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curl -X POST \
  'http://laksa150.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.statisticshowto.com/james-stein-estimator/\')), getAhrefsUnparsedNoserviceFromURL(\'https://www.statisticshowto.com/james-stein-estimator/\'))) FORMAT JSONEachRow'

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{"found_url":"https:\/\/www.statisticshowto.com\/james-stein-estimator\/","crawl_time":1776503490,"first_indexed_time":1586043045,"http_code":200,"src_unparsed":"com,statisticshowto!www,\/james-stein-estimator\/ s443","src_root_hash":"12746512124316119150","history_drop_reason":null,"meta_title":"James-Stein Estimator: Definition, Formulas - Statistics How To","meta_descriptions":["Simple definition for the James-Stein estimator: statistics terms in plain English. Formulas for the James-Stein estimator. Shrinking constant explained."],"attrs_boilerpipe_text":"Estimators\n> James-Stein Estimator\nWhat is the James-Stein Estimator?\nIt’s common in statistics to take averages to make predictions. For example, the\nsample mean\n(the\naverage\nscore from all samples) is used as an estimator for the\npopulation mean\n.\nJames-Stein estimators\nimprove upon these averages by\nshrinking\nthem towards a more central average.  The technique is named after Charles Stein and Willard James, who simplified Stein’s original 1956 method.\nCalculations\nThe basic steps are:\nCalculate the sample mean (X̄).\n“Shrink” individual scores towards (X̄); Reduce larger values and increase smaller values.\nEach of these individual shrunk values is a James-Stein estimator,\nz.\nThe basic formula for the James-Stein estimator is:\nz = x̄ + c(y – x̄)\nWhere:\n(y – x̄) = difference between an individual score and the sample mean,\nc = a shrinking factor.\nOther formulas exist, but they all have the shrinking factor in common.\nFor example, instead of the sample mean you could use the mean from a\nprior distribution\n(\nm\n). In that case, ȳ could be replaced by\nm\n. The shrinking factor’s value is calculate after collecting the sample data and is given by the formula:\nWhere:\nx = individual values,\nx̄ = sample mean,\nk = number of unknown means (must be 2 or more),\nσ\n2\n=\nvariance\n.\nThe shrinking factor’s value should be less than 1. For example, a value of .3 would shrink values by about 70 percent.\nJames-Stein Estimators vs. Sample Means\nThe James-Stein estimator is a significant departure from the “traditional” school of thought which states that the sample mean is the best estimator for the population mean. Stein and James proved that a better estimator than the “perfect” estimator exists, which seems to be somewhat of a paradox. However, the James-Stein estimator outperforms the sample mean when there are\nseveral\nunknown population means — not just  one. The means do not have to be related, so they have to be carefully chosen.\nCombining completely unrelated means will give you a result — but it will be a nonsensical one.\nBradley Efron and Carl Morris (1977) offer the extreme example of combining batting averages in baseball and proportions of imported cars; You can calculate a mean for these, but it will make no sense at all.\nReferences\nEfron, B. and Morris, C. (1977), “Stein’s Paradox in Statistics.” Scientific American. 236 (5): 119–127\nJames, W., and Stein, C., (1961). “Estimation with Quadratic Loss.” Proceedings of the Fourth Berkeley Symposium, Vol. 1 (Berkeley, California: University of California Press), pp. 361-379.\nStein C. (1956). “Inadmissibility of the usual estimator for the mean of a\nmultivariate normal distribution\n”. Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability. Vol. 1. University of California Press; Berkeley, CA, USA: pp. 197–208\nComments? Need to post a correction?\nPlease\nContact Us\n.","attrs_markdown":"[Skip to content](https:\/\/www.statisticshowto.com\/james-stein-estimator\/#content \"Skip to content\")\n\n[Statistics How To](https:\/\/www.statisticshowto.com\/)\n\nMenu\n\n- [Home](https:\/\/www.statisticshowto.com\/)\n- [Tables](https:\/\/www.statisticshowto.com\/tables\/)\n  - [Binomial Distribution Table](https:\/\/www.statisticshowto.com\/tables\/binomial-distribution-table\/)\n  - [F Table](https:\/\/www.statisticshowto.com\/tables\/f-table\/)\n  - [Inverse T Distribution Table](https:\/\/www.statisticshowto.com\/tables\/inverse-t-distribution-table\/)\n  - [PPMC Critical Values](https:\/\/www.statisticshowto.com\/tables\/ppmc-critical-values\/)\n  - [T-Distribution Table (One Tail and Two-Tails)](https:\/\/www.statisticshowto.com\/tables\/t-distribution-table\/)\n  - [Chi Squared Table (Right Tail)](https:\/\/www.statisticshowto.com\/tables\/chi-squared-table-right-tail\/)\n  - [Z-table (Right of Curve or Left)](https:\/\/www.statisticshowto.com\/tables\/z-table\/)\n- [Probability and Statistics](https:\/\/www.statisticshowto.com\/probability-and-statistics\/)\n  - 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For example, the [sample mean](https:\/\/www.statisticshowto.com\/probability-and-statistics\/statistics-definitions\/sample-mean\/) (the [average](https:\/\/www.statisticshowto.com\/arithmetic-mean\/) score from all samples) is used as an estimator for the [population mean](https:\/\/www.statisticshowto.com\/population-mean\/).  **James-Stein estimators** improve upon these averages by [shrinking](https:\/\/www.statisticshowto.com\/shrinkage-estimator\/) them towards a more central average. The technique is named after Charles Stein and Willard James, who simplified Stein’s original 1956 method.\n\n## Calculations\nThe basic steps are:\n\n1. Calculate the sample mean (X̄).\n2. “Shrink” individual scores towards (X̄); Reduce larger values and increase smaller values. **Each of these individual shrunk values is a James-Stein estimator,** z.\n\nThe basic formula for the James-Stein estimator is:  \n **z = x̄ + c(y – x̄)**  \n Where:\n\n- (y – x̄) = difference between an individual score and the sample mean,\n- c = a shrinking factor.\n\n**Other formulas exist, but they all have the shrinking factor in common.** For example, instead of the sample mean you could use the mean from a [prior distribution](https:\/\/www.statisticshowto.com\/prior-distribution\/) (*m*). In that case, ȳ could be replaced by *m*. The shrinking factor’s value is calculate after collecting the sample data and is given by the formula:  \n [![shrinking-factor james-stein estimator](https:\/\/www.statisticshowto.com\/wp-content\/uploads\/2016\/10\/shrinking-factor-james-stein-estimator.png)](https:\/\/www.statisticshowto.com\/wp-content\/uploads\/2016\/10\/shrinking-factor-james-stein-estimator.png)  \n   \n  \n Where:\n\n- x = individual values,\n- x̄ = sample mean,\n- k = number of unknown means (must be 2 or more),\n- σ2 = [variance](https:\/\/www.statisticshowto.com\/probability-and-statistics\/variance\/).\n\nThe shrinking factor’s value should be less than 1. For example, a value of .3 would shrink values by about 70 percent.\n\n## James-Stein Estimators vs. Sample Means\nThe James-Stein estimator is a significant departure from the “traditional” school of thought which states that the sample mean is the best estimator for the population mean. Stein and James proved that a better estimator than the “perfect” estimator exists, which seems to be somewhat of a paradox. However, the James-Stein estimator outperforms the sample mean when there are *several* unknown population means — not just one. The means do not have to be related, so they have to be carefully chosen. **Combining completely unrelated means will give you a result — but it will be a nonsensical one.** Bradley Efron and Carl Morris (1977) offer the extreme example of combining batting averages in baseball and proportions of imported cars; You can calculate a mean for these, but it will make no sense at all.\n\n## References\nEfron, B. and Morris, C. (1977), “Stein’s Paradox in Statistics.” Scientific American. 236 (5): 119–127  \n James, W., and Stein, C., (1961). “Estimation with Quadratic Loss.” Proceedings of the Fourth Berkeley Symposium, Vol. 1 (Berkeley, California: University of California Press), pp. 361-379.  \n Stein C. (1956). “Inadmissibility of the usual estimator for the mean of a [multivariate normal distribution](https:\/\/www.statisticshowto.com\/bivariate-normal-distribution\/)”. Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability. Vol. 1. University of California Press; Berkeley, CA, USA: pp. 197–208\n\n[Deterministic: Definition and Examples](https:\/\/www.statisticshowto.com\/deterministic\/)\n\n[Shrinkage Estimator: Definition, Examples](https:\/\/www.statisticshowto.com\/shrinkage-estimator\/)\n\n**Comments? Need to post a correction?** Please [***Contact Us***](https:\/\/www.statisticshowto.com\/contact\/).\n\nCheck out our **[\"Practically Cheating Statistics Handbook](https:\/\/www.statisticshowto.com\/?page_id=15743&preview=true\")**, which gives you hundreds of easy-to-follow answers in a convenient e-book.\n\n[![Statistics Handbook](https:\/\/www.statisticshowto.com\/wp-content\/uploads\/2024\/06\/BookCoverSHT-1.jpg)](https:\/\/www.statisticshowto.com\/?page_id=15743&preview=true)\n***\n\n**Readers are calling it suspenseful, haunting, and unforgettable. Check out Stephanie’s debut post-apocalyptic novel — available now on Amazon.**\n\n[![an image of S.E. Glen's book Fallout in Georgia](https:\/\/www.statisticshowto.com\/wp-content\/uploads\/2026\/02\/novel-ad.jpg)![an image of S.E. Glen's book Fallout in Georgia](https:\/\/www.statisticshowto.com\/wp-content\/uploads\/2026\/02\/novel-ad.jpg)](https:\/\/amzn.to\/3LLzICB)\n\nLooking for elementary [**statistics help**](https:\/\/www.statisticshowto.com\/help-with-statistics-equations\/)? You’ve come to the right place. **Statistics How To** has more than 1,000 articles and **videos** for elementary statistics, probability, AP and advanced statistics topics. **Looking for a specific topic?** Type it into the search box at the top of the page.\n\nLatest articles\n\n- [Hedges’ g: Definition, Formula](https:\/\/www.statisticshowto.com\/hedges-g\/)\n- [Kaiser-Meyer-Olkin (KMO) Test for Sampling Adequacy](https:\/\/www.statisticshowto.com\/kaiser-meyer-olkin\/)\n- [Benjamini-Hochberg Procedure](https:\/\/www.statisticshowto.com\/benjamini-hochberg-procedure\/)\n- [Tetrachoric Correlation: Definition, Examples, Formula](https:\/\/www.statisticshowto.com\/tetrachoric-correlation\/)\n- [Random Variable: What is it in Statistics?](https:\/\/www.statisticshowto.com\/random-variable\/)\n- [Maximum Likelihood and Maximum Likelihood Estimation](https:\/\/www.statisticshowto.com\/maximum-likelihood-estimation\/)\n- [What is a Parameter in Statistics?](https:\/\/www.statisticshowto.com\/what-is-a-parameter-in-statistics\/)\n- [Beta Level: Definition & Examples](https:\/\/www.statisticshowto.com\/beta-level\/)\n- [Pairwise Independent, Mutually Independent: Definition, Example](https:\/\/www.statisticshowto.com\/pairwise-independent-mutually\/)\n- [Population Mean Definition, Example, Formula](https:\/\/www.statisticshowto.com\/population-mean\/)\n- [Dispersion \/ Measures of Dispersion: Definition](https:\/\/www.statisticshowto.com\/dispersion\/)\n- [Serial Correlation \/ Autocorrelation: Definition, Tests](https:\/\/www.statisticshowto.com\/serial-correlation-autocorrelation\/)\n\n© 2026 [**Statistics How To**](https:\/\/www.statisticshowto.com\/) \\| [About Us](https:\/\/www.statisticshowto.com\/contact\/) \\| [Privacy Policy](https:\/\/www.statisticshowto.com\/privacy-policy\/) \\| [Terms of Use](https:\/\/www.statisticshowto.com\/terms-of-use\/)","attrs_readable_markdown":"[Estimators](https:\/\/www.statisticshowto.com\/estimator\/) \\> James-Stein Estimator\n\n## What is the James-Stein Estimator?\nIt’s common in statistics to take averages to make predictions. For example, the [sample mean](https:\/\/www.statisticshowto.com\/probability-and-statistics\/statistics-definitions\/sample-mean\/) (the [average](https:\/\/www.statisticshowto.com\/arithmetic-mean\/) score from all samples) is used as an estimator for the [population mean](https:\/\/www.statisticshowto.com\/population-mean\/).  **James-Stein estimators** improve upon these averages by [shrinking](https:\/\/www.statisticshowto.com\/shrinkage-estimator\/) them towards a more central average. The technique is named after Charles Stein and Willard James, who simplified Stein’s original 1956 method.\n\n## Calculations\nThe basic steps are:\n\n1. Calculate the sample mean (X̄).\n2. “Shrink” individual scores towards (X̄); Reduce larger values and increase smaller values. **Each of these individual shrunk values is a James-Stein estimator,** z.\n\nThe basic formula for the James-Stein estimator is:  \n **z = x̄ + c(y – x̄)**  \n Where:\n\n- (y – x̄) = difference between an individual score and the sample mean,\n- c = a shrinking factor.\n\n**Other formulas exist, but they all have the shrinking factor in common.** For example, instead of the sample mean you could use the mean from a [prior distribution](https:\/\/www.statisticshowto.com\/prior-distribution\/) (*m*). In that case, ȳ could be replaced by *m*. The shrinking factor’s value is calculate after collecting the sample data and is given by the formula:  \n [![shrinking-factor james-stein estimator](https:\/\/www.statisticshowto.com\/wp-content\/uploads\/2016\/10\/shrinking-factor-james-stein-estimator.png)](https:\/\/www.statisticshowto.com\/wp-content\/uploads\/2016\/10\/shrinking-factor-james-stein-estimator.png)\n\nWhere:\n\n- x = individual values,\n- x̄ = sample mean,\n- k = number of unknown means (must be 2 or more),\n- σ2 = [variance](https:\/\/www.statisticshowto.com\/probability-and-statistics\/variance\/).\n\nThe shrinking factor’s value should be less than 1. For example, a value of .3 would shrink values by about 70 percent.\n\n## James-Stein Estimators vs. Sample Means\nThe James-Stein estimator is a significant departure from the “traditional” school of thought which states that the sample mean is the best estimator for the population mean. Stein and James proved that a better estimator than the “perfect” estimator exists, which seems to be somewhat of a paradox. However, the James-Stein estimator outperforms the sample mean when there are *several* unknown population means — not just one. The means do not have to be related, so they have to be carefully chosen. **Combining completely unrelated means will give you a result — but it will be a nonsensical one.** Bradley Efron and Carl Morris (1977) offer the extreme example of combining batting averages in baseball and proportions of imported cars; You can calculate a mean for these, but it will make no sense at all.\n\n## References\nEfron, B. and Morris, C. (1977), “Stein’s Paradox in Statistics.” Scientific American. 236 (5): 119–127  \n James, W., and Stein, C., (1961). “Estimation with Quadratic Loss.” Proceedings of the Fourth Berkeley Symposium, Vol. 1 (Berkeley, California: University of California Press), pp. 361-379.  \n Stein C. (1956). “Inadmissibility of the usual estimator for the mean of a [multivariate normal distribution](https:\/\/www.statisticshowto.com\/bivariate-normal-distribution\/)”. Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability. Vol. 1. University of California Press; Berkeley, CA, USA: pp. 197–208\n\n**Comments? Need to post a correction?** Please [***Contact Us***](https:\/\/www.statisticshowto.com\/contact\/).","meta_canonical":null}

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Boilerpipe Text	Estimators > James-Stein Estimator What is the James-Stein Estimator? It’s common in statistics to take averages to make predictions. For example, the sample mean (the average score from all samples) is used as an estimator for the population mean . James-Stein estimators improve upon these averages by shrinking them towards a more central average. The technique is named after Charles Stein and Willard James, who simplified Stein’s original 1956 method. Calculations The basic steps are: Calculate the sample mean (X̄). “Shrink” individual scores towards (X̄); Reduce larger values and increase smaller values. Each of these individual shrunk values is a James-Stein estimator, z. The basic formula for the James-Stein estimator is: z = x̄ + c(y – x̄) Where: (y – x̄) = difference between an individual score and the sample mean, c = a shrinking factor. Other formulas exist, but they all have the shrinking factor in common. For example, instead of the sample mean you could use the mean from a prior distribution ( m ). In that case, ȳ could be replaced by m . The shrinking factor’s value is calculate after collecting the sample data and is given by the formula: Where: x = individual values, x̄ = sample mean, k = number of unknown means (must be 2 or more), σ 2 = variance . The shrinking factor’s value should be less than 1. For example, a value of .3 would shrink values by about 70 percent. James-Stein Estimators vs. Sample Means The James-Stein estimator is a significant departure from the “traditional” school of thought which states that the sample mean is the best estimator for the population mean. Stein and James proved that a better estimator than the “perfect” estimator exists, which seems to be somewhat of a paradox. However, the James-Stein estimator outperforms the sample mean when there are several unknown population means — not just one. The means do not have to be related, so they have to be carefully chosen. Combining completely unrelated means will give you a result — but it will be a nonsensical one. Bradley Efron and Carl Morris (1977) offer the extreme example of combining batting averages in baseball and proportions of imported cars; You can calculate a mean for these, but it will make no sense at all. References Efron, B. and Morris, C. (1977), “Stein’s Paradox in Statistics.” Scientific American. 236 (5): 119–127 James, W., and Stein, C., (1961). “Estimation with Quadratic Loss.” Proceedings of the Fourth Berkeley Symposium, Vol. 1 (Berkeley, California: University of California Press), pp. 361-379. Stein C. (1956). “Inadmissibility of the usual estimator for the mean of a multivariate normal distribution ”. Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability. Vol. 1. University of California Press; Berkeley, CA, USA: pp. 197–208 Comments? Need to post a correction? Please Contact Us .
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It’s common in statistics to take averages to make predictions. For example, the [sample mean](https://www.statisticshowto.com/probability-and-statistics/statistics-definitions/sample-mean/) (the [average](https://www.statisticshowto.com/arithmetic-mean/) score from all samples) is used as an estimator for the [population mean](https://www.statisticshowto.com/population-mean/). James-Stein estimators improve upon these averages by [shrinking](https://www.statisticshowto.com/shrinkage-estimator/) them towards a more central average. The technique is named after Charles Stein and Willard James, who simplified Stein’s original 1956 method. ## Calculations The basic steps are: 1. Calculate the sample mean (X̄). 2. “Shrink” individual scores towards (X̄); Reduce larger values and increase smaller values. Each of these individual shrunk values is a James-Stein estimator, z. The basic formula for the James-Stein estimator is: z = x̄ + c(y – x̄) Where: - (y – x̄) = difference between an individual score and the sample mean, - c = a shrinking factor. Other formulas exist, but they all have the shrinking factor in common. For example, instead of the sample mean you could use the mean from a [prior distribution](https://www.statisticshowto.com/prior-distribution/) (m). In that case, ȳ could be replaced by m. The shrinking factor’s value is calculate after collecting the sample data and is given by the formula: [![shrinking-factor james-stein estimator](https://www.statisticshowto.com/wp-content/uploads/2016/10/shrinking-factor-james-stein-estimator.png)](https://www.statisticshowto.com/wp-content/uploads/2016/10/shrinking-factor-james-stein-estimator.png) Where: - x = individual values, - x̄ = sample mean, - k = number of unknown means (must be 2 or more), - σ2 = [variance](https://www.statisticshowto.com/probability-and-statistics/variance/). The shrinking factor’s value should be less than 1. For example, a value of .3 would shrink values by about 70 percent. ## James-Stein Estimators vs. Sample Means The James-Stein estimator is a significant departure from the “traditional” school of thought which states that the sample mean is the best estimator for the population mean. Stein and James proved that a better estimator than the “perfect” estimator exists, which seems to be somewhat of a paradox. However, the James-Stein estimator outperforms the sample mean when there are several unknown population means — not just one. The means do not have to be related, so they have to be carefully chosen. Combining completely unrelated means will give you a result — but it will be a nonsensical one. Bradley Efron and Carl Morris (1977) offer the extreme example of combining batting averages in baseball and proportions of imported cars; You can calculate a mean for these, but it will make no sense at all. ## References Efron, B. and Morris, C. (1977), “Stein’s Paradox in Statistics.” Scientific American. 236 (5): 119–127 James, W., and Stein, C., (1961). “Estimation with Quadratic Loss.” Proceedings of the Fourth Berkeley Symposium, Vol. 1 (Berkeley, California: University of California Press), pp. 361-379. Stein C. (1956). “Inadmissibility of the usual estimator for the mean of a [multivariate normal distribution](https://www.statisticshowto.com/bivariate-normal-distribution/)”. Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability. Vol. 1. University of California Press; Berkeley, CA, USA: pp. 197–208 [Deterministic: Definition and Examples](https://www.statisticshowto.com/deterministic/) [Shrinkage Estimator: Definition, Examples](https://www.statisticshowto.com/shrinkage-estimator/) Comments? Need to post a correction? Please [*Contact Us](https://www.statisticshowto.com/contact/). Check out our ["Practically Cheating Statistics Handbook](https://www.statisticshowto.com/?page_id=15743&preview=true"), which gives you hundreds of easy-to-follow answers in a convenient e-book. [![Statistics Handbook](https://www.statisticshowto.com/wp-content/uploads/2024/06/BookCoverSHT-1.jpg)](https://www.statisticshowto.com/?page_id=15743&preview=true) Readers are calling it suspenseful, haunting, and unforgettable. Check out Stephanie’s debut post-apocalyptic novel — available now on Amazon. [![an image of S.E. Glen's book Fallout in Georgia](https://www.statisticshowto.com/wp-content/uploads/2026/02/novel-ad.jpg)![an image of S.E. 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Readable Markdown	[Estimators](https://www.statisticshowto.com/estimator/) \> James-Stein Estimator ## What is the James-Stein Estimator? It’s common in statistics to take averages to make predictions. For example, the [sample mean](https://www.statisticshowto.com/probability-and-statistics/statistics-definitions/sample-mean/) (the [average](https://www.statisticshowto.com/arithmetic-mean/) score from all samples) is used as an estimator for the [population mean](https://www.statisticshowto.com/population-mean/). James-Stein estimators improve upon these averages by [shrinking](https://www.statisticshowto.com/shrinkage-estimator/) them towards a more central average. The technique is named after Charles Stein and Willard James, who simplified Stein’s original 1956 method. ## Calculations The basic steps are: 1. Calculate the sample mean (X̄). 2. “Shrink” individual scores towards (X̄); Reduce larger values and increase smaller values. Each of these individual shrunk values is a James-Stein estimator, z. The basic formula for the James-Stein estimator is: z = x̄ + c(y – x̄) Where: - (y – x̄) = difference between an individual score and the sample mean, - c = a shrinking factor. Other formulas exist, but they all have the shrinking factor in common. For example, instead of the sample mean you could use the mean from a [prior distribution](https://www.statisticshowto.com/prior-distribution/) (m). In that case, ȳ could be replaced by m. The shrinking factor’s value is calculate after collecting the sample data and is given by the formula: [![shrinking-factor james-stein estimator](https://www.statisticshowto.com/wp-content/uploads/2016/10/shrinking-factor-james-stein-estimator.png)](https://www.statisticshowto.com/wp-content/uploads/2016/10/shrinking-factor-james-stein-estimator.png) Where: - x = individual values, - x̄ = sample mean, - k = number of unknown means (must be 2 or more), - σ2 = [variance](https://www.statisticshowto.com/probability-and-statistics/variance/). The shrinking factor’s value should be less than 1. For example, a value of .3 would shrink values by about 70 percent. ## James-Stein Estimators vs. Sample Means The James-Stein estimator is a significant departure from the “traditional” school of thought which states that the sample mean is the best estimator for the population mean. Stein and James proved that a better estimator than the “perfect” estimator exists, which seems to be somewhat of a paradox. However, the James-Stein estimator outperforms the sample mean when there are several unknown population means — not just one. The means do not have to be related, so they have to be carefully chosen. Combining completely unrelated means will give you a result — but it will be a nonsensical one. Bradley Efron and Carl Morris (1977) offer the extreme example of combining batting averages in baseball and proportions of imported cars; You can calculate a mean for these, but it will make no sense at all. ## References Efron, B. and Morris, C. (1977), “Stein’s Paradox in Statistics.” Scientific American. 236 (5): 119–127 James, W., and Stein, C., (1961). “Estimation with Quadratic Loss.” Proceedings of the Fourth Berkeley Symposium, Vol. 1 (Berkeley, California: University of California Press), pp. 361-379. Stein C. (1956). “Inadmissibility of the usual estimator for the mean of a [multivariate normal distribution](https://www.statisticshowto.com/bivariate-normal-distribution/)”. Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability. Vol. 1. University of California Press; Berkeley, CA, USA: pp. 197–208 Comments? Need to post a correction? Please [*Contact Us*](https://www.statisticshowto.com/contact/).
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