormula 2:\n\n![\\\\begin{equation\\*}R^2=1-\\\\dfrac{\\\\textup{RSS}}{\\\\textup{TSS}}\\\\end{equation\\*}](data:image\/svg+xml,%3Csvg%20xmlns='http:\/\/www.w3.org\/2000\/svg'%20viewBox='0%200%20117%2039'%3E%3C\/svg%3E)![\\\\begin{equation\\*}R^2=1-\\\\dfrac{\\\\textup{RSS}}{\\\\textup{TSS}}\\\\end{equation\\*}](https:\/\/www.scribbr.com\/wp-content\/ql-cache\/quicklatex.com-77060d9686c4e6200e15410694b0c221_l3.png)\n\nWhere:\n\n- RSS = sum of squared residuals\n- TSS = total sum of squares\n\nExample: Calculating *R*² using regression outputs\n\nAs part of performing a simple linear regression that predicts students’ exam scores (dependent variable) from their study time (independent variable), you calculate that:\n\n![\\\\begin{equation\\*}\\\\textup{RSS}=629.22\\\\end{equation\\*}](data:image\/svg+xml,%3Csvg%20xmlns='http:\/\/www.w3.org\/2000\/svg'%20viewBox='0%200%20110%2013'%3E%3C\/svg%3E)![\\\\begin{equation\\*}\\\\textup{RSS}=629.22\\\\end{equation\\*}](https:\/\/www.scribbr.com\/wp-content\/ql-cache\/quicklatex.com-9d3bed919c3705b3686559290d3b64a4_l3.png)\n\n![\\\\begin{equation\\*}\\\\textup{TSS}=2\\\\,187.04\\\\end{equation\\*}](data:image\/svg+xml,%3Csvg%20xmlns='http:\/\/www.w3.org\/2000\/svg'%20viewBox='0%200%20123%2014'%3E%3C\/svg%3E)![\\\\begin{equation\\*}\\\\textup{TSS}=2\\\\,187.04\\\\end{equation\\*}](https:\/\/www.scribbr.com\/wp-content\/ql-cache\/quicklatex.com-23b0842ab7eb241dd98d4073491126ce_l3.png)\n\nThese values can be used to calculate the coefficient of determination (*R*²) using Formula 2:\n\n![\\\\begin{equation\\*}R^2=1-\\\\dfrac{\\\\textup{RSS}}{\\\\textup{TSS}}\\\\end{equation\\*}](data:image\/svg+xml,%3Csvg%20xmlns='http:\/\/www.w3.org\/2000\/svg'%20viewBox='0%200%20117%2039'%3E%3C\/svg%3E)![\\\\begin{equation\\*}R^2=1-\\\\dfrac{\\\\textup{RSS}}{\\\\textup{TSS}}\\\\end{equation\\*}](https:\/\/www.scribbr.com\/wp-content\/ql-cache\/quicklatex.com-77060d9686c4e6200e15410694b0c221_l3.png)\n\n![\\\\begin{equation\\*}R^2=1-\\\\dfrac{629.22}{2\\\\,187.04}\\\\end{equation\\*}](data:image\/svg+xml,%3Csvg%20xmlns='http:\/\/www.w3.org\/2000\/svg'%20viewBox='0%200%20147%2039'%3E%3C\/svg%3E)![\\\\begin{equation\\*}R^2=1-\\\\dfrac{629.22}{2\\\\,187.04}\\\\end{equation\\*}](https:\/\/www.scribbr.com\/wp-content\/ql-cache\/quicklatex.com-25f7352ea18335eddc525ac74d0fea32_l3.png)\n\n![\\\\begin{equation\\*}R^2=1-0.29\\\\end{equation\\*}](data:image\/svg+xml,%3Csvg%20xmlns='http:\/\/www.w3.org\/2000\/svg'%20viewBox='0%200%20112%2018'%3E%3C\/svg%3E)![\\\\begin{equation\\*}R^2=1-0.29\\\\end{equation\\*}](https:\/\/www.scribbr.com\/wp-content\/ql-cache\/quicklatex.com-b74ead3cc867978ff7ec367f3830fba9_l3.png)\n\n![\\\\begin{equation\\*}R^2=0.71\\\\end{equation\\*}](data:image\/svg+xml,%3Csvg%20xmlns='http:\/\/www.w3.org\/2000\/svg'%20viewBox='0%200%2079%2018'%3E%3C\/svg%3E)![\\\\begin{equation\\*}R^2=0.71\\\\end{equation\\*}](https:\/\/www.scribbr.com\/wp-content\/ql-cache\/quicklatex.com-6e220f1a6c30feb37df1abf358c6dab4_l3.png)\n## Interpreting the coefficient of determination\nYou can interpret the coefficient of determination (*R*²) as the proportion of [variance](https:\/\/www.scribbr.com\/statistics\/variance\/) in the [dependent variable](https:\/\/www.scribbr.com\/methodology\/independent-and-dependent-variables\/#dependent) that is predicted by the statistical [model](https:\/\/www.scribbr.com\/frequently-asked-questions\/what-is-a-model\/).\n\nAnother way of thinking of it is that the *R*² is the proportion of variance that is shared between the independent and dependent variables.\n\nYou can also say that the *R*² is the proportion of variance “explained” or “accounted for” by the model. The proportion that remains (1 − *R*²) is the variance that is not predicted by the model.\n\nIf you prefer, you can write the *R*² as a percentage instead of a proportion. Simply multiply the proportion by 100.\n\n### *R*² as an effect size\nLastly, you can also interpret the *R*² as an [effect size](https:\/\/www.scribbr.com\/statistics\/effect-size\/): a measure of the strength of the relationship between the dependent and independent variables. Psychologist and statistician Jacob Cohen (1988) suggested the following rules of thumb for [simple linear regressions](https:\/\/www.scribbr.com\/statistics\/simple-linear-regression\/):\n\n| Minimum coefficient of determination (*R*²) value | Effect size interpretation |\n|---|---|\n| .01 | Small |\n| .09 | Medium |\n| .25 | Large |\n\nBe careful: the *R*² on its own can’t tell you anything about [causation](https:\/\/www.scribbr.com\/methodology\/correlation-vs-causation\/).\n\nExample: Interpreting *R*²\n\nA simple linear regression that predicts students’ exam scores (dependent variable) from their study time (independent variable) has an *R*² of .71. From this *R*² value, we know that:\n\n- 71% of the variance in students’ exam scores is predicted by their study time\n- 29% of the variance in student’s exam scores is unexplained by the model\n- The students’ study time has a large effect on their exam scores\n\nStudying longer may or may not [cause](https:\/\/www.scribbr.com\/methodology\/correlation-vs-causation\/) an improvement in the students’ scores. Although this causal relationship is very plausible, the *R*² alone can’t tell us why there’s a relationship between students’ study time and exam scores.\n\nFor example, students might find studying less frustrating when they understand the course material well, so they study longer.\n## Reporting the coefficient of determination\nIf you decide to include a coefficient of determination (*R*²) in your [research paper](https:\/\/www.scribbr.com\/category\/research-paper\/), [dissertation](https:\/\/www.scribbr.com\/category\/dissertation\/) or [thesis](https:\/\/www.scribbr.com\/dissertation\/thesis\/), you should report it in your [results section](https:\/\/www.scribbr.com\/dissertation\/results\/). You can follow these rules if you want to [report statistics in APA Style](https:\/\/www.scribbr.com\/apa-style\/numbers-and-statistics\/):\n\n- You should use “*r*²” for statistical models with one independent variable (such as simple linear regressions). Use “*R*²” for statistical models with multiple independent variables.\n- You don’t need to provide a reference or formula since the coefficient of determination is a commonly used statistic.\n- You should italicize *r*² and *R*² when reporting their values (but don’t italicize the ²).\n- You shouldn’t include a leading zero (a zero before the decimal point) since the coefficient of determination can’t be greater than one.\n- You should provide two significant digits after the decimal point.\n- Very often, the coefficient of determination is provided alongside related statistical results, such as the [*F* value](https:\/\/www.scribbr.com\/statistics\/test-statistic\/), [degrees of freedom](https:\/\/www.scribbr.com\/statistics\/degrees-of-freedom\/), and [*p* value](https:\/\/www.scribbr.com\/statistics\/p-value\/).\n\nExample: Reporting *r*² in APA Style\n\nStudents’ exam scores were predicted by their study time, *r*² = .71, *F*(1,32) = 7.33, *p* \\= .003.\n\n## Here's why students love Scribbr's proofreading services\n[Trustpilot](https:\/\/www.trustpilot.com\/review\/www.scribbr.com)\n\n[Discover proofreading & editing](https:\/\/www.scribbr.com\/proofreading-editing\/paper-editing-service\/?scr_source=knowledgebase&scr_medium=in-text&scr_campaign=proofreading&src_content=trustpilot)\n## Practice questions\npowered by [Typeform](https:\/\/admin.typeform.com\/signup?utm_campaign=N3apIC&utm_source=typeform.com-01D8JTMH8P14WJGAWB4RB1BJEB-pro&utm_medium=typeform&utm_content=typeform-embedded-poweredbytypeform&utm_term=EN)\n\n## Other interesting articles\nIf you want to know more about [statistics](https:\/\/www.scribbr.com\/category\/statistics\/), [methodology](https:\/\/www.scribbr.com\/category\/methodology\/), or [research bias](https:\/\/www.scribbr.com\/faq-category\/research-bias\/), make sure to check out some of our other articles with explanations and examples.\n\n**Statistics**\n\n- [Chi square test of independence](https:\/\/www.scribbr.com\/statistics\/chi-square-test-of-independence\/)\n- [Statistical power](https:\/\/www.scribbr.com\/statistics\/statistical-power\/)\n- [Descriptive statistics](https:\/\/www.scribbr.com\/statistics\/descriptive-statistics\/)\n- [Degrees of freedom](https:\/\/www.scribbr.com\/statistics\/degrees-of-freedom\/)\n- [Pearson correlation](https:\/\/www.scribbr.com\/statistics\/pearson-correlation-coefficient\/)\n- [Null hypothesis](https:\/\/www.scribbr.com\/statistics\/null-and-alternative-hypotheses\/)\n\n**Methodology**\n\n- [Double-blind study](https:\/\/www.scribbr.com\/methodology\/double-blind-study\/)\n- [Case-control study](https:\/\/www.scribbr.com\/methodology\/case-control-study\/)\n- [Research ethics](https:\/\/www.scribbr.com\/methodology\/research-ethics\/)\n- [Data collection](https:\/\/www.scribbr.com\/methodology\/data-collection\/)\n- [Hypothesis testing](https:\/\/www.scribbr.com\/statistics\/hypothesis-testing\/)\n- [Structured interviews](https:\/\/www.scribbr.com\/methodology\/structured-interview\/)\n\n**Research bias**\n\n- [Hawthorne effect](https:\/\/www.scribbr.com\/research-bias\/hawthorne-effect\/)\n- [Unconscious bias](https:\/\/www.scribbr.com\/research-bias\/unconscious-bias\/)\n- [Recall bias](https:\/\/www.scribbr.com\/research-bias\/recall-bias\/)\n- [Halo effect](https:\/\/www.scribbr.com\/research-bias\/halo-effect\/)\n- [Self-serving bias](https:\/\/www.scribbr.com\/research-bias\/self-serving-bias\/)\n- [Information bias](https:\/\/www.scribbr.com\/research-bias\/information-bias\/)\n## Frequently asked questions about the coefficient of determination\n[What is the definition of the coefficient of determination (R²)?](https:\/\/www.scribbr.com\/frequently-asked-questions\/definition-of-coefficient-of-determination\/)\n\nThe [coefficient of determination (R²)](https:\/\/www.scribbr.com\/statistics\/coefficient-of-determination\/) is a number between 0 and 1 that measures how well a statistical model predicts an outcome. You can interpret the R² as the proportion of variation in the [dependent variable](https:\/\/www.scribbr.com\/methodology\/independent-and-dependent-variables\/) that is predicted by the statistical model.\n\n[What is the formula for the coefficient of determination (R²)?](https:\/\/www.scribbr.com\/frequently-asked-questions\/what-is-the-formula-for-the-coefficient-of-determination-r%C2%B2\/)\n\nThere are two formulas you can use to calculate the [coefficient of determination (R²)](https:\/\/www.scribbr.com\/statistics\/coefficient-of-determination\/) of a [simple linear regression](https:\/\/www.scribbr.com\/statistics\/simple-linear-regression\/).\n\n**Formula 1:** ![R^2=(r)^2](data:image\/svg+xml,%3Csvg%20xmlns='http:\/\/www.w3.org\/2000\/svg'%20viewBox='0%200%2078%2022'%3E%3C\/svg%3E)![R^2=(r)^2](https:\/\/www.scribbr.com\/wp-content\/ql-cache\/quicklatex.com-06ecfc9eb09f339f2a6f8113909f3fe0_l3.png)\n\n**Formula 2:** ![R^2=1-\\\\dfrac{\\\\textup{RSS}}{\\\\textup{TSS}}](data:image\/svg+xml,%3Csvg%20xmlns='http:\/\/www.w3.org\/2000\/svg'%20viewBox='0%200%20117%2039'%3E%3C\/svg%3E)![R^2=1-\\\\dfrac{\\\\textup{RSS}}{\\\\textup{TSS}}](https:\/\/www.scribbr.com\/wp-content\/ql-cache\/quicklatex.com-5472ee9a4cc0c893784bf2e13e36d78b_l3.png)\n\n[How do I calculate the coefficient of determination (R²) in R?](https:\/\/www.scribbr.com\/frequently-asked-questions\/r%C2%B2-in-r\/)\n\nYou can use the summary() function to view the [R²](https:\/\/www.scribbr.com\/statistics\/coefficient-of-determination\/) of a linear model in R. You will see the “R-squared” near the bottom of the output.\n\n[How do I calculate the coefficient of determination (R²) in Excel?](https:\/\/www.scribbr.com\/frequently-asked-questions\/r%C2%B2-in-excel\/)\n\nYou can use the RSQ() function to calculate [R²](https:\/\/www.scribbr.com\/statistics\/coefficient-of-determination\/) in Excel. If your [dependent variable](https:\/\/www.scribbr.com\/methodology\/independent-and-dependent-variables\/) is in column A and your independent variable is in column B, then click any blank cell and type “RSQ(A:A,B:B)”.\n\n#### Cite this Scribbr article\nIf you want to cite this source, you can copy and paste the citation or click the “Cite this Scribbr article” button to automatically add the citation to our free Citation Generator.\n\n> Turney, S. (2023, June 22). Coefficient of Determination (R²) \\| Calculation & Interpretation. Scribbr. 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Revised on June 22, 2023.\n\nThe coefficient of determination is a number between 0 and 1 that measures how well a statistical [model](https:\/\/www.scribbr.com\/frequently-asked-questions\/what-is-a-model\/) predicts an outcome.\n\n| Coefficient of determination (*R*2) | Interpretation |\n|---|---|\n| 0 | The model **does not** predict the outcome. |\n| Between 0 and 1 | The model **partially** predicts the outcome. |\n| 1 | The model **perfectly** predicts the outcome. |\n\nThe coefficient of determination is often written as *R*2, which is pronounced as “r squared.” For [simple linear regressions](https:\/\/www.scribbr.com\/statistics\/simple-linear-regression\/), a lowercase *r* is usually used instead (*r*2).\n\n## Table of contents\n1. [What is the coefficient of determination?](https:\/\/www.scribbr.com\/statistics\/coefficient-of-determination\/#what)\n2. [Calculating the coefficient of determination](https:\/\/www.scribbr.com\/statistics\/coefficient-of-determination\/#calculating)\n3. [Interpreting the coefficient of determination](https:\/\/www.scribbr.com\/statistics\/coefficient-of-determination\/#interpreting)\n4. [Reporting the coefficient of determination](https:\/\/www.scribbr.com\/statistics\/coefficient-of-determination\/#reporting)\n5. [Practice questions](https:\/\/www.scribbr.com\/statistics\/coefficient-of-determination\/#quiz)\n6. [Other interesting articles](https:\/\/www.scribbr.com\/statistics\/coefficient-of-determination\/#other)\n7. [Frequently asked questions about the coefficient of determination](https:\/\/www.scribbr.com\/statistics\/coefficient-of-determination\/#frequently-asked-questions-about-the-coefficient-of-determination)\n## What is the coefficient of determination?\nThe coefficient of determination (*R*²) measures how well a statistical [model](https:\/\/www.scribbr.com\/frequently-asked-questions\/what-is-a-model\/) predicts an outcome. The outcome is represented by the model’s [dependent variable](https:\/\/www.scribbr.com\/methodology\/independent-and-dependent-variables\/#dependent).\n\nThe lowest possible value of *R*² is 0 and the highest possible value is 1. Put simply, the better a model is at making predictions, the closer its *R*² will be to 1.\n\nExample: Coefficient of determination\n\nImagine that you perform a simple linear regression that predicts students’ exam scores (dependent variable) from their time spent studying ([independent variable](https:\/\/www.scribbr.com\/methodology\/independent-and-dependent-variables\/#independent)).\n\n- If the *R*2 is 0, the linear regression model doesn’t allow you to predict exam scores any better than simply estimating that everyone has an [average](https:\/\/www.scribbr.com\/statistics\/mean\/) exam score.\n- If the *R*2 is between 0 and 1, the model allows you to partially predict exam scores. The model’s estimates are not perfect, but they’re better than simply using the average exam score.\n- If the *R*2 is 1, the model allows you to perfectly predict anyone’s exam score.\n\nMore technically, *R*2 is a measure of goodness of fit. It is the proportion of [variance](https:\/\/www.scribbr.com\/statistics\/variance\/) in the dependent variable that is explained by the model.\n\nGraphing your linear regression data usually gives you a good clue as to whether its *R*2 is high or low. For example, the graphs below show two sets of simulated data:\n\n- The observations are shown as dots.\n- The model’s predictions (the line of best fit) are shown as a black line.\n- The distance between the observations and their predicted values (the residuals) are shown as purple lines.\n\nYou can see in the first dataset that when the *R*2 is high, the observations are close to the model’s predictions. In other words, most points are close to the line of best fit:\n\n![Coefficient of determination (R2) = 0.9](https:\/\/www.scribbr.com\/wp-content\/uploads\/2022\/04\/Coefficient-of-determination_0.9.webp)\n\n**Note:** The coefficient of determination is always positive, even when the [correlation](https:\/\/www.scribbr.com\/frequently-asked-questions\/what-is-a-correlation\/) is negative.\n\nIn contrast, you can see in the second dataset that when the *R*2 is low, the observations are far from the model’s predictions. In other words, when the *R*2 is low, many points are far from the line of best fit:\n\n![Coefficient of determination (R2) = 0.2](https:\/\/www.scribbr.com\/wp-content\/uploads\/2022\/04\/Coefficient-of-determination_0.2.webp)\n\n### Receive feedback on language, structure, and formatting\nProfessional editors proofread and edit your paper by focusing on:\n\n- Academic style\n- Vague sentences\n- Grammar\n- Style consistency\n\n[See an example](https:\/\/www.scribbr.com\/proofreading-editing\/paper-editing-service\/?scr_source=knowledgebase&scr_medium=in-text&scr_campaign=proofreading&src_content=feedback+language+structure+layout)\n\n![](https:\/\/www.scribbr.com\/wp-content\/uploads\/2018\/01\/dissertation-proofreading-service.png)\n## Calculating the coefficient of determination\nYou can choose between two formulas to calculate the coefficient of determination (*R*²) of a simple linear regression. The first formula is specific to [simple linear regressions](https:\/\/www.scribbr.com\/statistics\/simple-linear-regression\/), and the second formula can be used to calculate the *R*² of many types of statistical models.\n\n### Formula 1: Using the correlation coefficient\nFormula 1:\n\n![\\\\begin{equation\\*}R^2=(r)^2\\\\end{equation\\*}](https:\/\/www.scribbr.com\/wp-content\/ql-cache\/quicklatex.com-8c4d82b2fe20f208da8c31b613e5fd89_l3.png)\n\nWhere *r* = [Pearson correlation coefficient](https:\/\/www.scribbr.com\/statistics\/pearson-correlation-coefficient\/)\n\nExample: Calculating *R*² using the correlation coefficient\n\nYou are studying the relationship between heart rate and age in children, and you find that the two variables have a negative Pearson correlation:\n\n![\\\\begin{equation\\*}r=-0.28\\\\end{equation\\*}](https:\/\/www.scribbr.com\/wp-content\/ql-cache\/quicklatex.com-028980c08bb3f318f52836e30ee9cfa9_l3.png)\n\nThis value can be used to calculate the coefficient of determination (*R*²) using Formula 1:\n\n![\\\\begin{equation\\*}R^2=(r)^2\\\\end{equation\\*}](https:\/\/www.scribbr.com\/wp-content\/ql-cache\/quicklatex.com-8c4d82b2fe20f208da8c31b613e5fd89_l3.png)\n\n![\\\\begin{equation\\*}R^2=(-0.28)^2\\\\end{equation\\*}](https:\/\/www.scribbr.com\/wp-content\/ql-cache\/quicklatex.com-335562cda79b94c68f75c3b43bf7a341_l3.png)\n\n![\\\\begin{equation\\*}R^2=0.08\\\\end{equation\\*}](https:\/\/www.scribbr.com\/wp-content\/ql-cache\/quicklatex.com-7c06a31c5618b67f96fec79ffb74e1e3_l3.png)\n### Formula 2: Using the regression outputs\nFormula 2:\n\n![\\\\begin{equation\\*}R^2=1-\\\\dfrac{\\\\textup{RSS}}{\\\\textup{TSS}}\\\\end{equation\\*}](https:\/\/www.scribbr.com\/wp-content\/ql-cache\/quicklatex.com-77060d9686c4e6200e15410694b0c221_l3.png)\n\nWhere:\n\n- RSS = sum of squared residuals\n- TSS = total sum of squares\n\nExample: Calculating *R*² using regression outputs\n\nAs part of performing a simple linear regression that predicts students’ exam scores (dependent variable) from their study time (independent variable), you calculate that:\n\n![\\\\begin{equation\\*}\\\\textup{RSS}=629.22\\\\end{equation\\*}](https:\/\/www.scribbr.com\/wp-content\/ql-cache\/quicklatex.com-9d3bed919c3705b3686559290d3b64a4_l3.png)\n\n![\\\\begin{equation\\*}\\\\textup{TSS}=2\\\\,187.04\\\\end{equation\\*}](https:\/\/www.scribbr.com\/wp-content\/ql-cache\/quicklatex.com-23b0842ab7eb241dd98d4073491126ce_l3.png)\n\nThese values can be used to calculate the coefficient of determination (*R*²) using Formula 2:\n\n![\\\\begin{equation\\*}R^2=1-\\\\dfrac{\\\\textup{RSS}}{\\\\textup{TSS}}\\\\end{equation\\*}](https:\/\/www.scribbr.com\/wp-content\/ql-cache\/quicklatex.com-77060d9686c4e6200e15410694b0c221_l3.png)\n\n![\\\\begin{equation\\*}R^2=1-\\\\dfrac{629.22}{2\\\\,187.04}\\\\end{equation\\*}](https:\/\/www.scribbr.com\/wp-content\/ql-cache\/quicklatex.com-25f7352ea18335eddc525ac74d0fea32_l3.png)\n\n![\\\\begin{equation\\*}R^2=1-0.29\\\\end{equation\\*}](https:\/\/www.scribbr.com\/wp-content\/ql-cache\/quicklatex.com-b74ead3cc867978ff7ec367f3830fba9_l3.png)\n\n![\\\\begin{equation\\*}R^2=0.71\\\\end{equation\\*}](https:\/\/www.scribbr.com\/wp-content\/ql-cache\/quicklatex.com-6e220f1a6c30feb37df1abf358c6dab4_l3.png)\n## Interpreting the coefficient of determination\nYou can interpret the coefficient of determination (*R*²) as the proportion of [variance](https:\/\/www.scribbr.com\/statistics\/variance\/) in the [dependent variable](https:\/\/www.scribbr.com\/methodology\/independent-and-dependent-variables\/#dependent) that is predicted by the statistical [model](https:\/\/www.scribbr.com\/frequently-asked-questions\/what-is-a-model\/).\n\nAnother way of thinking of it is that the *R*² is the proportion of variance that is shared between the independent and dependent variables.\n\nYou can also say that the *R*² is the proportion of variance “explained” or “accounted for” by the model. The proportion that remains (1 − *R*²) is the variance that is not predicted by the model.\n\nIf you prefer, you can write the *R*² as a percentage instead of a proportion. Simply multiply the proportion by 100.\n\n### *R*² as an effect size\nLastly, you can also interpret the *R*² as an [effect size](https:\/\/www.scribbr.com\/statistics\/effect-size\/): a measure of the strength of the relationship between the dependent and independent variables. Psychologist and statistician Jacob Cohen (1988) suggested the following rules of thumb for [simple linear regressions](https:\/\/www.scribbr.com\/statistics\/simple-linear-regression\/):\n\n| Minimum coefficient of determination (*R*²) value | Effect size interpretation |\n|---|---|\n| .01 | Small |\n| .09 | Medium |\n| .25 | Large |\n\nBe careful: the *R*² on its own can’t tell you anything about [causation](https:\/\/www.scribbr.com\/methodology\/correlation-vs-causation\/).\n\nExample: Interpreting *R*²\n\nA simple linear regression that predicts students’ exam scores (dependent variable) from their study time (independent variable) has an *R*² of .71. From this *R*² value, we know that:\n\n- 71% of the variance in students’ exam scores is predicted by their study time\n- 29% of the variance in student’s exam scores is unexplained by the model\n- The students’ study time has a large effect on their exam scores\n\nStudying longer may or may not [cause](https:\/\/www.scribbr.com\/methodology\/correlation-vs-causation\/) an improvement in the students’ scores. Although this causal relationship is very plausible, the *R*² alone can’t tell us why there’s a relationship between students’ study time and exam scores.\n\nFor example, students might find studying less frustrating when they understand the course material well, so they study longer.\n## Reporting the coefficient of determination\nIf you decide to include a coefficient of determination (*R*²) in your [research paper](https:\/\/www.scribbr.com\/category\/research-paper\/), [dissertation](https:\/\/www.scribbr.com\/category\/dissertation\/) or [thesis](https:\/\/www.scribbr.com\/dissertation\/thesis\/), you should report it in your [results section](https:\/\/www.scribbr.com\/dissertation\/results\/). You can follow these rules if you want to [report statistics in APA Style](https:\/\/www.scribbr.com\/apa-style\/numbers-and-statistics\/):\n\n- You should use “*r*²” for statistical models with one independent variable (such as simple linear regressions). Use “*R*²” for statistical models with multiple independent variables.\n- You don’t need to provide a reference or formula since the coefficient of determination is a commonly used statistic.\n- You should italicize *r*² and *R*² when reporting their values (but don’t italicize the ²).\n- You shouldn’t include a leading zero (a zero before the decimal point) since the coefficient of determination can’t be greater than one.\n- You should provide two significant digits after the decimal point.\n- Very often, the coefficient of determination is provided alongside related statistical results, such as the [*F* value](https:\/\/www.scribbr.com\/statistics\/test-statistic\/), [degrees of freedom](https:\/\/www.scribbr.com\/statistics\/degrees-of-freedom\/), and [*p* value](https:\/\/www.scribbr.com\/statistics\/p-value\/).\n\nExample: Reporting *r*² in APA Style\n\nStudents’ exam scores were predicted by their study time, *r*² = .71, *F*(1,32) = 7.33, *p* \\= .003.\n## Practice questions\n## Other interesting articles\nIf you want to know more about [statistics](https:\/\/www.scribbr.com\/category\/statistics\/), [methodology](https:\/\/www.scribbr.com\/category\/methodology\/), or [research bias](https:\/\/www.scribbr.com\/faq-category\/research-bias\/), make sure to check out some of our other articles with explanations and examples.\n\n**Statistics**\n\n- [Chi square test of independence](https:\/\/www.scribbr.com\/statistics\/chi-square-test-of-independence\/)\n- [Statistical power](https:\/\/www.scribbr.com\/statistics\/statistical-power\/)\n- [Descriptive statistics](https:\/\/www.scribbr.com\/statistics\/descriptive-statistics\/)\n- [Degrees of freedom](https:\/\/www.scribbr.com\/statistics\/degrees-of-freedom\/)\n- [Pearson correlation](https:\/\/www.scribbr.com\/statistics\/pearson-correlation-coefficient\/)\n- [Null hypothesis](https:\/\/www.scribbr.com\/statistics\/null-and-alternative-hypotheses\/)\n\n**Methodology**\n\n- [Double-blind study](https:\/\/www.scribbr.com\/methodology\/double-blind-study\/)\n- [Case-control study](https:\/\/www.scribbr.com\/methodology\/case-control-study\/)\n- [Research ethics](https:\/\/www.scribbr.com\/methodology\/research-ethics\/)\n- [Data collection](https:\/\/www.scribbr.com\/methodology\/data-collection\/)\n- [Hypothesis testing](https:\/\/www.scribbr.com\/statistics\/hypothesis-testing\/)\n- [Structured interviews](https:\/\/www.scribbr.com\/methodology\/structured-interview\/)\n## Frequently asked questions about the coefficient of determination\n#### Cite this Scribbr article\nIf you want to cite this source, you can copy and paste the citation or click the “Cite this Scribbr article” button to automatically add the citation to our free Citation Generator.\n\n> Turney, S. (2023, June 22). Coefficient of Determination (R²) \\| Calculation & Interpretation. Scribbr. Retrieved April 2, 2026, from https:\/\/www.scribbr.com\/statistics\/coefficient-of-determination\/\n\nYou have already voted. Thanks :-) Your vote is saved :-) Processing your vote...","meta_canonical":null}

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Boilerpipe Text	Published on April 22, 2022 by Shaun Turney . Revised on June 22, 2023. The coefficient of determination is a number between 0 and 1 that measures how well a statistical model predicts an outcome. Interpreting the coefficient of determination Coefficient of determination ( R 2 ) Interpretation 0 The model does not predict the outcome. Between 0 and 1 The model partially predicts the outcome. 1 The model perfectly predicts the outcome. The coefficient of determination is often written as R 2 , which is pronounced as “r squared.” For simple linear regressions , a lowercase r is usually used instead ( r 2 ). Table of contents What is the coefficient of determination? Calculating the coefficient of determination Interpreting the coefficient of determination Reporting the coefficient of determination Practice questions Other interesting articles Frequently asked questions about the coefficient of determination What is the coefficient of determination? The coefficient of determination ( R ²) measures how well a statistical model predicts an outcome. The outcome is represented by the model’s dependent variable . The lowest possible value of R ² is 0 and the highest possible value is 1. Put simply, the better a model is at making predictions, the closer its R ² will be to 1. Example: Coefficient of determination Imagine that you perform a simple linear regression that predicts students’ exam scores (dependent variable) from their time spent studying ( independent variable ). If the R 2 is 0, the linear regression model doesn’t allow you to predict exam scores any better than simply estimating that everyone has an average exam score. If the R 2 is between 0 and 1, the model allows you to partially predict exam scores. The model’s estimates are not perfect, but they’re better than simply using the average exam score. If the R 2 is 1, the model allows you to perfectly predict anyone’s exam score. More technically, R 2 is a measure of goodness of fit. It is the proportion of variance in the dependent variable that is explained by the model. Graphing your linear regression data usually gives you a good clue as to whether its R 2 is high or low. For example, the graphs below show two sets of simulated data: The observations are shown as dots. The model’s predictions (the line of best fit) are shown as a black line. The distance between the observations and their predicted values (the residuals) are shown as purple lines. You can see in the first dataset that when the R 2 is high, the observations are close to the model’s predictions . In other words, most points are close to the line of best fit: Note: The coefficient of determination is always positive, even when the correlation is negative. In contrast, you can see in the second dataset that when the R 2 is low, the observations are far from the model’s predictions . In other words, when the R 2 is low, many points are far from the line of best fit: Receive feedback on language, structure, and formatting Professional editors proofread and edit your paper by focusing on: Academic style Vague sentences Grammar Style consistency See an example Calculating the coefficient of determination You can choose between two formulas to calculate the coefficient of determination ( R ²) of a simple linear regression. The first formula is specific to simple linear regressions , and the second formula can be used to calculate the R ² of many types of statistical models. Formula 1: Using the correlation coefficient Formula 1: Where r = Pearson correlation coefficient Example: Calculating R ² using the correlation coefficient You are studying the relationship between heart rate and age in children, and you find that the two variables have a negative Pearson correlation: This value can be used to calculate the coefficient of determination ( R ²) using Formula 1: Formula 2: Using the regression outputs Formula 2: Where: RSS = sum of squared residuals TSS = total sum of squares Example: Calculating R ² using regression outputs As part of performing a simple linear regression that predicts students’ exam scores (dependent variable) from their study time (independent variable), you calculate that: These values can be used to calculate the coefficient of determination ( R ²) using Formula 2: Interpreting the coefficient of determination You can interpret the coefficient of determination ( R ²) as the proportion of variance in the dependent variable that is predicted by the statistical model . Another way of thinking of it is that the R ² is the proportion of variance that is shared between the independent and dependent variables. You can also say that the R ² is the proportion of variance “explained” or “accounted for” by the model. The proportion that remains (1 − R ²) is the variance that is not predicted by the model. If you prefer, you can write the R ² as a percentage instead of a proportion. Simply multiply the proportion by 100. R ² as an effect size Lastly, you can also interpret the R ² as an effect size : a measure of the strength of the relationship between the dependent and independent variables. Psychologist and statistician Jacob Cohen (1988) suggested the following rules of thumb for simple linear regressions : R² as an effect size Minimum coefficient of determination ( R ²) value Effect size interpretation .01 Small .09 Medium .25 Large Be careful: the R ² on its own can’t tell you anything about causation . Example: Interpreting R ² A simple linear regression that predicts students’ exam scores (dependent variable) from their study time (independent variable) has an R ² of .71. From this R ² value, we know that: 71% of the variance in students’ exam scores is predicted by their study time 29% of the variance in student’s exam scores is unexplained by the model The students’ study time has a large effect on their exam scores Studying longer may or may not cause an improvement in the students’ scores. Although this causal relationship is very plausible, the R ² alone can’t tell us why there’s a relationship between students’ study time and exam scores. For example, students might find studying less frustrating when they understand the course material well, so they study longer. Reporting the coefficient of determination If you decide to include a coefficient of determination ( R ²) in your research paper , dissertation or thesis , you should report it in your results section . You can follow these rules if you want to report statistics in APA Style : You should use “ r ²” for statistical models with one independent variable (such as simple linear regressions). Use “ R ²” for statistical models with multiple independent variables. You don’t need to provide a reference or formula since the coefficient of determination is a commonly used statistic. You should italicize r ² and R ² when reporting their values (but don’t italicize the ²). You shouldn’t include a leading zero (a zero before the decimal point) since the coefficient of determination can’t be greater than one. You should provide two significant digits after the decimal point. Very often, the coefficient of determination is provided alongside related statistical results, such as the F value , degrees of freedom , and p value . Example: Reporting r ² in APA Style Students’ exam scores were predicted by their study time, r ² = .71, F (1,32) = 7.33, p = .003. Practice questions Other interesting articles If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples. Statistics Chi square test of independence Statistical power Descriptive statistics Degrees of freedom Pearson correlation Null hypothesis Methodology Double-blind study Case-control study Research ethics Data collection Hypothesis testing Structured interviews Frequently asked questions about the coefficient of determination Cite this Scribbr article If you want to cite this source, you can copy and paste the citation or click the “Cite this Scribbr article” button to automatically add the citation to our free Citation Generator. Turney, S. (2023, June 22). Coefficient of Determination (R²) \| Calculation & Interpretation. Scribbr. Retrieved April 2, 2026, from https://www.scribbr.com/statistics/coefficient-of-determination/ You have already voted. 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Coefficient of Determination (R²) \\| Calculation & Interpretation ### Statistics - [Statistical analysis step by step](https://www.scribbr.com/category/statistics/) - [Data collection](https://www.scribbr.com/methodology/data-collection/ "Data Collection \| Definition, Methods & Examples") - [Data collection guide](https://www.scribbr.com/methodology/data-collection/ "Data Collection \| Definition, Methods & Examples") - [Experimental design](https://www.scribbr.com/methodology/experimental-design/ "Guide to Experimental Design \| Overview, Steps, & Examples") - [Populations and samples](https://www.scribbr.com/methodology/population-vs-sample/ "Population vs. Sample \| Definitions, Differences & Examples") - [Populations and samples](https://www.scribbr.com/methodology/population-vs-sample/ "Population vs. Sample \| Definitions, Differences & Examples") - [Sampling methods](https://www.scribbr.com/methodology/sampling-methods/ "Sampling Methods \| Types, Techniques & Examples") - [Types of variables](https://www.scribbr.com/methodology/types-of-variables/ "Types of Variables in Research & Statistics \| Examples") - 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(Examples)") - [Overview of effect sizes](https://www.scribbr.com/statistics/effect-size/ "What is Effect Size and Why Does It Matter? 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Try for free](https://www.scribbr.com/paraphrasing-tool/?scr_source=knowledgebase&scr_medium=sidebar&scr_campaign=minimalistic-sidebar) [![](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%2032%2032'%3E%3C/svg%3E) ![](https://www.scribbr.com/wp-content/uploads/2024/09/grammer-checker.svg) Grammar Checker Eliminate grammar errors and improve your writing with our free AI-powered grammar checker. Try for free](https://www.scribbr.com/grammar-checker/?scr_source=knowledgebase&scr_medium=sidebar&scr_campaign=minimalistic-sidebar) # Coefficient of Determination (R²) \\| Calculation & Interpretation Published on April 22, 2022 by [Shaun Turney](https://www.scribbr.com/author/shaunt/ "All articles by Shaun Turney"). Revised on June 22, 2023. The coefficient of determination is a number between 0 and 1 that measures how well a statistical [model](https://www.scribbr.com/frequently-asked-questions/what-is-a-model/) predicts an outcome. \| Coefficient of determination (R2) \| Interpretation \| \|---\|---\| \| 0 \| The model does not predict the outcome. \| \| Between 0 and 1 \| The model partially predicts the outcome. \| \| 1 \| The model perfectly predicts the outcome. \| The coefficient of determination is often written as R2, which is pronounced as “r squared.” For [simple linear regressions](https://www.scribbr.com/statistics/simple-linear-regression/), a lowercase r is usually used instead (r2). ## Table of contents 1. [What is the coefficient of determination?](https://www.scribbr.com/statistics/coefficient-of-determination/#what) 2. [Calculating the coefficient of determination](https://www.scribbr.com/statistics/coefficient-of-determination/#calculating) 3. [Interpreting the coefficient of determination](https://www.scribbr.com/statistics/coefficient-of-determination/#interpreting) 4. [Reporting the coefficient of determination](https://www.scribbr.com/statistics/coefficient-of-determination/#reporting) 5. [Practice questions](https://www.scribbr.com/statistics/coefficient-of-determination/#quiz) 6. [Other interesting articles](https://www.scribbr.com/statistics/coefficient-of-determination/#other) 7. [Frequently asked questions about the coefficient of determination](https://www.scribbr.com/statistics/coefficient-of-determination/#frequently-asked-questions-about-the-coefficient-of-determination) ## What is the coefficient of determination? The coefficient of determination (R²) measures how well a statistical [model](https://www.scribbr.com/frequently-asked-questions/what-is-a-model/) predicts an outcome. The outcome is represented by the model’s [dependent variable](https://www.scribbr.com/methodology/independent-and-dependent-variables/#dependent). The lowest possible value of R² is 0 and the highest possible value is 1. Put simply, the better a model is at making predictions, the closer its R² will be to 1. Example: Coefficient of determination Imagine that you perform a simple linear regression that predicts students’ exam scores (dependent variable) from their time spent studying ([independent variable](https://www.scribbr.com/methodology/independent-and-dependent-variables/#independent)). - If the R2 is 0, the linear regression model doesn’t allow you to predict exam scores any better than simply estimating that everyone has an [average](https://www.scribbr.com/statistics/mean/) exam score. - If the R2 is between 0 and 1, the model allows you to partially predict exam scores. The model’s estimates are not perfect, but they’re better than simply using the average exam score. - If the R2 is 1, the model allows you to perfectly predict anyone’s exam score. More technically, R2 is a measure of goodness of fit. It is the proportion of [variance](https://www.scribbr.com/statistics/variance/) in the dependent variable that is explained by the model. Graphing your linear regression data usually gives you a good clue as to whether its R2 is high or low. For example, the graphs below show two sets of simulated data: - The observations are shown as dots. - The model’s predictions (the line of best fit) are shown as a black line. - The distance between the observations and their predicted values (the residuals) are shown as purple lines. You can see in the first dataset that when the R2 is high, the observations are close to the model’s predictions. In other words, most points are close to the line of best fit: ![Coefficient of determination (R2) = 0.9](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20660%20480'%3E%3C/svg%3E)![Coefficient of determination (R2) = 0.9](https://www.scribbr.com/wp-content/uploads/2022/04/Coefficient-of-determination_0.9.webp) Note: The coefficient of determination is always positive, even when the [correlation](https://www.scribbr.com/frequently-asked-questions/what-is-a-correlation/) is negative. In contrast, you can see in the second dataset that when the R2 is low, the observations are far from the model’s predictions. In other words, when the R2 is low, many points are far from the line of best fit: ![Coefficient of determination (R2) = 0.2](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20660%20480'%3E%3C/svg%3E)![Coefficient of determination (R2) = 0.2](https://www.scribbr.com/wp-content/uploads/2022/04/Coefficient-of-determination_0.2.webp) ### Receive feedback on language, structure, and formatting Professional editors proofread and edit your paper by focusing on: - Academic style - Vague sentences - Grammar - Style consistency [See an example](https://www.scribbr.com/proofreading-editing/paper-editing-service/?scr_source=knowledgebase&scr_medium=in-text&scr_campaign=proofreading&src_content=feedback+language+structure+layout) ![](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20830%20660'%3E%3C/svg%3E) ![](https://www.scribbr.com/wp-content/uploads/2018/01/dissertation-proofreading-service.png) ## Calculating the coefficient of determination You can choose between two formulas to calculate the coefficient of determination (R²) of a simple linear regression. The first formula is specific to [simple linear regressions](https://www.scribbr.com/statistics/simple-linear-regression/), and the second formula can be used to calculate the R² of many types of statistical models. ### Formula 1: Using the correlation coefficient Formula 1: ![\\begin{equation\}R^2=(r)^2\\end{equation\}](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%2078%2023'%3E%3C/svg%3E)![\\begin{equation\}R^2=(r)^2\\end{equation\}](https://www.scribbr.com/wp-content/ql-cache/quicklatex.com-8c4d82b2fe20f208da8c31b613e5fd89_l3.png) Where r = [Pearson correlation coefficient](https://www.scribbr.com/statistics/pearson-correlation-coefficient/) Example: Calculating R² using the correlation coefficient You are studying the relationship between heart rate and age in children, and you find that the two variables have a negative Pearson correlation: ![\\begin{equation\}r=-0.28\\end{equation\}](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%2082%2013'%3E%3C/svg%3E)![\\begin{equation\}r=-0.28\\end{equation\}](https://www.scribbr.com/wp-content/ql-cache/quicklatex.com-028980c08bb3f318f52836e30ee9cfa9_l3.png) This value can be used to calculate the coefficient of determination (R²) using Formula 1: ![\\begin{equation\}R^2=(r)^2\\end{equation\}](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%2078%2023'%3E%3C/svg%3E)![\\begin{equation\}R^2=(r)^2\\end{equation\}](https://www.scribbr.com/wp-content/ql-cache/quicklatex.com-8c4d82b2fe20f208da8c31b613e5fd89_l3.png) ![\\begin{equation\}R^2=(-0.28)^2\\end{equation\}](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20117%2023'%3E%3C/svg%3E)![\\begin{equation\}R^2=(-0.28)^2\\end{equation\}](https://www.scribbr.com/wp-content/ql-cache/quicklatex.com-335562cda79b94c68f75c3b43bf7a341_l3.png) ![\\begin{equation\}R^2=0.08\\end{equation\}](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%2080%2018'%3E%3C/svg%3E)![\\begin{equation\}R^2=0.08\\end{equation\}](https://www.scribbr.com/wp-content/ql-cache/quicklatex.com-7c06a31c5618b67f96fec79ffb74e1e3_l3.png) ### Formula 2: Using the regression outputs Formula 2: ![\\begin{equation\}R^2=1-\\dfrac{\\textup{RSS}}{\\textup{TSS}}\\end{equation\}](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20117%2039'%3E%3C/svg%3E)![\\begin{equation\}R^2=1-\\dfrac{\\textup{RSS}}{\\textup{TSS}}\\end{equation\}](https://www.scribbr.com/wp-content/ql-cache/quicklatex.com-77060d9686c4e6200e15410694b0c221_l3.png) Where: - RSS = sum of squared residuals - TSS = total sum of squares Example: Calculating R² using regression outputs As part of performing a simple linear regression that predicts students’ exam scores (dependent variable) from their study time (independent variable), you calculate that: ![\\begin{equation\}\\textup{RSS}=629.22\\end{equation\}](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20110%2013'%3E%3C/svg%3E)![\\begin{equation\}\\textup{RSS}=629.22\\end{equation\}](https://www.scribbr.com/wp-content/ql-cache/quicklatex.com-9d3bed919c3705b3686559290d3b64a4_l3.png) ![\\begin{equation\}\\textup{TSS}=2\\,187.04\\end{equation\}](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20123%2014'%3E%3C/svg%3E)![\\begin{equation\}\\textup{TSS}=2\\,187.04\\end{equation\}](https://www.scribbr.com/wp-content/ql-cache/quicklatex.com-23b0842ab7eb241dd98d4073491126ce_l3.png) These values can be used to calculate the coefficient of determination (R²) using Formula 2: ![\\begin{equation\}R^2=1-\\dfrac{\\textup{RSS}}{\\textup{TSS}}\\end{equation\}](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20117%2039'%3E%3C/svg%3E)![\\begin{equation\}R^2=1-\\dfrac{\\textup{RSS}}{\\textup{TSS}}\\end{equation\}](https://www.scribbr.com/wp-content/ql-cache/quicklatex.com-77060d9686c4e6200e15410694b0c221_l3.png) ![\\begin{equation\}R^2=1-\\dfrac{629.22}{2\\,187.04}\\end{equation\}](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20147%2039'%3E%3C/svg%3E)![\\begin{equation\}R^2=1-\\dfrac{629.22}{2\\,187.04}\\end{equation\}](https://www.scribbr.com/wp-content/ql-cache/quicklatex.com-25f7352ea18335eddc525ac74d0fea32_l3.png) ![\\begin{equation\}R^2=1-0.29\\end{equation\}](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20112%2018'%3E%3C/svg%3E)![\\begin{equation\}R^2=1-0.29\\end{equation\}](https://www.scribbr.com/wp-content/ql-cache/quicklatex.com-b74ead3cc867978ff7ec367f3830fba9_l3.png) ![\\begin{equation\}R^2=0.71\\end{equation\}](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%2079%2018'%3E%3C/svg%3E)![\\begin{equation\}R^2=0.71\\end{equation\}](https://www.scribbr.com/wp-content/ql-cache/quicklatex.com-6e220f1a6c30feb37df1abf358c6dab4_l3.png) ## Interpreting the coefficient of determination You can interpret the coefficient of determination (R²) as the proportion of [variance](https://www.scribbr.com/statistics/variance/) in the [dependent variable](https://www.scribbr.com/methodology/independent-and-dependent-variables/#dependent) that is predicted by the statistical [model](https://www.scribbr.com/frequently-asked-questions/what-is-a-model/). Another way of thinking of it is that the R² is the proportion of variance that is shared between the independent and dependent variables. You can also say that the R² is the proportion of variance “explained” or “accounted for” by the model. The proportion that remains (1 − R²) is the variance that is not predicted by the model. If you prefer, you can write the R² as a percentage instead of a proportion. Simply multiply the proportion by 100. ### R² as an effect size Lastly, you can also interpret the R² as an [effect size](https://www.scribbr.com/statistics/effect-size/): a measure of the strength of the relationship between the dependent and independent variables. Psychologist and statistician Jacob Cohen (1988) suggested the following rules of thumb for [simple linear regressions](https://www.scribbr.com/statistics/simple-linear-regression/): \| Minimum coefficient of determination (R²) value \| Effect size interpretation \| \|---\|---\| \| .01 \| Small \| \| .09 \| Medium \| \| .25 \| Large \| Be careful: the R² on its own can’t tell you anything about [causation](https://www.scribbr.com/methodology/correlation-vs-causation/). Example: Interpreting R² A simple linear regression that predicts students’ exam scores (dependent variable) from their study time (independent variable) has an R² of .71. From this R² value, we know that: - 71% of the variance in students’ exam scores is predicted by their study time - 29% of the variance in student’s exam scores is unexplained by the model - The students’ study time has a large effect on their exam scores Studying longer may or may not [cause](https://www.scribbr.com/methodology/correlation-vs-causation/) an improvement in the students’ scores. Although this causal relationship is very plausible, the R² alone can’t tell us why there’s a relationship between students’ study time and exam scores. For example, students might find studying less frustrating when they understand the course material well, so they study longer. ## Reporting the coefficient of determination If you decide to include a coefficient of determination (R²) in your [research paper](https://www.scribbr.com/category/research-paper/), [dissertation](https://www.scribbr.com/category/dissertation/) or [thesis](https://www.scribbr.com/dissertation/thesis/), you should report it in your [results section](https://www.scribbr.com/dissertation/results/). You can follow these rules if you want to [report statistics in APA Style](https://www.scribbr.com/apa-style/numbers-and-statistics/): - You should use “r²” for statistical models with one independent variable (such as simple linear regressions). Use “R²” for statistical models with multiple independent variables. - You don’t need to provide a reference or formula since the coefficient of determination is a commonly used statistic. - You should italicize r² and R² when reporting their values (but don’t italicize the ²). - You shouldn’t include a leading zero (a zero before the decimal point) since the coefficient of determination can’t be greater than one. - You should provide two significant digits after the decimal point. - Very often, the coefficient of determination is provided alongside related statistical results, such as the [F value](https://www.scribbr.com/statistics/test-statistic/), [degrees of freedom](https://www.scribbr.com/statistics/degrees-of-freedom/), and [p value](https://www.scribbr.com/statistics/p-value/). Example: Reporting r² in APA Style Students’ exam scores were predicted by their study time, r² = .71, F(1,32) = 7.33, p \= .003. ## Here's why students love Scribbr's proofreading services [Trustpilot](https://www.trustpilot.com/review/www.scribbr.com) [Discover proofreading & editing](https://www.scribbr.com/proofreading-editing/paper-editing-service/?scr_source=knowledgebase&scr_medium=in-text&scr_campaign=proofreading&src_content=trustpilot) ## Practice questions powered by [Typeform](https://admin.typeform.com/signup?utm_campaign=N3apIC&utm_source=typeform.com-01D8JTMH8P14WJGAWB4RB1BJEB-pro&utm_medium=typeform&utm_content=typeform-embedded-poweredbytypeform&utm_term=EN) ## Other interesting articles If you want to know more about [statistics](https://www.scribbr.com/category/statistics/), [methodology](https://www.scribbr.com/category/methodology/), or [research bias](https://www.scribbr.com/faq-category/research-bias/), make sure to check out some of our other articles with explanations and examples. Statistics - [Chi square test of independence](https://www.scribbr.com/statistics/chi-square-test-of-independence/) - [Statistical power](https://www.scribbr.com/statistics/statistical-power/) - [Descriptive statistics](https://www.scribbr.com/statistics/descriptive-statistics/) - [Degrees of freedom](https://www.scribbr.com/statistics/degrees-of-freedom/) - [Pearson correlation](https://www.scribbr.com/statistics/pearson-correlation-coefficient/) - [Null hypothesis](https://www.scribbr.com/statistics/null-and-alternative-hypotheses/) Methodology - [Double-blind study](https://www.scribbr.com/methodology/double-blind-study/) - [Case-control study](https://www.scribbr.com/methodology/case-control-study/) - [Research ethics](https://www.scribbr.com/methodology/research-ethics/) - [Data collection](https://www.scribbr.com/methodology/data-collection/) - [Hypothesis testing](https://www.scribbr.com/statistics/hypothesis-testing/) - [Structured interviews](https://www.scribbr.com/methodology/structured-interview/) Research bias - [Hawthorne effect](https://www.scribbr.com/research-bias/hawthorne-effect/) - [Unconscious bias](https://www.scribbr.com/research-bias/unconscious-bias/) - [Recall bias](https://www.scribbr.com/research-bias/recall-bias/) - [Halo effect](https://www.scribbr.com/research-bias/halo-effect/) - [Self-serving bias](https://www.scribbr.com/research-bias/self-serving-bias/) - [Information bias](https://www.scribbr.com/research-bias/information-bias/) ## Frequently asked questions about the coefficient of determination [What is the definition of the coefficient of determination (R²)?](https://www.scribbr.com/frequently-asked-questions/definition-of-coefficient-of-determination/) The [coefficient of determination (R²)](https://www.scribbr.com/statistics/coefficient-of-determination/) is a number between 0 and 1 that measures how well a statistical model predicts an outcome. You can interpret the R² as the proportion of variation in the [dependent variable](https://www.scribbr.com/methodology/independent-and-dependent-variables/) that is predicted by the statistical model. [What is the formula for the coefficient of determination (R²)?](https://www.scribbr.com/frequently-asked-questions/what-is-the-formula-for-the-coefficient-of-determination-r%C2%B2/) There are two formulas you can use to calculate the [coefficient of determination (R²)](https://www.scribbr.com/statistics/coefficient-of-determination/) of a [simple linear regression](https://www.scribbr.com/statistics/simple-linear-regression/). Formula 1: ![R^2=(r)^2](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%2078%2022'%3E%3C/svg%3E)![R^2=(r)^2](https://www.scribbr.com/wp-content/ql-cache/quicklatex.com-06ecfc9eb09f339f2a6f8113909f3fe0_l3.png) Formula 2: ![R^2=1-\\dfrac{\\textup{RSS}}{\\textup{TSS}}](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20117%2039'%3E%3C/svg%3E)![R^2=1-\\dfrac{\\textup{RSS}}{\\textup{TSS}}](https://www.scribbr.com/wp-content/ql-cache/quicklatex.com-5472ee9a4cc0c893784bf2e13e36d78b_l3.png) [How do I calculate the coefficient of determination (R²) in R?](https://www.scribbr.com/frequently-asked-questions/r%C2%B2-in-r/) You can use the summary() function to view the [R²](https://www.scribbr.com/statistics/coefficient-of-determination/) of a linear model in R. You will see the “R-squared” near the bottom of the output. [How do I calculate the coefficient of determination (R²) in Excel?](https://www.scribbr.com/frequently-asked-questions/r%C2%B2-in-excel/) You can use the RSQ() function to calculate [R²](https://www.scribbr.com/statistics/coefficient-of-determination/) in Excel. If your [dependent variable](https://www.scribbr.com/methodology/independent-and-dependent-variables/) is in column A and your independent variable is in column B, then click any blank cell and type “RSQ(A:A,B:B)”. #### Cite this Scribbr article If you want to cite this source, you can copy and paste the citation or click the “Cite this Scribbr article” button to automatically add the citation to our free Citation Generator. > Turney, S. (2023, June 22). Coefficient of Determination (R²) \\| Calculation & Interpretation. Scribbr. 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[Scribbr customers enjoy an ad-free experience\!](https://www.scribbr.com/frequently-asked-questions/ad-free-experience/) ## Other students also liked [Correlation Coefficient \\| Types, Formulas & Examples A correlation coefficient tells you the strength of the relationship between variables using a single number between -1 and 1. 479](https://www.scribbr.com/statistics/correlation-coefficient/) [Simple Linear Regression \\| An Easy Introduction & Examples Simple linear regression is a model that describes the relationship between one dependent and one independent variable using a straight line. 912](https://www.scribbr.com/statistics/simple-linear-regression/) [What is Effect Size and Why Does It Matter? 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Readable Markdown	Published on April 22, 2022 by [Shaun Turney](https://www.scribbr.com/author/shaunt/ "All articles by Shaun Turney"). Revised on June 22, 2023. The coefficient of determination is a number between 0 and 1 that measures how well a statistical [model](https://www.scribbr.com/frequently-asked-questions/what-is-a-model/) predicts an outcome. \| Coefficient of determination (R2) \| Interpretation \| \|---\|---\| \| 0 \| The model does not predict the outcome. \| \| Between 0 and 1 \| The model partially predicts the outcome. \| \| 1 \| The model perfectly predicts the outcome. \| The coefficient of determination is often written as R2, which is pronounced as “r squared.” For [simple linear regressions](https://www.scribbr.com/statistics/simple-linear-regression/), a lowercase r is usually used instead (r2). ## Table of contents 1. [What is the coefficient of determination?](https://www.scribbr.com/statistics/coefficient-of-determination/#what) 2. [Calculating the coefficient of determination](https://www.scribbr.com/statistics/coefficient-of-determination/#calculating) 3. [Interpreting the coefficient of determination](https://www.scribbr.com/statistics/coefficient-of-determination/#interpreting) 4. [Reporting the coefficient of determination](https://www.scribbr.com/statistics/coefficient-of-determination/#reporting) 5. [Practice questions](https://www.scribbr.com/statistics/coefficient-of-determination/#quiz) 6. [Other interesting articles](https://www.scribbr.com/statistics/coefficient-of-determination/#other) 7. [Frequently asked questions about the coefficient of determination](https://www.scribbr.com/statistics/coefficient-of-determination/#frequently-asked-questions-about-the-coefficient-of-determination) ## What is the coefficient of determination? The coefficient of determination (R²) measures how well a statistical [model](https://www.scribbr.com/frequently-asked-questions/what-is-a-model/) predicts an outcome. The outcome is represented by the model’s [dependent variable](https://www.scribbr.com/methodology/independent-and-dependent-variables/#dependent). The lowest possible value of R² is 0 and the highest possible value is 1. Put simply, the better a model is at making predictions, the closer its R² will be to 1. Example: Coefficient of determination Imagine that you perform a simple linear regression that predicts students’ exam scores (dependent variable) from their time spent studying ([independent variable](https://www.scribbr.com/methodology/independent-and-dependent-variables/#independent)). - If the R2 is 0, the linear regression model doesn’t allow you to predict exam scores any better than simply estimating that everyone has an [average](https://www.scribbr.com/statistics/mean/) exam score. - If the R2 is between 0 and 1, the model allows you to partially predict exam scores. The model’s estimates are not perfect, but they’re better than simply using the average exam score. - If the R2 is 1, the model allows you to perfectly predict anyone’s exam score. More technically, R2 is a measure of goodness of fit. It is the proportion of [variance](https://www.scribbr.com/statistics/variance/) in the dependent variable that is explained by the model. Graphing your linear regression data usually gives you a good clue as to whether its R2 is high or low. For example, the graphs below show two sets of simulated data: - The observations are shown as dots. - The model’s predictions (the line of best fit) are shown as a black line. - The distance between the observations and their predicted values (the residuals) are shown as purple lines. You can see in the first dataset that when the R2 is high, the observations are close to the model’s predictions. In other words, most points are close to the line of best fit: ![Coefficient of determination (R2) = 0.9](https://www.scribbr.com/wp-content/uploads/2022/04/Coefficient-of-determination_0.9.webp) Note: The coefficient of determination is always positive, even when the [correlation](https://www.scribbr.com/frequently-asked-questions/what-is-a-correlation/) is negative. In contrast, you can see in the second dataset that when the R2 is low, the observations are far from the model’s predictions. In other words, when the R2 is low, many points are far from the line of best fit: ![Coefficient of determination (R2) = 0.2](https://www.scribbr.com/wp-content/uploads/2022/04/Coefficient-of-determination_0.2.webp) ### Receive feedback on language, structure, and formatting Professional editors proofread and edit your paper by focusing on: - Academic style - Vague sentences - Grammar - Style consistency [See an example](https://www.scribbr.com/proofreading-editing/paper-editing-service/?scr_source=knowledgebase&scr_medium=in-text&scr_campaign=proofreading&src_content=feedback+language+structure+layout) ![](https://www.scribbr.com/wp-content/uploads/2018/01/dissertation-proofreading-service.png) ## Calculating the coefficient of determination You can choose between two formulas to calculate the coefficient of determination (R²) of a simple linear regression. The first formula is specific to [simple linear regressions](https://www.scribbr.com/statistics/simple-linear-regression/), and the second formula can be used to calculate the R² of many types of statistical models. ### Formula 1: Using the correlation coefficient Formula 1: ![\\begin{equation\}R^2=(r)^2\\end{equation\}](https://www.scribbr.com/wp-content/ql-cache/quicklatex.com-8c4d82b2fe20f208da8c31b613e5fd89_l3.png) Where r = [Pearson correlation coefficient](https://www.scribbr.com/statistics/pearson-correlation-coefficient/) Example: Calculating R² using the correlation coefficient You are studying the relationship between heart rate and age in children, and you find that the two variables have a negative Pearson correlation: ![\\begin{equation\}r=-0.28\\end{equation\}](https://www.scribbr.com/wp-content/ql-cache/quicklatex.com-028980c08bb3f318f52836e30ee9cfa9_l3.png) This value can be used to calculate the coefficient of determination (R²) using Formula 1: ![\\begin{equation\}R^2=(r)^2\\end{equation\}](https://www.scribbr.com/wp-content/ql-cache/quicklatex.com-8c4d82b2fe20f208da8c31b613e5fd89_l3.png) ![\\begin{equation\}R^2=(-0.28)^2\\end{equation\}](https://www.scribbr.com/wp-content/ql-cache/quicklatex.com-335562cda79b94c68f75c3b43bf7a341_l3.png) ![\\begin{equation\}R^2=0.08\\end{equation\}](https://www.scribbr.com/wp-content/ql-cache/quicklatex.com-7c06a31c5618b67f96fec79ffb74e1e3_l3.png) ### Formula 2: Using the regression outputs Formula 2: ![\\begin{equation\}R^2=1-\\dfrac{\\textup{RSS}}{\\textup{TSS}}\\end{equation\}](https://www.scribbr.com/wp-content/ql-cache/quicklatex.com-77060d9686c4e6200e15410694b0c221_l3.png) Where: - RSS = sum of squared residuals - TSS = total sum of squares Example: Calculating R² using regression outputs As part of performing a simple linear regression that predicts students’ exam scores (dependent variable) from their study time (independent variable), you calculate that: ![\\begin{equation\}\\textup{RSS}=629.22\\end{equation\}](https://www.scribbr.com/wp-content/ql-cache/quicklatex.com-9d3bed919c3705b3686559290d3b64a4_l3.png) ![\\begin{equation\}\\textup{TSS}=2\\,187.04\\end{equation\}](https://www.scribbr.com/wp-content/ql-cache/quicklatex.com-23b0842ab7eb241dd98d4073491126ce_l3.png) These values can be used to calculate the coefficient of determination (R²) using Formula 2: ![\\begin{equation\}R^2=1-\\dfrac{\\textup{RSS}}{\\textup{TSS}}\\end{equation\}](https://www.scribbr.com/wp-content/ql-cache/quicklatex.com-77060d9686c4e6200e15410694b0c221_l3.png) ![\\begin{equation\}R^2=1-\\dfrac{629.22}{2\\,187.04}\\end{equation\}](https://www.scribbr.com/wp-content/ql-cache/quicklatex.com-25f7352ea18335eddc525ac74d0fea32_l3.png) ![\\begin{equation\}R^2=1-0.29\\end{equation\}](https://www.scribbr.com/wp-content/ql-cache/quicklatex.com-b74ead3cc867978ff7ec367f3830fba9_l3.png) ![\\begin{equation\}R^2=0.71\\end{equation\}](https://www.scribbr.com/wp-content/ql-cache/quicklatex.com-6e220f1a6c30feb37df1abf358c6dab4_l3.png) ## Interpreting the coefficient of determination You can interpret the coefficient of determination (R²) as the proportion of [variance](https://www.scribbr.com/statistics/variance/) in the [dependent variable](https://www.scribbr.com/methodology/independent-and-dependent-variables/#dependent) that is predicted by the statistical [model](https://www.scribbr.com/frequently-asked-questions/what-is-a-model/). Another way of thinking of it is that the R² is the proportion of variance that is shared between the independent and dependent variables. You can also say that the R² is the proportion of variance “explained” or “accounted for” by the model. The proportion that remains (1 − R²) is the variance that is not predicted by the model. If you prefer, you can write the R² as a percentage instead of a proportion. Simply multiply the proportion by 100. ### R² as an effect size Lastly, you can also interpret the R² as an [effect size](https://www.scribbr.com/statistics/effect-size/): a measure of the strength of the relationship between the dependent and independent variables. Psychologist and statistician Jacob Cohen (1988) suggested the following rules of thumb for [simple linear regressions](https://www.scribbr.com/statistics/simple-linear-regression/): \| Minimum coefficient of determination (R²) value \| Effect size interpretation \| \|---\|---\| \| .01 \| Small \| \| .09 \| Medium \| \| .25 \| Large \| Be careful: the R² on its own can’t tell you anything about [causation](https://www.scribbr.com/methodology/correlation-vs-causation/). Example: Interpreting R² A simple linear regression that predicts students’ exam scores (dependent variable) from their study time (independent variable) has an R² of .71. From this R² value, we know that: - 71% of the variance in students’ exam scores is predicted by their study time - 29% of the variance in student’s exam scores is unexplained by the model - The students’ study time has a large effect on their exam scores Studying longer may or may not [cause](https://www.scribbr.com/methodology/correlation-vs-causation/) an improvement in the students’ scores. Although this causal relationship is very plausible, the R² alone can’t tell us why there’s a relationship between students’ study time and exam scores. For example, students might find studying less frustrating when they understand the course material well, so they study longer. ## Reporting the coefficient of determination If you decide to include a coefficient of determination (R²) in your [research paper](https://www.scribbr.com/category/research-paper/), [dissertation](https://www.scribbr.com/category/dissertation/) or [thesis](https://www.scribbr.com/dissertation/thesis/), you should report it in your [results section](https://www.scribbr.com/dissertation/results/). You can follow these rules if you want to [report statistics in APA Style](https://www.scribbr.com/apa-style/numbers-and-statistics/): - You should use “r²” for statistical models with one independent variable (such as simple linear regressions). Use “R²” for statistical models with multiple independent variables. - You don’t need to provide a reference or formula since the coefficient of determination is a commonly used statistic. - You should italicize r² and R² when reporting their values (but don’t italicize the ²). - You shouldn’t include a leading zero (a zero before the decimal point) since the coefficient of determination can’t be greater than one. - You should provide two significant digits after the decimal point. - Very often, the coefficient of determination is provided alongside related statistical results, such as the [F value](https://www.scribbr.com/statistics/test-statistic/), [degrees of freedom](https://www.scribbr.com/statistics/degrees-of-freedom/), and [p value](https://www.scribbr.com/statistics/p-value/). Example: Reporting r² in APA Style Students’ exam scores were predicted by their study time, r² = .71, F(1,32) = 7.33, p \= .003. ## Practice questions ## Other interesting articles If you want to know more about [statistics](https://www.scribbr.com/category/statistics/), [methodology](https://www.scribbr.com/category/methodology/), or [research bias](https://www.scribbr.com/faq-category/research-bias/), make sure to check out some of our other articles with explanations and examples. Statistics - [Chi square test of independence](https://www.scribbr.com/statistics/chi-square-test-of-independence/) - [Statistical power](https://www.scribbr.com/statistics/statistical-power/) - [Descriptive statistics](https://www.scribbr.com/statistics/descriptive-statistics/) - [Degrees of freedom](https://www.scribbr.com/statistics/degrees-of-freedom/) - [Pearson correlation](https://www.scribbr.com/statistics/pearson-correlation-coefficient/) - [Null hypothesis](https://www.scribbr.com/statistics/null-and-alternative-hypotheses/) Methodology - [Double-blind study](https://www.scribbr.com/methodology/double-blind-study/) - [Case-control study](https://www.scribbr.com/methodology/case-control-study/) - [Research ethics](https://www.scribbr.com/methodology/research-ethics/) - [Data collection](https://www.scribbr.com/methodology/data-collection/) - [Hypothesis testing](https://www.scribbr.com/statistics/hypothesis-testing/) - [Structured interviews](https://www.scribbr.com/methodology/structured-interview/) ## Frequently asked questions about the coefficient of determination #### Cite this Scribbr article If you want to cite this source, you can copy and paste the citation or click the “Cite this Scribbr article” button to automatically add the citation to our free Citation Generator. > Turney, S. 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