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curl -X POST \
  'http://laksa080.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.statology.org/central-limit-theorem-conditions/\')), getAhrefsUnparsedNoserviceFromURL(\'https://www.statology.org/central-limit-theorem-conditions/\'))) FORMAT JSONEachRow'

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{"found_url":"https:\/\/www.statology.org\/central-limit-theorem-conditions\/","crawl_time":1776246516,"first_indexed_time":1611067980,"http_code":200,"src_unparsed":"org,statology!www,\/central-limit-theorem-conditions\/ s443","src_root_hash":"6121278954233998080","history_drop_reason":null,"meta_title":"Central Limit Theorem: The Four Conditions to Meet","meta_descriptions":["This tutorial explains the four conditions that must be met in order to apply the central limit theorem."],"attrs_boilerpipe_text":"The\ncentral limit theorem\nstates that\nthe sampling distribution of a sample mean\nis approximately normal if the sample size is large enough,\neven if the population distribution is not normal\n.\nIn order to apply the central limit theorem, there are four conditions that must be met:\n1.\n \nRandomization\n: The data must be sampled randomly such that every member in a population has an equal probability of being selected to be in the sample.\n2. Independence:\nThe sample values must be independent of each other.\n3. The 10% Condition:\nWhen the sample is drawn without replacement, the sample size should be no larger than 10% of the population.\n4. Large Sample Condition:\nThe sample size needs to be sufficiently large.\nThis tutorial provides a brief explanation of each condition.\nCondition 1: Randomization\nIn order to apply the central limit theorem, the data that we use must be sampled randomly from the population by using a\nprobability sampling method\n.\nIn statistics, there are two types of\nsampling methods\n:\n1. Probability sampling methods:\nSampling methods in which every member in a population has an equal probability of being selected to be in the sample. Examples include:\nSimple random sample\nStratified random sample\nCluster random sample\nSystematic random sample\n2. Non-probability sampling methods:\nSampling methods in which every member in a population does not have an equal probability of being selected to be in the sample. Examples include:\nConvenience sample\nVoluntary response sample\nSnowball sample\nPurposive sample\nIt’s important that a probability sampling method is used to obtain the sample because this maximizes the chances that we obtain a sample that is\nrepresentative of the population\n.\nCondition 2: Independence\nIn order to apply the central limit theorem, we must also assume that each of the sample values is independent of each other.\nThat is, the occurrence of one event does not affect the occurrence of any other event.\nThis assumption is often met if we use a probability sampling method because these types of sampling methods choose\nobservations\nto be included in the sample completely independently of each other. \nCondition 3: The 10% Condition\nWhen the sample is drawn without replacement (which is almost always the case), the sample size must be no larger than 10% of the total population.\nFor example:\nIf our population size is 500, then our sample size should be no larger than 50.\nIf our population size is 1,000 then our sample size should be no larger than 100.\nIf our population size is 50,000, then our sample size should be no larger than 5,000.\nAnd so on.\nCondition 4: Large Sample Condition\nLastly, in order to apply the central limit theorem our sample size must be sufficiently large.\nIn general, we consider “sufficiently large” to be 30 or larger. However, this number can vary a bit based on the underlying shape of the population distribution.\nIn particular:\nIf the population distribution is symmetric, sometimes a sample size as small as 15 is sufficient.\nIf the population distribution is skewed, generally a sample size of at least 30 is needed.\nIf the population distribution is extremely skewed, then a sample size of 40 or higher may be necessary.\nDepending on the shape of the population distribution, you may require more or less than a sample size of 30 in order for the Central Limit Theorem to apply.\nHey there. My name is Zach Bobbitt. I have a Masters of Science degree in Applied Statistics and I’ve worked on machine learning algorithms for professional businesses in both healthcare and retail. I’m passionate about statistics, machine learning, and data visualization and I created Statology to be a resource for both students and teachers alike.  My goal with this site is to help you learn statistics through using simple terms, plenty of real-world examples, and helpful illustrations.","attrs_markdown":"[![Statology](https:\/\/www.statology.org\/wp-content\/uploads\/2024\/07\/StatologyLogo_OnWhite.png)](https:\/\/www.statology.org\/)\n\n- [About](https:\/\/www.statology.org\/about\/)\n- [Course](https:\/\/www.statology.org\/course-register\/)\n- [Basic Stats](https:\/\/www.statology.org\/tutorials\/)\n- [Machine Learning](https:\/\/www.statology.org\/machine-learning-tutorials\/)\n- [Software Tutorials]()\n  - [Excel](https:\/\/www.statology.org\/excel-guides\/)\n  - [Google Sheets](https:\/\/www.statology.org\/google-sheets-guides\/)\n  - [MongoDB](https:\/\/www.statology.org\/mongodb-guides\/)\n  - [MySQL](https:\/\/www.statology.org\/mysql-guides\/)\n  - [Power BI](https:\/\/www.statology.org\/power-bi-guides\/)\n  - [PySpark](https:\/\/www.statology.org\/pyspark-guides\/)\n  - [Python](https:\/\/www.statology.org\/python-guides\/)\n  - [R](https:\/\/www.statology.org\/r-guides\/)\n  - [SAS](https:\/\/www.statology.org\/sas-guides\/)\n  - [SPSS](https:\/\/www.statology.org\/spss-guides\/)\n  - [Stata](https:\/\/www.statology.org\/stata-guides\/)\n  - [TI-84](https:\/\/www.statology.org\/ti-84-guides\/)\n  - [VBA](https:\/\/www.statology.org\/vba-guides\/)\n- [Tools]()\n  - [Calculators](https:\/\/www.statology.org\/calculators\/)\n  - [Critical Value Tables](https:\/\/www.statology.org\/tables\/)\n  - [Glossary](https:\/\/www.statology.org\/glossary\/)\n\n# Central Limit Theorem: The Four Conditions to Meet\nby [Zach Bobbitt](https:\/\/www.statology.org\/author\/admin\/)\n\nLast updated on\n\n[Last updated on April 12, 2021](https:\/\/www.statology.org\/central-limit-theorem-conditions\/)\n\n***\nThe [central limit theorem](https:\/\/www.statology.org\/central-limit-theorem\/) states that [the sampling distribution of a sample mean](https:\/\/www.statology.org\/sampling-distribution\/) is approximately normal if the sample size is large enough, *even if the population distribution is not normal*.\n\nIn order to apply the central limit theorem, there are four conditions that must be met:\n\n**1\\.** **Randomization**: The data must be sampled randomly such that every member in a population has an equal probability of being selected to be in the sample.\n\n**2\\. Independence:** The sample values must be independent of each other.\n\n**3\\. The 10% Condition:** When the sample is drawn without replacement, the sample size should be no larger than 10% of the population.\n\n**4\\. Large Sample Condition:** The sample size needs to be sufficiently large.\n\nThis tutorial provides a brief explanation of each condition.\n\n### **Condition 1: Randomization**\nIn order to apply the central limit theorem, the data that we use must be sampled randomly from the population by using a **probability sampling method**.\n\nIn statistics, there are two types of [sampling methods](https:\/\/www.statology.org\/sampling-methods\/):\n\n**1\\. Probability sampling methods:** Sampling methods in which every member in a population has an equal probability of being selected to be in the sample. Examples include:\n\n- Simple random sample\n- Stratified random sample\n- Cluster random sample\n- Systematic random sample\n\n**2\\. Non-probability sampling methods:** Sampling methods in which every member in a population does not have an equal probability of being selected to be in the sample. Examples include:\n\n- Convenience sample\n- [Voluntary response sample](https:\/\/www.statology.org\/voluntary-response-sample\/)\n- [Snowball sample](https:\/\/www.statology.org\/snowball-sampling\/)\n- Purposive sample\n\nIt’s important that a probability sampling method is used to obtain the sample because this maximizes the chances that we obtain a sample that is [representative of the population](https:\/\/www.statology.org\/representative-sample\/).\n\n### **Condition 2: Independence**\nIn order to apply the central limit theorem, we must also assume that each of the sample values is independent of each other. That is, the occurrence of one event does not affect the occurrence of any other event.\n\nThis assumption is often met if we use a probability sampling method because these types of sampling methods choose [observations](https:\/\/www.statology.org\/observation-in-statistics\/) to be included in the sample completely independently of each other.\n\n### **Condition 3: The 10% Condition**\nWhen the sample is drawn without replacement (which is almost always the case), the sample size must be no larger than 10% of the total population.\n\nFor example:\n\n- If our population size is 500, then our sample size should be no larger than 50.\n- If our population size is 1,000 then our sample size should be no larger than 100.\n- If our population size is 50,000, then our sample size should be no larger than 5,000.\n\nAnd so on.\n\n### **Condition 4: Large Sample Condition**\nLastly, in order to apply the central limit theorem our sample size must be sufficiently large.\n\nIn general, we consider “sufficiently large” to be 30 or larger. However, this number can vary a bit based on the underlying shape of the population distribution.\n\nIn particular:\n\n- If the population distribution is symmetric, sometimes a sample size as small as 15 is sufficient.\n- If the population distribution is skewed, generally a sample size of at least 30 is needed.\n- If the population distribution is extremely skewed, then a sample size of 40 or higher may be necessary.\n\nDepending on the shape of the population distribution, you may require more or less than a sample size of 30 in order for the Central Limit Theorem to apply.\n\nPosted in [Programming](https:\/\/www.statology.org\/category\/programming\/)\n\n![](https:\/\/www.statology.org\/wp-content\/uploads\/2023\/08\/statology_gravatar-scaled.jpg)\n\n[Zach Bobbitt](https:\/\/www.statology.org\/author\/admin\/)\n\nHey there. My name is Zach Bobbitt. I have a Masters of Science degree in Applied Statistics and I’ve worked on machine learning algorithms for professional businesses in both healthcare and retail. I’m passionate about statistics, machine learning, and data visualization and I created Statology to be a resource for both students and teachers alike. My goal with this site is to help you learn statistics through using simple terms, plenty of real-world examples, and helpful illustrations.\n\n## Post navigation\n[Prev Pandas: How to Sum Columns Based on a Condition](https:\/\/www.statology.org\/pandas-sum-column-with-condition\/)\n\n[Next Correlation Matrix Calculator](https:\/\/www.statology.org\/correlation-matrix-calculator\/)\n### Leave a Reply [Cancel reply](https:\/\/www.statology.org\/central-limit-theorem-conditions\/#respond)\n\n## Search\n## ABOUT STATOLOGY\n[![](https:\/\/www.statology.org\/wp-content\/uploads\/2023\/08\/statology_circle-150x150.png)](https:\/\/www.statology.org\/about\/)Statology makes learning statistics easy by explaining topics in simple and straightforward ways. Our team of writers have over 40 years of experience in the fields of Machine Learning, AI and Statistics. **[Learn more about our team here.](https:\/\/www.statology.org\/about\/)**\n\n## Featured Posts\n- [![](https:\/\/www.statology.org\/wp-content\/uploads\/2026\/04\/sta-chugani-amaras-law-explained-technology-forecasts-systematically-wrong-feature-150x150.png)](https:\/\/www.statology.org\/amaras-law-explained-why-our-technology-forecasts-are-systematically-wrong\/)\n  [Amara’s Law Explained: Why Our Technology Forecasts Are Systematically Wrong](https:\/\/www.statology.org\/amaras-law-explained-why-our-technology-forecasts-are-systematically-wrong\/)\n  April 14, 2026\n- [![](https:\/\/www.statology.org\/wp-content\/uploads\/2026\/03\/5-Ways-to-Go-Beyond-Correlation-and-Start-Inferring-Causality-150x150.png)](https:\/\/www.statology.org\/5-visualizations-that-reveal-statistical-signal\/)\n  [5 Visualizations That Reveal Statistical Signal](https:\/\/www.statology.org\/5-visualizations-that-reveal-statistical-signal\/)\n  April 13, 2026\n- [![](https:\/\/www.statology.org\/wp-content\/uploads\/2026\/04\/sta-chugani-distance-metric-clustering-decision-tree-feature-150x150.png)](https:\/\/www.statology.org\/choosing-the-right-distance-metric-for-clustering-a-decision-tree-approach\/)\n  [Choosing the Right Distance Metric for Clustering: A Decision Tree Approach](https:\/\/www.statology.org\/choosing-the-right-distance-metric-for-clustering-a-decision-tree-approach\/)\n  April 10, 2026\n- [![](https:\/\/www.statology.org\/wp-content\/uploads\/2026\/04\/sta-chugani-textblob-python-simple-visual-feature-150x150.png)](https:\/\/www.statology.org\/how-to-analyze-word-frequencies-with-textblob-in-python\/)\n  [How to Analyze Word Frequencies with TextBlob in Python](https:\/\/www.statology.org\/how-to-analyze-word-frequencies-with-textblob-in-python\/)\n  April 9, 2026\n- [![](https:\/\/www.statology.org\/wp-content\/uploads\/2026\/04\/sta-chugani-raw-data-insight-copilot-microsoft-stack-feature-150x150.png)](https:\/\/www.statology.org\/from-raw-data-to-insight-with-copilot-across-the-microsoft-stack\/)\n  [From Raw Data to Insight with Copilot Across the Microsoft Stack](https:\/\/www.statology.org\/from-raw-data-to-insight-with-copilot-across-the-microsoft-stack\/)\n  April 9, 2026\n- [![](https:\/\/www.statology.org\/wp-content\/uploads\/2026\/04\/sta-chugani-textblob-python-simple-visual-feature-150x150.png)](https:\/\/www.statology.org\/how-to-measure-subjectivity-in-text-with-textblob-in-python\/)\n  [How to Measure Subjectivity in Text with TextBlob in Python](https:\/\/www.statology.org\/how-to-measure-subjectivity-in-text-with-textblob-in-python\/)\n  April 8, 2026\n\n## Statology Study\n**[Statology Study](https:\/\/www.statology.org\/study-register\/)** is the ultimate online statistics study guide that helps you study and practice all of the core concepts taught in any elementary statistics course and makes your life so much easier as a student.\n\n[![statology study](https:\/\/www.statology.org\/wp-content\/uploads\/2021\/01\/statology_study_cover1.png)](https:\/\/www.statology.org\/study-register\/)\n\n## Introduction to Statistics Course\n**Introduction to Statistics** is our premier online video course that teaches you all of the topics covered in introductory statistics. **[Get started](https:\/\/www.statology.org\/course-register\/)**  with our course today.\n\n[![introduction to statistics](https:\/\/www.statology.org\/wp-content\/uploads\/2022\/06\/Intro-to-Statistics-Cover-Photo-2.jpg)](https:\/\/www.statology.org\/course-register\/)\n\n## You Might Also Like\n- [The Large Sample Condition: Definition & Example](https:\/\/www.statology.org\/large-sample-condition\/)\n- [5 Examples of Using the Central Limit Theorem in Real Life](https:\/\/www.statology.org\/central-limit-theorem-real-life-examples\/)\n- [How to Apply the Central Limit Theorem in Excel](https:\/\/www.statology.org\/central-limit-theorem-excel\/)\n- [How to Apply the Central Limit Theorem on TI-84 Calculator](https:\/\/www.statology.org\/central-limit-theorem-ti-84\/)\n- [Central Limit Theorem Visualizer](https:\/\/www.statology.org\/central-limit-theorem-visualizer\/)\n- [The 6 Confidence Interval Assumptions to Check](https:\/\/www.statology.org\/confidence-interval-assumptions\/)\n\n© 2025 [Statology](https:\/\/www.statology.org\/) \\| [Privacy Policy](https:\/\/www.guidingtechmedia.com\/privacy\/) \\| [Terms of Use](https:\/\/www.guidingtechmedia.com\/terms-of-use\/)\n\nWisteria Theme by [WPFriendship](https:\/\/wpfriendship.com\/ \"WPFriendship\") ⋅ Powered by [WordPress](https:\/\/wordpress.org\/ \"WordPress\")","attrs_readable_markdown":"***\nThe [central limit theorem](https:\/\/www.statology.org\/central-limit-theorem\/) states that [the sampling distribution of a sample mean](https:\/\/www.statology.org\/sampling-distribution\/) is approximately normal if the sample size is large enough, *even if the population distribution is not normal*.\n\nIn order to apply the central limit theorem, there are four conditions that must be met:\n\n**1\\.** **Randomization**: The data must be sampled randomly such that every member in a population has an equal probability of being selected to be in the sample.\n\n**2\\. Independence:** The sample values must be independent of each other.\n\n**3\\. The 10% Condition:** When the sample is drawn without replacement, the sample size should be no larger than 10% of the population.\n\n**4\\. Large Sample Condition:** The sample size needs to be sufficiently large.\n\nThis tutorial provides a brief explanation of each condition.\n\n### **Condition 1: Randomization**\nIn order to apply the central limit theorem, the data that we use must be sampled randomly from the population by using a **probability sampling method**.\n\nIn statistics, there are two types of [sampling methods](https:\/\/www.statology.org\/sampling-methods\/):\n\n**1\\. Probability sampling methods:** Sampling methods in which every member in a population has an equal probability of being selected to be in the sample. Examples include:\n\n- Simple random sample\n- Stratified random sample\n- Cluster random sample\n- Systematic random sample\n\n**2\\. Non-probability sampling methods:** Sampling methods in which every member in a population does not have an equal probability of being selected to be in the sample. Examples include:\n\n- Convenience sample\n- [Voluntary response sample](https:\/\/www.statology.org\/voluntary-response-sample\/)\n- [Snowball sample](https:\/\/www.statology.org\/snowball-sampling\/)\n- Purposive sample\n\nIt’s important that a probability sampling method is used to obtain the sample because this maximizes the chances that we obtain a sample that is [representative of the population](https:\/\/www.statology.org\/representative-sample\/).\n\n### **Condition 2: Independence**\nIn order to apply the central limit theorem, we must also assume that each of the sample values is independent of each other. That is, the occurrence of one event does not affect the occurrence of any other event.\n\nThis assumption is often met if we use a probability sampling method because these types of sampling methods choose [observations](https:\/\/www.statology.org\/observation-in-statistics\/) to be included in the sample completely independently of each other.\n\n### **Condition 3: The 10% Condition**\nWhen the sample is drawn without replacement (which is almost always the case), the sample size must be no larger than 10% of the total population.\n\nFor example:\n\n- If our population size is 500, then our sample size should be no larger than 50.\n- If our population size is 1,000 then our sample size should be no larger than 100.\n- If our population size is 50,000, then our sample size should be no larger than 5,000.\n\nAnd so on.\n\n### **Condition 4: Large Sample Condition**\nLastly, in order to apply the central limit theorem our sample size must be sufficiently large.\n\nIn general, we consider “sufficiently large” to be 30 or larger. However, this number can vary a bit based on the underlying shape of the population distribution.\n\nIn particular:\n\n- If the population distribution is symmetric, sometimes a sample size as small as 15 is sufficient.\n- If the population distribution is skewed, generally a sample size of at least 30 is needed.\n- If the population distribution is extremely skewed, then a sample size of 40 or higher may be necessary.\n\nDepending on the shape of the population distribution, you may require more or less than a sample size of 30 in order for the Central Limit Theorem to apply.\n\n![](https:\/\/www.statology.org\/wp-content\/uploads\/2023\/08\/statology_gravatar-scaled.jpg)\n\nHey there. My name is Zach Bobbitt. I have a Masters of Science degree in Applied Statistics and I’ve worked on machine learning algorithms for professional businesses in both healthcare and retail. I’m passionate about statistics, machine learning, and data visualization and I created Statology to be a resource for both students and teachers alike. My goal with this site is to help you learn statistics through using simple terms, plenty of real-world examples, and helpful illustrations.","meta_canonical":null}

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Boilerpipe Text	The central limit theorem states that the sampling distribution of a sample mean is approximately normal if the sample size is large enough, even if the population distribution is not normal . In order to apply the central limit theorem, there are four conditions that must be met: 1. Randomization : The data must be sampled randomly such that every member in a population has an equal probability of being selected to be in the sample. 2. Independence: The sample values must be independent of each other. 3. The 10% Condition: When the sample is drawn without replacement, the sample size should be no larger than 10% of the population. 4. Large Sample Condition: The sample size needs to be sufficiently large. This tutorial provides a brief explanation of each condition. Condition 1: Randomization In order to apply the central limit theorem, the data that we use must be sampled randomly from the population by using a probability sampling method . In statistics, there are two types of sampling methods : 1. Probability sampling methods: Sampling methods in which every member in a population has an equal probability of being selected to be in the sample. Examples include: Simple random sample Stratified random sample Cluster random sample Systematic random sample 2. Non-probability sampling methods: Sampling methods in which every member in a population does not have an equal probability of being selected to be in the sample. Examples include: Convenience sample Voluntary response sample Snowball sample Purposive sample It’s important that a probability sampling method is used to obtain the sample because this maximizes the chances that we obtain a sample that is representative of the population . Condition 2: Independence In order to apply the central limit theorem, we must also assume that each of the sample values is independent of each other. That is, the occurrence of one event does not affect the occurrence of any other event. This assumption is often met if we use a probability sampling method because these types of sampling methods choose observations to be included in the sample completely independently of each other. Condition 3: The 10% Condition When the sample is drawn without replacement (which is almost always the case), the sample size must be no larger than 10% of the total population. For example: If our population size is 500, then our sample size should be no larger than 50. If our population size is 1,000 then our sample size should be no larger than 100. If our population size is 50,000, then our sample size should be no larger than 5,000. And so on. Condition 4: Large Sample Condition Lastly, in order to apply the central limit theorem our sample size must be sufficiently large. In general, we consider “sufficiently large” to be 30 or larger. However, this number can vary a bit based on the underlying shape of the population distribution. In particular: If the population distribution is symmetric, sometimes a sample size as small as 15 is sufficient. If the population distribution is skewed, generally a sample size of at least 30 is needed. If the population distribution is extremely skewed, then a sample size of 40 or higher may be necessary. Depending on the shape of the population distribution, you may require more or less than a sample size of 30 in order for the Central Limit Theorem to apply. Hey there. My name is Zach Bobbitt. I have a Masters of Science degree in Applied Statistics and I’ve worked on machine learning algorithms for professional businesses in both healthcare and retail. I’m passionate about statistics, machine learning, and data visualization and I created Statology to be a resource for both students and teachers alike. My goal with this site is to help you learn statistics through using simple terms, plenty of real-world examples, and helpful illustrations.
Markdown	[![Statology](https://www.statology.org/wp-content/uploads/2024/07/StatologyLogo_OnWhite.png)](https://www.statology.org/) - [About](https://www.statology.org/about/) - [Course](https://www.statology.org/course-register/) - [Basic Stats](https://www.statology.org/tutorials/) - [Machine Learning](https://www.statology.org/machine-learning-tutorials/) - [Software Tutorials]() - [Excel](https://www.statology.org/excel-guides/) - [Google Sheets](https://www.statology.org/google-sheets-guides/) - [MongoDB](https://www.statology.org/mongodb-guides/) - [MySQL](https://www.statology.org/mysql-guides/) - [Power BI](https://www.statology.org/power-bi-guides/) - [PySpark](https://www.statology.org/pyspark-guides/) - [Python](https://www.statology.org/python-guides/) - [R](https://www.statology.org/r-guides/) - [SAS](https://www.statology.org/sas-guides/) - [SPSS](https://www.statology.org/spss-guides/) - [Stata](https://www.statology.org/stata-guides/) - [TI-84](https://www.statology.org/ti-84-guides/) - [VBA](https://www.statology.org/vba-guides/) - [Tools]() - [Calculators](https://www.statology.org/calculators/) - [Critical Value Tables](https://www.statology.org/tables/) - [Glossary](https://www.statology.org/glossary/) # Central Limit Theorem: The Four Conditions to Meet by [Zach Bobbitt](https://www.statology.org/author/admin/) Last updated on [Last updated on April 12, 2021](https://www.statology.org/central-limit-theorem-conditions/) *** The [central limit theorem](https://www.statology.org/central-limit-theorem/) states that [the sampling distribution of a sample mean](https://www.statology.org/sampling-distribution/) is approximately normal if the sample size is large enough, even if the population distribution is not normal. In order to apply the central limit theorem, there are four conditions that must be met: 1\. Randomization: The data must be sampled randomly such that every member in a population has an equal probability of being selected to be in the sample. 2\. Independence: The sample values must be independent of each other. 3\. The 10% Condition: When the sample is drawn without replacement, the sample size should be no larger than 10% of the population. 4\. Large Sample Condition: The sample size needs to be sufficiently large. This tutorial provides a brief explanation of each condition. ### Condition 1: Randomization In order to apply the central limit theorem, the data that we use must be sampled randomly from the population by using a probability sampling method. In statistics, there are two types of [sampling methods](https://www.statology.org/sampling-methods/): 1\. Probability sampling methods: Sampling methods in which every member in a population has an equal probability of being selected to be in the sample. Examples include: - Simple random sample - Stratified random sample - Cluster random sample - Systematic random sample 2\. Non-probability sampling methods: Sampling methods in which every member in a population does not have an equal probability of being selected to be in the sample. Examples include: - Convenience sample - [Voluntary response sample](https://www.statology.org/voluntary-response-sample/) - [Snowball sample](https://www.statology.org/snowball-sampling/) - Purposive sample It’s important that a probability sampling method is used to obtain the sample because this maximizes the chances that we obtain a sample that is [representative of the population](https://www.statology.org/representative-sample/). ### Condition 2: Independence In order to apply the central limit theorem, we must also assume that each of the sample values is independent of each other. That is, the occurrence of one event does not affect the occurrence of any other event. This assumption is often met if we use a probability sampling method because these types of sampling methods choose [observations](https://www.statology.org/observation-in-statistics/) to be included in the sample completely independently of each other. ### Condition 3: The 10% Condition When the sample is drawn without replacement (which is almost always the case), the sample size must be no larger than 10% of the total population. For example: - If our population size is 500, then our sample size should be no larger than 50. - If our population size is 1,000 then our sample size should be no larger than 100. - If our population size is 50,000, then our sample size should be no larger than 5,000. And so on. ### Condition 4: Large Sample Condition Lastly, in order to apply the central limit theorem our sample size must be sufficiently large. In general, we consider “sufficiently large” to be 30 or larger. However, this number can vary a bit based on the underlying shape of the population distribution. In particular: - If the population distribution is symmetric, sometimes a sample size as small as 15 is sufficient. - If the population distribution is skewed, generally a sample size of at least 30 is needed. - If the population distribution is extremely skewed, then a sample size of 40 or higher may be necessary. Depending on the shape of the population distribution, you may require more or less than a sample size of 30 in order for the Central Limit Theorem to apply. Posted in [Programming](https://www.statology.org/category/programming/) ![](https://www.statology.org/wp-content/uploads/2023/08/statology_gravatar-scaled.jpg) [Zach Bobbitt](https://www.statology.org/author/admin/) Hey there. My name is Zach Bobbitt. I have a Masters of Science degree in Applied Statistics and I’ve worked on machine learning algorithms for professional businesses in both healthcare and retail. 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Readable Markdown	*** The [central limit theorem](https://www.statology.org/central-limit-theorem/) states that [the sampling distribution of a sample mean](https://www.statology.org/sampling-distribution/) is approximately normal if the sample size is large enough, even if the population distribution is not normal. In order to apply the central limit theorem, there are four conditions that must be met: 1\. Randomization: The data must be sampled randomly such that every member in a population has an equal probability of being selected to be in the sample. 2\. Independence: The sample values must be independent of each other. 3\. The 10% Condition: When the sample is drawn without replacement, the sample size should be no larger than 10% of the population. 4\. Large Sample Condition: The sample size needs to be sufficiently large. This tutorial provides a brief explanation of each condition. ### Condition 1: Randomization In order to apply the central limit theorem, the data that we use must be sampled randomly from the population by using a probability sampling method. In statistics, there are two types of [sampling methods](https://www.statology.org/sampling-methods/): 1\. Probability sampling methods: Sampling methods in which every member in a population has an equal probability of being selected to be in the sample. Examples include: - Simple random sample - Stratified random sample - Cluster random sample - Systematic random sample 2\. Non-probability sampling methods: Sampling methods in which every member in a population does not have an equal probability of being selected to be in the sample. Examples include: - Convenience sample - [Voluntary response sample](https://www.statology.org/voluntary-response-sample/) - [Snowball sample](https://www.statology.org/snowball-sampling/) - Purposive sample It’s important that a probability sampling method is used to obtain the sample because this maximizes the chances that we obtain a sample that is [representative of the population](https://www.statology.org/representative-sample/). ### Condition 2: Independence In order to apply the central limit theorem, we must also assume that each of the sample values is independent of each other. That is, the occurrence of one event does not affect the occurrence of any other event. This assumption is often met if we use a probability sampling method because these types of sampling methods choose [observations](https://www.statology.org/observation-in-statistics/) to be included in the sample completely independently of each other. ### Condition 3: The 10% Condition When the sample is drawn without replacement (which is almost always the case), the sample size must be no larger than 10% of the total population. For example: - If our population size is 500, then our sample size should be no larger than 50. - If our population size is 1,000 then our sample size should be no larger than 100. - If our population size is 50,000, then our sample size should be no larger than 5,000. And so on. ### Condition 4: Large Sample Condition Lastly, in order to apply the central limit theorem our sample size must be sufficiently large. In general, we consider “sufficiently large” to be 30 or larger. However, this number can vary a bit based on the underlying shape of the population distribution. In particular: - If the population distribution is symmetric, sometimes a sample size as small as 15 is sufficient. - If the population distribution is skewed, generally a sample size of at least 30 is needed. - If the population distribution is extremely skewed, then a sample size of 40 or higher may be necessary. Depending on the shape of the population distribution, you may require more or less than a sample size of 30 in order for the Central Limit Theorem to apply. ![](https://www.statology.org/wp-content/uploads/2023/08/statology_gravatar-scaled.jpg) Hey there. My name is Zach Bobbitt. I have a Masters of Science degree in Applied Statistics and I’ve worked on machine learning algorithms for professional businesses in both healthcare and retail. I’m passionate about statistics, machine learning, and data visualization and I created Statology to be a resource for both students and teachers alike. My goal with this site is to help you learn statistics through using simple terms, plenty of real-world examples, and helpful illustrations.
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