{"id":65468,"date":"2026-09-29T12:26:52","date_gmt":"2026-09-29T09:26:52","guid":{"rendered":"https:\/\/wpcalc.com\/en\/?p=65468"},"modified":"2026-09-29T15:19:11","modified_gmt":"2026-09-29T12:19:11","slug":"confidence-interval-calculator","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/statistics\/confidence-interval-calculator\/","title":{"rendered":"Confidence Interval Calculator"},"content":{"rendered":"<section class=\"not-prose calculator-box\" aria-labelledby=\"calculator-title\"><div class=\"flex items-baseline sm:items-center justify-between gap-2 sm:gap-3\"><div class=\"flex flex-col sm:flex-row sm:items-center gap-2\"><span class=\"icon icon-calculator text-primary-700! text-base! dark:text-primary-300!\" aria-hidden=\"true\"><\/span><h2 id=\"calculator-title\" class=\"text-lg font-display font-bold\">Compute a Confidence Interval<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"65468\" data-favorite-title=\"Confidence Interval\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/statistics\/confidence-interval-calculator\/\" data-favorite-excerpt=\"Compute a confidence interval for a population mean or proportion. Enter your sample statistics, pick a confidence level, and get the margin of error and the lower and upper limits.\" aria-label=\"Add to Favorites\" data-favorite-icon=\"icon icon-wpcalc-sample-size\">\n    <span class=\"icon icon-shape-star-empty\" aria-hidden=\"true\"><\/span>\n  <\/button>\n  <div class=\"calcdash-favorite__tip\" aria-hidden=\"true\">\n   <span class=\"favorite-tooltip\" id=\"favorite-tooltip\"><\/span>\n  <\/div>\n  <span class=\"sr-only\" role=\"status\" id=\"favorite-status\"><\/span>\n<\/div><\/div><form action=\"https:\/\/wpcalc.com\/en\/statistics\/confidence-interval-calculator\/\" method=\"POST\" class=\"calculator\" id=\"calculator-65468\" data-post=\"65468\"><fieldset class=\"fieldset-input is-1\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field\" id=\"input-1\"><label for=\"field-1\">Estimate type<\/label><div class=\"grid grid-cols-1\"><select id=\"field-1\" name=\"estimate_type\"><option value=\"mean\" selected>Sample mean<\/option><option value=\"prop\">Sample proportion<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><div class=\"field has-term\" id=\"input-2\"><label for=\"field-2\">Sample mean (x\u0304)<\/label><div class=\"term\">x\u0304<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"sample_mean_x\" id=\"field-2\" aria-describedby=\"calc-info-2\" placeholder=\"e.g. 100\" step=\"any\" value=\"100\"\/><small id=\"calc-info-2\">For a proportion, enter the sample proportion as a decimal, e.g. 0.55 for 55%.<\/small><\/div><div class=\"field has-term\" id=\"input-3\" hidden><label for=\"field-3\">Sample standard deviation (s)<\/label><div class=\"term\">s<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"sample_standard_deviation_s\" id=\"field-3\" aria-describedby=\"calc-info-3\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 15\" min=\"0\" value=\"15\"\/><small id=\"calc-info-3\">Sample SD of the data. Must be greater than 0. Not used for proportions.<\/small><\/div><div class=\"field has-term\" id=\"input-4\"><label for=\"field-4\">Sample size (n)<\/label><div class=\"term\">n<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"sample_size_n\" id=\"field-4\" aria-describedby=\"calc-info-4\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 25\" min=\"2\" value=\"25\"\/><small id=\"calc-info-4\">Number of observations. Must be an integer of at least 2.<\/small><\/div><div class=\"field\" id=\"input-5\"><label for=\"field-5\">Confidence level<\/label><div class=\"grid grid-cols-1\"><select id=\"field-5\" name=\"confidence_level\"><option value=\"80\">80%<\/option><option value=\"90\">90%<\/option><option value=\"95\" selected>95%<\/option><option value=\"98\">98%<\/option><option value=\"99\">99%<\/option><option value=\"99.9\">99.9%<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><div class=\"field\" id=\"input-6\"><label for=\"field-6\">Critical value method<\/label><div class=\"grid grid-cols-1\"><select id=\"field-6\" name=\"critical_value_method\"><option value=\"auto\" selected>Automatic (t when n &lt; 30, z when n \u2265 30)<\/option><option value=\"t\">Student t (n \u2212 1 degrees of freedom)<\/option><option value=\"z\">Normal z (use when \u03c3 is known)<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"65468\">Calculate<\/button><button type=\"reset\">Reset<\/button><\/div><div class=\"field is-checkbox hidden!\" id=\"field-auto-calc\"><input type=\"checkbox\" id=\"auto-calc\"><label for=\"auto-calc\">Calculate automatically<\/label><small>If enabled, the result will update automatically when you change any value.<\/small><\/div><div class=\"fieldset-result is-hidden\" aria-labelledby=\"result-title\" aria-live=\"polite\" role=\"region\"> <h3 class=\"result-title bg-gradient-to-r from-primary-50 to-gray-50 dark:from-primary-900 dark:to-gray-800\" id=\"result-title\"><span class=\"icon icon-s-pulse\" aria-hidden=\"true\"><\/span> Your Results<\/h3><div class=\"result-box\"><div class=\"field-result\" id=\"output-1\"><span class=\"field-title\"><span>Margin of error<\/span><span class=\"result-term\">ME<\/span><\/span><span class=\"field-value\" id=\"result-1\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-2\"><span class=\"field-title\"><span>Lower confidence limit<\/span><span class=\"result-term\">LCL<\/span><\/span><span class=\"field-value\" id=\"result-2\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-3\"><span class=\"field-title\"><span>Upper confidence limit<\/span><span class=\"result-term\">UCL<\/span><\/span><span class=\"field-value\" id=\"result-3\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-4\"><span class=\"field-title\"><span>Standard error<\/span><span class=\"result-term\">SE<\/span><\/span><span class=\"field-value\" id=\"result-4\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-5\"><span class=\"field-title\"><span>Critical value<\/span><span class=\"result-term\">z \/ t<\/span><\/span><span class=\"field-value\" id=\"result-5\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result is-column\" id=\"output-6\"><span class=\"field-title\"><span>Interval &amp; details<\/span><\/span><span class=\"field-value\" id=\"result-6\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><\/div><div class=\"result-actions\"><button type=\"button\" class=\"copy-link\"><svg viewBox=\"0 0 20 20\" fill=\"currentColor\" aria-hidden=\"true\"><path d=\"M12.232 4.232a2.5 2.5 0 0 1 3.536 3.536l-1.225 1.224a.75.75 0 0 0 1.061 1.06l1.224-1.224a4 4 0 0 0-5.656-5.656l-3 3a4 4 0 0 0 .225 5.865.75.75 0 0 0 .977-1.138 2.5 2.5 0 0 1-.142-3.667l3-3Z\"\/><path d=\"M11.603 7.963a.75.75 0 0 0-.977 1.138 2.5 2.5 0 0 1 .142 3.667l-3 3a2.5 2.5 0 0 1-3.536-3.536l1.225-1.224a.75.75 0 0 0-1.061-1.06l-1.224 1.224a4 4 0 1 0 5.656 5.656l3-3a4 4 0 0 0-.225-5.865Z\"\/><\/svg><span class=\"copy-link__text\">Copy link to this calculation<\/span><\/button><span class=\"sr-only\" role=\"status\" id=\"copy-link-status\"><\/span><\/div><\/div><a href=\"#respond\" class=\"hidden transition-opacity duration-300 opacity-0 w-50 text-sm justify-center items-center gap-2 px-4 py-2 rounded bg-gray-200 text-gray-700 hover:bg-gray-300\" id=\"leave-comment\"><span class=\"icon icon-comments\" aria-hidden=\"true\"><\/span>Leave a Comment<\/a><\/form><\/section><section id=\"calc-reactions\" class=\"not-prose hidden my-12 bg-gradient-to-r from-primary-50 to-gray-50 border border-indigo-100 rounded-xl px-6 py-4 shadow-sm\" data-post=\"65468\" aria-live=\"polite\" aria-labelledby=\"calc-reactions-title\"><p id=\"calc-reactions-title\" class=\"text-sm text-gray-500 text-center mb-2\">How did this result make you feel?<\/p><div role=\"group\" aria-labelledby=\"calc-reactions-title\" class=\"grid grid-cols-3 sm:flex sm:flex-row sm:justify-around gap-4 sm:items-center sm:flex-wrap\"><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"like\"><span class=\"reaction-count\" data-reaction-count=\"like\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude00 <\/span><span class=\"reaction-description font-medium\">Like <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"helpful\"><span class=\"reaction-count\" data-reaction-count=\"helpful\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\udca1 <\/span><span class=\"reaction-description font-medium\">Helpful <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"confused\"><span class=\"reaction-count\" data-reaction-count=\"confused\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude15 <\/span><span class=\"reaction-description font-medium\">Confused <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"disappointed\"><span class=\"reaction-count\" data-reaction-count=\"disappointed\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude1e <\/span><span class=\"reaction-description font-medium\">Disappointed <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"inaccurate\"><span class=\"reaction-count\" data-reaction-count=\"inaccurate\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\u274c <\/span><span class=\"reaction-description font-medium\">Inaccurate <\/span><\/button><\/div><div id=\"reaction-message\" class=\"hidden mt-8 rounded-md border border-primary-100 bg-white\/50 backdrop-blur-sm px-4 py-2 text-sm text-center shadow-sm\"><\/div><\/section><div class=\"calcdash-ad not-prose calcdash-ad--calc_result calcdash-ad--in-article\"><ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-1721569815777345\" data-ad-slot=\"6317458308\" data-ad-layout=\"in-article\" data-ad-format=\"fluid\"><\/ins><script>(adsbygoogle = window.adsbygoogle || []).push({});<\/script><\/div>\n<style id=\"wpcoderpro-inline-1\">.formula-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section class=\"formula-box\" aria-labelledby=\"formula-title\">\r\n\t<div id=\"formula-title\" class=\"not-prose font-display mb-4\"><span class=\"icon icon-formula\" aria-hidden=\"true\"><\/span><h2>Confidence Interval Formula<\/h2><\/div>\r\n\t<figure class=\"not-prose formula\">\r\n\t\t<figcaption class=\"formula-title\">Formula<\/figcaption>\r\n\t\t<div class=\"text-base text-gray-800\" id=\"formula\">\n\\\\[ \\\\text{CI for a mean:} \\\\quad \\\\bar{x} \\\\pm t_{\\\\alpha\/2,\\\\,n-1}\\\\cdot\\\\frac{s}{\\\\sqrt{n}} \\\\qquad \\\\text{CI for a proportion:} \\\\quad \\\\hat{p} \\\\pm z_{\\\\alpha\/2}\\\\cdot\\\\sqrt{\\\\frac{\\\\hat{p}(1-\\\\hat{p})}{n}} \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\\\(\\\\bar{x}\\) \u2014 sample mean (point estimate for the population mean)<\/li>\n<li>\\\\(\\\\hat{p}\\) \u2014 sample proportion, entered as a decimal (e.g. 0.55 for 55%)<\/li>\n<li>\\\\(s\\) \u2014 sample standard deviation<\/li>\n<li>\\\\(n\\) \u2014 sample size (number of observations)<\/li>\n<li>\\\\(t_{\\\\alpha\/2,\\\\,n-1}\\) \u2014 critical value from Student&rsquo;s t distribution with \\\\(n-1\\) degrees of freedom (used when \\\\(s\\) estimates an unknown \\\\(\\\\sigma\\))<\/li>\n<li>\\\\(z_{\\\\alpha\/2}\\) \u2014 critical value from the standard normal distribution<\/li>\n<li>\\\\(\\\\alpha = 1 - \\\\text{confidence level}\\) \u2014 significance level<\/li>\n<\/ul>\n<p>The formula builds the interval as the point estimate plus or minus the margin of error, where the margin of error is a critical value multiplied by the standard error \\\\(SE = s\/\\\\sqrt{n}\\) (mean) or \\\\(SE = \\\\sqrt{\\\\hat{p}(1-\\\\hat{p})\/n}\\) (proportion).<\/p>\n<p>\r\n<\/section>\n<style id=\"wpcoderpro-inline-459\">.calculation-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section id=\"calculation\" class=\"calculation-box\" aria-labelledby=\"calculation-title\">\r\n\t<div id=\"calculation-title\" class=\"not-prose font-display\"><span class=\"icon icon-unordered-list\" aria-hidden=\"true\"><\/span><h2>Confidence Interval &ndash; Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> a sample of 25 measurements has a mean of 100 and a standard deviation of 15. Find the 95% confidence interval for the population mean. Because \\\\(n &lt; 30\\) and the population SD is unknown, the t distribution with \\\\(n - 1 = 24\\) degrees of freedom is used.<\/p>\n<p>\\\\[ SE = \\\\frac{s}{\\\\sqrt{n}} = \\\\frac{15}{\\\\sqrt{25}} = 3 \\]<\/p>\n<p>\\\\[ t_{0.025,\\\\,24} \\\\approx 2.0639, \\\\quad ME = 2.0639 \\\\times 3 \\\\approx 6.1917 \\]<\/p>\n<p>\\\\[ \\\\text{CI} = 100 \\\\pm 6.1917 = (93.8083,\\\\; 106.1917) \\]<\/p>\n<p><strong>Result:<\/strong> the 95% confidence interval is approximately (93.81, 106.19), with a margin of error of &plusmn;6.19. You can reproduce this with the calculator&rsquo;s default values.<\/p>\n<p>\n\r\n<\/section>\n<h2>How to Use the Confidence Interval Calculator<\/h2>\n<ol>\n<li>Choose the <strong>Estimate type<\/strong>: &ldquo;Sample mean&rdquo; for averages of measured data, or &ldquo;Sample proportion&rdquo; for percentages or counts of successes.<\/li>\n<li>Enter the <strong>Sample mean (x&#772;)<\/strong>. For a proportion, type the value as a decimal between 0 and 1, e.g. 0.55 for 55%.<\/li>\n<li>If you selected a mean, enter the <strong>Sample standard deviation (s)<\/strong>. This field is hidden for proportions.<\/li>\n<li>Enter the <strong>Sample size (n)<\/strong> &mdash; a whole number of at least 2.<\/li>\n<li>Select the <strong>Confidence level<\/strong>: 80%, 90%, 95%, 98%, 99% or 99.9%. 95% is the most common choice.<\/li>\n<li>Pick the <strong>Critical value method<\/strong>. The default &ldquo;Automatic&rdquo; mode uses Student&rsquo;s t when \\\\(n &lt; 30\\) and the normal z value when \\\\(n \\\\ge 30\\). Choose &ldquo;Normal z&rdquo; only when the population standard deviation \\\\(\\\\sigma\\) is known.<\/li>\n<li>Read the results: <strong>Margin of error<\/strong>, <strong>Lower confidence limit<\/strong>, <strong>Upper confidence limit<\/strong>, <strong>Standard error<\/strong>, the <strong>Critical value<\/strong> used, and a details block with the full interval and interpretation.<\/li>\n<\/ol>\n<h2>What a Confidence Interval Means<\/h2>\n<p>A confidence interval is a range of plausible values for an unknown population parameter, such as a mean or a proportion, based on a single sample. The sample mean or sample proportion is only a point estimate &mdash; it is unlikely to hit the true value exactly. The interval surrounds that estimate with a margin of error that reflects how much the estimate could plausibly shift if you drew a different sample.<\/p>\n<p>The confidence level describes the long-run behavior of the procedure, not the probability for one specific interval. If you repeated the sampling many times and computed an interval each time, about 95% of those intervals (at the 95% level) would contain the true population value. Once an interval is computed, it either contains the parameter or it does not; the uncertainty is about the method, not the single number pair.<\/p>\n<p>The width of the interval carries most of the practical information. A narrow interval means the estimate is precise; a wide interval means the data leave considerable doubt. The width grows with the standard deviation and with the critical value (higher confidence &rarr; wider interval), and shrinks as the sample size grows &mdash; quadrupling \\\\(n\\) roughly halves the margin of error, because \\\\(SE\\) is divided by \\\\(\\\\sqrt{n}\\).<\/p>\n<h2>Reference: Common Critical Values (Two-Tailed)<\/h2>\n<table>\n<thead>\n<tr>\n<th>Confidence level<\/th>\n<th>z critical value<\/th>\n<th>t (df = 24, for comparison)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>80%<\/td>\n<td>1.2816<\/td>\n<td>1.3178<\/td>\n<\/tr>\n<tr>\n<td>90%<\/td>\n<td>1.6449<\/td>\n<td>1.7109<\/td>\n<\/tr>\n<tr>\n<td>95%<\/td>\n<td>1.9600<\/td>\n<td>2.0639<\/td>\n<\/tr>\n<tr>\n<td>98%<\/td>\n<td>2.3263<\/td>\n<td>2.4922<\/td>\n<\/tr>\n<tr>\n<td>99%<\/td>\n<td>2.5758<\/td>\n<td>2.7969<\/td>\n<\/tr>\n<tr>\n<td>99.9%<\/td>\n<td>3.2905<\/td>\n<td>3.7454<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Assumptions, Limitations and Tips<\/h2>\n<ul>\n<li><strong>Random sampling.<\/strong> The interval is only valid for the population actually sampled. Non-random or convenience samples invalidate the coverage guarantee.<\/li>\n<li><strong>t vs. z.<\/strong> When the population SD is unknown and estimated by \\\\(s\\), Student&rsquo;s t is the correct distribution; it accounts for the extra uncertainty in estimating \\\\(\\\\sigma\\) and gives a wider, more honest interval, especially for small \\\\(n\\). Use z only when \\\\(\\\\sigma\\) is genuinely known or when \\\\(n\\) is large enough that the difference is negligible.<\/li>\n<li><strong>Normality.<\/strong> For small samples the mean-based interval assumes the underlying data are approximately normal. For \\\\(n \\\\ge 30\\), the central limit theorem makes this assumption largely unnecessary unless the data are extremely skewed.<\/li>\n<li><strong>Proportions.<\/strong> The Wald interval used for proportions relies on a normal approximation. It works well when \\\\(n\\\\hat{p} \\\\ge 5\\) and \\\\(n(1-\\\\hat{p}) \\\\ge 5\\); the calculator warns you when this condition fails. For small samples or proportions near 0 or 1, a Wilson score interval is more reliable.<\/li>\n<li><strong>Higher confidence costs width.<\/strong> Moving from 95% to 99% widens the interval noticeably. Choose the level that matches how serious an error would be in your application, not simply the most impressive number.<\/li>\n<li><strong>Don&rsquo;t confuse precision with confidence.<\/strong> A 99% interval from 10 observations may be so wide it is practically useless. Increasing \\\\(n\\), not the confidence level, is the way to get a narrow interval.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Compute a confidence interval for a population mean or proportion. Enter your sample statistics, pick a confidence level, and get the margin of error and the lower and upper limits.<\/p>\n","protected":false},"author":3168,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[74],"tags":[80],"class_list":["post-65468","post","type-post","status-publish","format-standard","hentry","category-statistics","tag-sample-size"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/65468","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/users\/3168"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/comments?post=65468"}],"version-history":[{"count":3,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/65468\/revisions"}],"predecessor-version":[{"id":65684,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/65468\/revisions\/65684"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=65468"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=65468"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=65468"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}