How to calculate confidence interval without standard deviation in Excel

{"appState":{"pageLoadApiCallsStatus":true},"articleState":{"article":{"headers":{"creationTime":"2016-03-26T15:35:46+00:00","modifiedTime":"2021-07-12T14:10:21+00:00","timestamp":"2022-06-22T19:36:37+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"//dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"//dummies-api.dummies.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Statistics","_links":{"self":"//dummies-api.dummies.com/v2/categories/33728"},"slug":"statistics","categoryId":33728}],"title":"How to Calculate a Confidence Interval for a Population Mean with Unknown Standard Deviation or Small Sample","strippedTitle":"how to calculate a confidence interval for a population mean with unknown standard deviation or small sample","slug":"how-to-calculate-a-confidence-interval-for-a-population-mean-with-unknown-standard-deviation-andor-small-sample-size","canonicalUrl":"","seo":{"metaDescription":"Here's how to calculate a confidence interval (CI) for the average of a population, even if the standard deviation is unknown or the sample is small.","noIndex":0,"noFollow":0},"content":"You can calculate a confidence interval (CI) for the mean, or average, of a population even if the standard deviation is unknown or the sample size is small. When a statistical characteristic that’s being measured (such as income, IQ, price, height, quantity, or weight) is <i>numerical,</i> most people want to estimate the mean (average) value for the population. You estimate the population mean,\r\n\r\n<img src=\"//sg.cdnki.com/how-to-calculate-confidence-interval-without-standard-deviation-in-excel---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzM2MTAzMi5pbWFnZTAucG5n.webp\" alt=\"image0.png\" width=\"17\" height=\"17\" />\r\n\r\nby using a sample mean,\r\n\r\n<img src=\"//sg.cdnki.com/how-to-calculate-confidence-interval-without-standard-deviation-in-excel---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzM2MTAzMy5pbWFnZTEucG5n.webp\" alt=\"image1.png\" width=\"16\" height=\"23\" />\r\n\r\nplus or minus a margin of error. The result is called a <i>confidence interval for the population mean, </i>\r\n\r\n<img src=\"//sg.cdnki.com/how-to-calculate-confidence-interval-without-standard-deviation-in-excel---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzM2MTAzNC5pbWFnZTIucG5n.webp\" alt=\"image2.png\" width=\"17\" height=\"17\" />\r\n\r\nIn many situations, you don’t know\r\n\r\n<img src=\"//sg.cdnki.com/how-to-calculate-confidence-interval-without-standard-deviation-in-excel---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzM2MTAzNS5pbWFnZTMucG5n.webp\" alt=\"image3.png\" width=\"17\" height=\"16\" />\r\n\r\nso you estimate it with the sample standard deviation, <i>s. </i>But if the sample size is small (less than 30), and you can’t be sure your data came from a normal distribution. (In the latter case, the Central Limit Theorem can’t be used.) In either situation, you can’t use a <i>z*-</i>value from the standard normal (<i>Z</i>-) distribution as your critical value anymore; you have to use a larger critical value than that, because of not knowing what\r\n\r\n<img src=\"//sg.cdnki.com/how-to-calculate-confidence-interval-without-standard-deviation-in-excel---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzM2MTAzNi5pbWFnZTQucG5n.webp\" alt=\"image4.png\" width=\"15\" height=\"15\" />\r\n\r\nis and/or having less data.\r\n\r\nThe formula for a confidence interval for one population mean in this case is\r\n\r\n<img src=\"//sg.cdnki.com/how-to-calculate-confidence-interval-without-standard-deviation-in-excel---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzM2MTAzNy5pbWFnZTUucG5n.webp\" alt=\"image5.png\" width=\"168\" height=\"40\" />\r\n\r\nis the critical <i>t*</i>-value from the <i>t</i>-distribution with <i>n </i>– 1 degrees of freedom (where <i>n</i> is the sample size).\r\n<h2 id=\"tab1\" >The t-table</h2>\r\n<img class=\"alignnone size-full wp-image-287045\" src=\"//sg.cdnki.com/how-to-calculate-confidence-interval-without-standard-deviation-in-excel---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzL3QtdGFibGUucG5n.webp\" alt=\"t-table\" width=\"710\" height=\"1056\" />\r\n\r\nThe <i>t*-</i>values for common confidence levels are found using the last row of the <i>t-</i>table above.\r\n<p class=\"Warning\">The <i>t</i>-distribution has a shape similar to the <i>Z</i>-distribution except it’s flatter and more spread out. For small values of <i>n</i> and a specific confidence level, the critical values on the <i>t</i>-distribution are larger than on the <i>Z-</i>distribution, so when you use the critical values from the<i> t</i>-distribution, the margin of error for your confidence interval will be wider. As the values of <i>n</i> get larger, the <i>t*</i>-values are closer to <i>z*</i>-values.</p>\r\nTo calculate a CI for the population mean (average), under these conditions, do the following:\r\n<ol class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\">Determine the confidence level and degrees of freedom and then find the appropriate <i>t*</i>-value.</p>\r\n<p class=\"child-para\">Refer to the preceding t-table.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Find the sample mean</p>\r\n<img src=\"//sg.cdnki.com/how-to-calculate-confidence-interval-without-standard-deviation-in-excel---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzM2MTAzOS5pbWFnZTcucG5n.webp\" alt=\"image7.png\" width=\"24\" height=\"29\" />\r\n<p class=\"child-para\">and the sample standard deviation (<i>s</i>) for the sample.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Multiply <i>t*</i> times <i>s</i> and divide that by the square root of <i>n</i>.</p>\r\n<p class=\"child-para\">This calculation gives you the margin of error.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Take</p>\r\n<img src=\"//sg.cdnki.com/how-to-calculate-confidence-interval-without-standard-deviation-in-excel---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzM2MTA0MC5pbWFnZTgucG5n.webp\" alt=\"image8.png\" width=\"13\" height=\"20\" />\r\n<p class=\"child-para\">plus or minus the margin of error to obtain the CI.</p>\r\n<p class=\"child-para\">The lower end of the CI is</p>\r\n<img src=\"//sg.cdnki.com/how-to-calculate-confidence-interval-without-standard-deviation-in-excel---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzM2MTA0MS5pbWFnZTkucG5n.webp\" alt=\"image9.png\" width=\"13\" height=\"20\" />\r\n<p class=\"child-para\">minus the margin of error, whereas the upper end of the CI is</p>\r\n<img src=\"//sg.cdnki.com/how-to-calculate-confidence-interval-without-standard-deviation-in-excel---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzM2MTA0Mi5pbWFnZTEwLnBuZw==.webp\" alt=\"image10.png\" width=\"13\" height=\"20\" />\r\n<p class=\"child-para\">plus the margin of error.</p>\r\n</li>\r\n</ol>\r\n<h2 id=\"tab2\" >Here's an example of how this works</h2>\r\nFor example, suppose you work for the Department of Natural Resources and you want to estimate, with 95 percent confidence, the mean (average) length of all walleye fingerlings in a fish hatchery pond. You take a random sample of 10 fingerlings and determine that the average length is 7.5 inches and the sample standard deviation is 2.3 inches.\r\n<ol class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\">Because you want a 95 percent confidence interval, you determine your <i>t*</i>-value as follows:</p>\r\n<p class=\"child-para\">The <i>t*</i>-value comes from a <i>t</i>-distribution with 10 – 1 = 9 degrees of freedom. This <i>t*-</i>value is found by looking at the <i>t</i>-table. Look in the last row where the confidence levels are located, and find the confidence level of 95 percent; this marks the column you need. Then find the row corresponding to <i>df</i> = 9. Intersect the row and column, and you find <i>t*</i> = 2.262. This is the <i>t*-</i>value for a 95 percent confidence interval for the mean with a sample size of 10. (Notice this is larger than the <i>z</i>*-value, which would be 1.96 for the same confidence interval.)</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">You know that the average length is 7.5 inches, the sample standard deviation is 2.3 inches, and the sample size is 10. This means</p>\r\n<img src=\"//sg.cdnki.com/how-to-calculate-confidence-interval-without-standard-deviation-in-excel---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzM2MTA0My5pbWFnZTExLnBuZw==.webp\" alt=\"image11.png\" width=\"179\" height=\"23\" /></li>\r\n \t<li>\r\n<p class=\"first-para\">Multiply 2.262 times 2.3 divided by the square root of 10. The margin of error is, therefore,</p>\r\n<img src=\"//sg.cdnki.com/how-to-calculate-confidence-interval-without-standard-deviation-in-excel---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzM2MTA0NC5pbWFnZTEyLnBuZw==.webp\" alt=\"image12.png\" width=\"161\" height=\"45\" /></li>\r\n \t<li>\r\n<p class=\"first-para\">Your 95 percent confidence interval for the mean length of all walleye fingerlings in this fish hatchery pond is</p>\r\n<img src=\"//sg.cdnki.com/how-to-calculate-confidence-interval-without-standard-deviation-in-excel---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzM2MTA0NS5pbWFnZTEzLnBuZw==.webp\" alt=\"image13.png\" width=\"236\" height=\"19\" />\r\n<p class=\"child-para\">(The lower end of the interval is 7.5 – 1.645 = 5.86 inches; the upper end is 7.5 + 1.645 = 9.15 inches.)</p>\r\n</li>\r\n</ol>\r\nNotice this confidence interval is wider than it would be for a large sample size. In addition to having a larger critical value (<i>t*</i> versus <i>z*</i>), the smaller sample size increases the margin of error, because <i>n</i> is in its denominator.\r\n\r\nWith a smaller sample size, you don’t have as much information to “guess” at the population mean. Hence keeping with 95 percent confidence, you need a wider interval than you would have needed with a larger sample size in order to be 95 percent confident that the population mean falls in your interval.\r\n<h2 id=\"tab3\" >Now, say it in a way others can understand</h2>\r\n<p class=\"Remember\">After you calculate a confidence interval, make sure you always interpret it in words a non-statistician would understand. That is, talk about the results in terms of what the person in the problem is trying to find out — statisticians call this interpreting the results “in the context of the problem.”</p>\r\n<p class=\"Remember\">In this example you can say: “With 95 percent confidence, the average length of walleye fingerlings in this entire fish hatchery pond is between 5.86 and 9.15 inches, based on my sample data.” (Always be sure to include appropriate units.)</p>","description":"You can calculate a confidence interval (CI) for the mean, or average, of a population even if the standard deviation is unknown or the sample size is small. When a statistical characteristic that’s being measured (such as income, IQ, price, height, quantity, or weight) is <i>numerical,</i> most people want to estimate the mean (average) value for the population. You estimate the population mean,\r\n\r\n<img src=\"//www.dummies.com/wp-content/uploads/361032.image0.png\" alt=\"image0.png\" width=\"17\" height=\"17\" />\r\n\r\nby using a sample mean,\r\n\r\n<img src=\"//www.dummies.com/wp-content/uploads/361033.image1.png\" alt=\"image1.png\" width=\"16\" height=\"23\" />\r\n\r\nplus or minus a margin of error. The result is called a <i>confidence interval for the population mean, </i>\r\n\r\n<img src=\"//www.dummies.com/wp-content/uploads/361034.image2.png\" alt=\"image2.png\" width=\"17\" height=\"17\" />\r\n\r\nIn many situations, you don’t know\r\n\r\n<img src=\"//www.dummies.com/wp-content/uploads/361035.image3.png\" alt=\"image3.png\" width=\"17\" height=\"16\" />\r\n\r\nso you estimate it with the sample standard deviation, <i>s. </i>But if the sample size is small (less than 30), and you can’t be sure your data came from a normal distribution. (In the latter case, the Central Limit Theorem can’t be used.) In either situation, you can’t use a <i>z*-</i>value from the standard normal (<i>Z</i>-) distribution as your critical value anymore; you have to use a larger critical value than that, because of not knowing what\r\n\r\n<img src=\"//www.dummies.com/wp-content/uploads/361036.image4.png\" alt=\"image4.png\" width=\"15\" height=\"15\" />\r\n\r\nis and/or having less data.\r\n\r\nThe formula for a confidence interval for one population mean in this case is\r\n\r\n<img src=\"//www.dummies.com/wp-content/uploads/361037.image5.png\" alt=\"image5.png\" width=\"168\" height=\"40\" />\r\n\r\nis the critical <i>t*</i>-value from the <i>t</i>-distribution with <i>n </i>– 1 degrees of freedom (where <i>n</i> is the sample size).\r\n<h2 id=\"tab1\" >The t-table</h2>\r\n<img class=\"alignnone size-full wp-image-287045\" src=\"//www.dummies.com/wp-content/uploads/t-table.png\" alt=\"t-table\" width=\"710\" height=\"1056\" />\r\n\r\nThe <i>t*-</i>values for common confidence levels are found using the last row of the <i>t-</i>table above.\r\n<p class=\"Warning\">The <i>t</i>-distribution has a shape similar to the <i>Z</i>-distribution except it’s flatter and more spread out. For small values of <i>n</i> and a specific confidence level, the critical values on the <i>t</i>-distribution are larger than on the <i>Z-</i>distribution, so when you use the critical values from the<i> t</i>-distribution, the margin of error for your confidence interval will be wider. As the values of <i>n</i> get larger, the <i>t*</i>-values are closer to <i>z*</i>-values.</p>\r\nTo calculate a CI for the population mean (average), under these conditions, do the following:\r\n<ol class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\">Determine the confidence level and degrees of freedom and then find the appropriate <i>t*</i>-value.</p>\r\n<p class=\"child-para\">Refer to the preceding t-table.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Find the sample mean</p>\r\n<img src=\"//www.dummies.com/wp-content/uploads/361039.image7.png\" alt=\"image7.png\" width=\"24\" height=\"29\" />\r\n<p class=\"child-para\">and the sample standard deviation (<i>s</i>) for the sample.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Multiply <i>t*</i> times <i>s</i> and divide that by the square root of <i>n</i>.</p>\r\n<p class=\"child-para\">This calculation gives you the margin of error.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Take</p>\r\n<img src=\"//www.dummies.com/wp-content/uploads/361040.image8.png\" alt=\"image8.png\" width=\"13\" height=\"20\" />\r\n<p class=\"child-para\">plus or minus the margin of error to obtain the CI.</p>\r\n<p class=\"child-para\">The lower end of the CI is</p>\r\n<img src=\"//www.dummies.com/wp-content/uploads/361041.image9.png\" alt=\"image9.png\" width=\"13\" height=\"20\" />\r\n<p class=\"child-para\">minus the margin of error, whereas the upper end of the CI is</p>\r\n<img src=\"//www.dummies.com/wp-content/uploads/361042.image10.png\" alt=\"image10.png\" width=\"13\" height=\"20\" />\r\n<p class=\"child-para\">plus the margin of error.</p>\r\n</li>\r\n</ol>\r\n<h2 id=\"tab2\" >Here's an example of how this works</h2>\r\nFor example, suppose you work for the Department of Natural Resources and you want to estimate, with 95 percent confidence, the mean (average) length of all walleye fingerlings in a fish hatchery pond. You take a random sample of 10 fingerlings and determine that the average length is 7.5 inches and the sample standard deviation is 2.3 inches.\r\n<ol class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\">Because you want a 95 percent confidence interval, you determine your <i>t*</i>-value as follows:</p>\r\n<p class=\"child-para\">The <i>t*</i>-value comes from a <i>t</i>-distribution with 10 – 1 = 9 degrees of freedom. This <i>t*-</i>value is found by looking at the <i>t</i>-table. Look in the last row where the confidence levels are located, and find the confidence level of 95 percent; this marks the column you need. Then find the row corresponding to <i>df</i> = 9. Intersect the row and column, and you find <i>t*</i> = 2.262. This is the <i>t*-</i>value for a 95 percent confidence interval for the mean with a sample size of 10. (Notice this is larger than the <i>z</i>*-value, which would be 1.96 for the same confidence interval.)</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">You know that the average length is 7.5 inches, the sample standard deviation is 2.3 inches, and the sample size is 10. This means</p>\r\n<img src=\"//www.dummies.com/wp-content/uploads/361043.image11.png\" alt=\"image11.png\" width=\"179\" height=\"23\" /></li>\r\n \t<li>\r\n<p class=\"first-para\">Multiply 2.262 times 2.3 divided by the square root of 10. The margin of error is, therefore,</p>\r\n<img src=\"//www.dummies.com/wp-content/uploads/361044.image12.png\" alt=\"image12.png\" width=\"161\" height=\"45\" /></li>\r\n \t<li>\r\n<p class=\"first-para\">Your 95 percent confidence interval for the mean length of all walleye fingerlings in this fish hatchery pond is</p>\r\n<img src=\"//www.dummies.com/wp-content/uploads/361045.image13.png\" alt=\"image13.png\" width=\"236\" height=\"19\" />\r\n<p class=\"child-para\">(The lower end of the interval is 7.5 – 1.645 = 5.86 inches; the upper end is 7.5 + 1.645 = 9.15 inches.)</p>\r\n</li>\r\n</ol>\r\nNotice this confidence interval is wider than it would be for a large sample size. In addition to having a larger critical value (<i>t*</i> versus <i>z*</i>), the smaller sample size increases the margin of error, because <i>n</i> is in its denominator.\r\n\r\nWith a smaller sample size, you don’t have as much information to “guess” at the population mean. Hence keeping with 95 percent confidence, you need a wider interval than you would have needed with a larger sample size in order to be 95 percent confident that the population mean falls in your interval.\r\n<h2 id=\"tab3\" >Now, say it in a way others can understand</h2>\r\n<p class=\"Remember\">After you calculate a confidence interval, make sure you always interpret it in words a non-statistician would understand. That is, talk about the results in terms of what the person in the problem is trying to find out — statisticians call this interpreting the results “in the context of the problem.”</p>\r\n<p class=\"Remember\">In this example you can say: “With 95 percent confidence, the average length of walleye fingerlings in this entire fish hatchery pond is between 5.86 and 9.15 inches, based on my sample data.” (Always be sure to include appropriate units.)</p>","blurb":"","authors":[{"authorId":9121,"name":"Deborah J. Rumsey","slug":"deborah-j-rumsey","description":"Deborah Rumsey, PhD, is an auxiliary faculty member and program specialist in department of statistics at The Ohio State University. An author of several Dummies books, she is a fellow of the American Statistical Association.","_links":{"self":"//dummies-api.dummies.com/v2/authors/9121"}}],"primaryCategoryTaxonomy":{"categoryId":33728,"title":"Statistics","slug":"statistics","_links":{"self":"//dummies-api.dummies.com/v2/categories/33728"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[{"label":"The t-table","target":"#tab1"},{"label":"Here's an example of how this works","target":"#tab2"},{"label":"Now, say it in a way others can understand","target":"#tab3"}],"relatedArticles":{"fromBook":[{"articleId":208650,"title":"Statistics For Dummies Cheat Sheet","slug":"statistics-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"//dummies-api.dummies.com/v2/articles/208650"}},{"articleId":188342,"title":"Checking Out Statistical Confidence Interval Critical Values","slug":"checking-out-statistical-confidence-interval-critical-values","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"//dummies-api.dummies.com/v2/articles/188342"}},{"articleId":188341,"title":"Handling Statistical Hypothesis Tests","slug":"handling-statistical-hypothesis-tests","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"//dummies-api.dummies.com/v2/articles/188341"}},{"articleId":188343,"title":"Statistically Figuring Sample Size","slug":"statistically-figuring-sample-size","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"//dummies-api.dummies.com/v2/articles/188343"}},{"articleId":188336,"title":"Surveying Statistical Confidence Intervals","slug":"surveying-statistical-confidence-intervals","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"//dummies-api.dummies.com/v2/articles/188336"}}],"fromCategory":[{"articleId":263501,"title":"10 Steps to a Better Math Grade with Statistics","slug":"10-steps-to-a-better-math-grade-with-statistics","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"//dummies-api.dummies.com/v2/articles/263501"}},{"articleId":263495,"title":"Statistics and Histograms","slug":"statistics-and-histograms","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"//dummies-api.dummies.com/v2/articles/263495"}},{"articleId":263492,"title":"What is Categorical Data and How is It Summarized?","slug":"what-is-categorical-data-and-how-is-it-summarized","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"//dummies-api.dummies.com/v2/articles/263492"}},{"articleId":209320,"title":"Statistics II For Dummies Cheat Sheet","slug":"statistics-ii-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"//dummies-api.dummies.com/v2/articles/209320"}},{"articleId":209293,"title":"SPSS For Dummies Cheat Sheet","slug":"spss-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"//dummies-api.dummies.com/v2/articles/209293"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":282603,"slug":"statistics-for-dummies-2nd-edition","isbn":"9781119293521","categoryList":["academics-the-arts","math","statistics"],"amazon":{"default":"//www.amazon.com/gp/product/1119293529/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"//www.amazon.ca/gp/product/1119293529/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"//www.tkqlhce.com/click-9208661-13710633?url=//www.chapters.indigo.ca/en-ca/books/product/1119293529-item.html&cjsku=978111945484","gb":"//www.amazon.co.uk/gp/product/1119293529/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"//www.amazon.de/gp/product/1119293529/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"//www.dummies.com/wp-content/uploads/statistics-for-dummies-2nd-edition-cover-9781119293521-203x255.jpg","width":203,"height":255},"title":"Statistics For Dummies","testBankPinActivationLink":"","bookOutOfPrint":true,"authorsInfo":"<p><p><b><b data-author-id=\"34805\">Deborah J. Rumsey</b>, PhD,</b> is an Auxiliary Professor and Statistics Education Specialist at The Ohio State University. She is the author of <i>Statistics For Dummies, Statistics II For Dummies, Statistics Workbook For Dummies,</i> and <i>Probability For Dummies.</i></p>","authors":[{"authorId":34805,"name":"Deborah J. Rumsey","slug":"deborah-j.-rumsey","description":" <p><b>Deborah J. Rumsey, PhD,</b> is an Auxiliary Professor and Statistics Education Specialist at The Ohio State University. She is the author of <i>Statistics For Dummies, Statistics II For Dummies, Statistics Workbook For Dummies,</i> and <i>Probability For Dummies.</i> ","_links":{"self":"//dummies-api.dummies.com/v2/authors/34805"}}],"_links":{"self":"//dummies-api.dummies.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;statistics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119293521&quot;]}]\" id=\"du-slot-62b36f4561c91\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;statistics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119293521&quot;]}]\" id=\"du-slot-62b36f4562415\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0},"sponsorAd":null,"sponsorEbookTitle":null,"sponsorEbookLink":null,"sponsorEbookImage":null},"primaryLearningPath":"Advance","lifeExpectancy":"Two years","lifeExpectancySetFrom":"2021-07-08T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":169357},"articleLoadedStatus":"success"},"listState":{"list":{},"objectTitle":"","status":"initial","pageType":null,"objectId":null,"page":1,"sortField":"time","sortOrder":1,"categoriesIds":[],"articleTypes":[],"filterData":{},"filterDataLoadedStatus":"initial","pageSize":10},"adsState":{"pageScripts":{"headers":{"timestamp":"2022-08-23T12:59:06+00:00"},"adsId":0,"data":{"scripts":[{"pages":["all"],"location":"header","script":"<!--Optimizely Script-->\r\n<script src=\"//cdn.optimizely.com/js/10563184655.js\"></script>","enabled":false},{"pages":["all"],"location":"header","script":"<!-- comScore Tag -->\r\n<script>var _comscore = _comscore || [];_comscore.push({ c1: \"2\", c2: \"15097263\" });(function() {var s = document.createElement(\"script\"), el = document.getElementsByTagName(\"script\")[0]; s.async = true;s.src = (document.location.protocol == \"\" ? \"//sb\" : \"//b\") + \".scorecardresearch.com/beacon.js\";el.parentNode.insertBefore(s, el);})();</script><noscript><img src=\"//sb.scorecardresearch.com/p?c1=2&c2=15097263&cv=2.0&cj=1\" /></noscript>\r\n<!-- / comScore Tag -->","enabled":true},{"pages":["all"],"location":"footer","script":"<!--BEGIN QUALTRICS WEBSITE FEEDBACK SNIPPET-->\r\n<script type='text/javascript'>\r\n(function(){var g=function(e,h,f,g){\r\nthis.get=function(a){for(var a=a+\"=\",c=document.cookie.split(\";\"),b=0,e=c.length;b<e;b++){for(var d=c[b];\" \"==d.charAt(0);)d=d.substring(1,d.length);if(0==d.indexOf(a))return d.substring(a.length,d.length)}return null};\r\nthis.set=function(a,c){var b=\"\",b=new Date;b.setTime(b.getTime()+6048E5);b=\"; expires=\"+b.toGMTString();document.cookie=a+\"=\"+c+b+\"; path=/; \"};\r\nthis.check=function(){var a=this.get(f);if(a)a=a.split(\":\");else if(100!=e)\"v\"==h&&(e=Math.random()>=e/100?0:100),a=[h,e,0],this.set(f,a.join(\":\"));else return!0;var c=a[1];if(100==c)return!0;switch(a[0]){case \"v\":return!1;case \"r\":return c=a[2]%Math.floor(100/c),a[2]++,this.set(f,a.join(\":\")),!c}return!0};\r\nthis.go=function(){if(this.check()){var a=document.createElement(\"script\");a.type=\"text/javascript\";a.src=g;document.body&&document.body.appendChild(a)}};\r\nthis.start=function(){var t=this;\"complete\"!==document.readyState?window.addEventListener?window.addEventListener(\"load\",function(){t.go()},!1):window.attachEvent&&window.attachEvent(\"onload\",function(){t.go()}):t.go()};};\r\ntry{(new g(100,\"r\",\"QSI_S_ZN_5o5yqpvMVjgDOuN\",\"//zn5o5yqpvmvjgdoun-wiley.siteintercept.qualtrics.com/SIE/?Q_ZID=ZN_5o5yqpvMVjgDOuN\")).start()}catch(i){}})();\r\n</script><div id='ZN_5o5yqpvMVjgDOuN'><!--DO NOT REMOVE-CONTENTS PLACED HERE--></div>\r\n<!--END WEBSITE FEEDBACK SNIPPET-->","enabled":false},{"pages":["all"],"location":"header","script":"<!-- Hotjar Tracking Code for //www.dummies.com -->\r\n<script>\r\n (function(h,o,t,j,a,r){\r\n h.hj=h.hj||function(){(h.hj.q=h.hj.q||[]).push(arguments)};\r\n h._hjSettings={hjid:257151,hjsv:6};\r\n a=o.getElementsByTagName('head')[0];\r\n r=o.createElement('script');r.async=1;\r\n r.src=t+h._hjSettings.hjid+j+h._hjSettings.hjsv;\r\n a.appendChild(r);\r\n })(window,document,'//static.hotjar.com/c/hotjar-','.js?sv=');\r\n</script>","enabled":false},{"pages":["article"],"location":"header","script":"<!-- //Connect Container: dummies --> <script src=\"//get.s-onetag.com/bffe21a1-6bb8-4928-9449-7beadb468dae/tag.min.js\" async defer></script>","enabled":true},{"pages":["homepage"],"location":"header","script":"<meta name=\"facebook-domain-verification\" content=\"irk8y0irxf718trg3uwwuexg6xpva0\" />","enabled":true},{"pages":["homepage","article","category","search"],"location":"footer","script":"<!-- Facebook Pixel Code -->\r\n<noscript>\r\n<img height=\"1\" width=\"1\" src=\"//www.facebook.com/tr?id=256338321977984&ev=PageView&noscript=1\"/>\r\n</noscript>\r\n<!-- End Facebook Pixel Code -->","enabled":true}]}},"pageScriptsLoadedStatus":"success"},"navigationState":{"navigationCollections":[{"collectionId":287568,"title":"BYOB (Be Your Own Boss)","hasSubCategories":false,"url":"/collection/for-the-entry-level-entrepreneur-287568"},{"collectionId":293237,"title":"Be a Rad Dad","hasSubCategories":false,"url":"/collection/be-the-best-dad-293237"},{"collectionId":294090,"title":"Contemplating the Cosmos","hasSubCategories":false,"url":"/collection/theres-something-about-space-294090"},{"collectionId":287563,"title":"For Those Seeking Peace of Mind","hasSubCategories":false,"url":"/collection/for-those-seeking-peace-of-mind-287563"},{"collectionId":287570,"title":"For the Aspiring Aficionado","hasSubCategories":false,"url":"/collection/for-the-bougielicious-287570"},{"collectionId":291903,"title":"For the Budding Cannabis Enthusiast","hasSubCategories":false,"url":"/collection/for-the-budding-cannabis-enthusiast-291903"},{"collectionId":291934,"title":"For the Exam-Season Crammer","hasSubCategories":false,"url":"/collection/for-the-exam-season-crammer-291934"},{"collectionId":287569,"title":"For the Hopeless Romantic","hasSubCategories":false,"url":"/collection/for-the-hopeless-romantic-287569"},{"collectionId":287567,"title":"For the Unabashed Hippie","hasSubCategories":false,"url":"/collection/for-the-unabashed-hippie-287567"},{"collectionId":292186,"title":"Just DIY It","hasSubCategories":false,"url":"/collection/just-diy-it-292186"}],"navigationCollectionsLoadedStatus":"success","navigationCategories":{"books":{"0":{"data":[{"categoryId":33512,"title":"Technology","hasSubCategories":true,"url":"/category/books/technology-33512"},{"categoryId":33662,"title":"Academics & The Arts","hasSubCategories":true,"url":"/category/books/academics-the-arts-33662"},{"categoryId":33809,"title":"Home, Auto, & Hobbies","hasSubCategories":true,"url":"/category/books/home-auto-hobbies-33809"},{"categoryId":34038,"title":"Body, Mind, & Spirit","hasSubCategories":true,"url":"/category/books/body-mind-spirit-34038"},{"categoryId":34224,"title":"Business, Careers, & Money","hasSubCategories":true,"url":"/category/books/business-careers-money-34224"}],"breadcrumbs":[],"categoryTitle":"Level 0 Category","mainCategoryUrl":"/category/books/level-0-category-0"}},"articles":{"0":{"data":[{"categoryId":33512,"title":"Technology","hasSubCategories":true,"url":"/category/articles/technology-33512"},{"categoryId":33662,"title":"Academics & The Arts","hasSubCategories":true,"url":"/category/articles/academics-the-arts-33662"},{"categoryId":33809,"title":"Home, Auto, & Hobbies","hasSubCategories":true,"url":"/category/articles/home-auto-hobbies-33809"},{"categoryId":34038,"title":"Body, Mind, & Spirit","hasSubCategories":true,"url":"/category/articles/body-mind-spirit-34038"},{"categoryId":34224,"title":"Business, Careers, & Money","hasSubCategories":true,"url":"/category/articles/business-careers-money-34224"}],"breadcrumbs":[],"categoryTitle":"Level 0 Category","mainCategoryUrl":"/category/articles/level-0-category-0"}}},"navigationCategoriesLoadedStatus":"success"},"searchState":{"searchList":[],"searchStatus":"initial","relatedArticlesList":[],"relatedArticlesStatus":"initial"},"routeState":{"name":"Article3","path":"/article/academics-the-arts/math/statistics/how-to-calculate-a-confidence-interval-for-a-population-mean-with-unknown-standard-deviation-andor-small-sample-size-169357/","hash":"","query":{},"params":{"category1":"academics-the-arts","category2":"math","category3":"statistics","article":"how-to-calculate-a-confidence-interval-for-a-population-mean-with-unknown-standard-deviation-andor-small-sample-size-169357"},"fullPath":"/article/academics-the-arts/math/statistics/how-to-calculate-a-confidence-interval-for-a-population-mean-with-unknown-standard-deviation-andor-small-sample-size-169357/","meta":{"routeType":"article","breadcrumbInfo":{"suffix":"Articles","baseRoute":"/category/articles"},"prerenderWithAsyncData":true},"from":{"name":null,"path":"/","hash":"","query":{},"params":{},"fullPath":"/","meta":{}}},"dropsState":{"submitEmailResponse":false,"status":"initial"},"sfmcState":{"status":"initial"},"profileState":{"auth":{},"userOptions":{},"status":"initial"}}