{"id":65464,"date":"2026-09-29T12:26:43","date_gmt":"2026-09-29T09:26:43","guid":{"rendered":"https:\/\/wpcalc.com\/en\/?p=65464"},"modified":"2026-09-29T15:19:25","modified_gmt":"2026-09-29T12:19:25","slug":"z-score-calculator","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/statistics\/z-score-calculator\/","title":{"rendered":"Z-Score 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\">Calculate Your Z-Score<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"65464\" data-favorite-title=\"Z-Score\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/statistics\/z-score-calculator\/\" data-favorite-excerpt=\"Convert any raw value into a z-score using the mean and standard deviation, or let the calculator derive them from your own data set. You also get the percentile and tail probabilities.\" aria-label=\"Add to Favorites\" data-favorite-icon=\"icon icon-wpcalc-distribution-histogram\">\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\/z-score-calculator\/\" method=\"POST\" class=\"calculator\" id=\"calculator-65464\" data-post=\"65464\"><fieldset class=\"fieldset-input is-1\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field\" id=\"input-1\"><label for=\"field-1\">Input mode<\/label><div class=\"grid grid-cols-1\"><select id=\"field-1\" name=\"input_mode\"><option value=\"direct\" selected>Mean &amp; standard deviation known<\/option><option value=\"data\">From a data set<\/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\">Raw value (x)<\/label><div class=\"term\">x<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"raw_value_x\" id=\"field-2\" aria-describedby=\"calc-info-2\" placeholder=\"e.g. 110\" step=\"any\" value=\"110\"\/><small id=\"calc-info-2\">The value you want to convert to a z-score.<\/small><\/div><div class=\"field has-term\" id=\"input-3\" hidden><label for=\"field-3\">Population mean (\u03bc)<\/label><div class=\"term\">\u03bc<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"population_mean_\" id=\"field-3\" placeholder=\"e.g. 100\" step=\"any\" value=\"100\"\/><\/div><div class=\"field has-term\" id=\"input-4\" hidden><label for=\"field-4\">Standard deviation (\u03c3)<\/label><div class=\"term\">\u03c3<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"standard_deviation_\" id=\"field-4\" aria-describedby=\"calc-info-4\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 15\" min=\"0\" value=\"15\"\/><small id=\"calc-info-4\">Must be greater than 0.<\/small><\/div><div class=\"field is-single\" id=\"input-5\" hidden><label for=\"field-5\">Data set<\/label><textarea id=\"field-5\" name=\"data_set\">85, 90, 95, 100, 105, 110, 115, 120, 125, 130<\/textarea><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"65464\">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>Z-score<\/span><span class=\"result-term\">z<\/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>Percentile \u2014 P(X \u2264 x)<\/span><span class=\"result-term\">\u03a6(z)<\/span><\/span><span class=\"field-value\" id=\"result-2\" data-unit=\"%\"><\/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>Probability of a higher value \u2014 P(X &gt; x)<\/span><span class=\"result-term\">1 \u2212 \u03a6(z)<\/span><\/span><span class=\"field-value\" id=\"result-3\" data-unit=\"%\"><\/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-4\"><span class=\"field-title\"><span>Interpretation<\/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><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=\"65464\" 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>Z-Score 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\\[ z = \\frac{x - \\mu}{\\sigma} \\]<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(x\\) \u2014 the raw value you want to standardize<\/li>\n<li>\\(\\mu\\) \u2014 the mean of the population (or of your data set)<\/li>\n<li>\\(\\sigma\\) \u2014 the standard deviation of the population (or the sample standard deviation, \\(s\\))<\/li>\n<li>\\(z\\) \u2014 the z-score, i.e. the number of standard deviations between \\(x\\) and the mean<\/li>\n<\/ul>\n<p>The formula measures the distance from the value to the mean and expresses it in units of standard deviation. A positive z means the value is above the mean, a negative z means it is below.<\/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>Z-Score \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> an exam is scored with a mean of \\(\\mu = 100\\) and a standard deviation of \\(\\sigma = 15\\). A student scores \\(x = 110\\). How many standard deviations above the mean is that, and what percentile does it correspond to?<\/p>\n<p>\\[ z = \\frac{x - \\mu}{\\sigma} = \\frac{110 - 100}{15} = \\frac{10}{15} \\approx 0.6667 \\]<\/p>\n<p>\\[ \\Phi(0.6667) \\approx 0.7475 \\quad\\Rightarrow\\quad P(X \\le 110) \\approx 74.75\\% \\]<\/p>\n<p><strong>Result:<\/strong> \\(z \\approx 0.6667\\). The score is about two-thirds of a standard deviation above the mean; roughly 74.75% of students score at or below 110, and about 25.25% score higher.<\/p>\n<p>\n\r\n<\/section>\n<h2>How to use the calculator<\/h2>\n<ol>\n<li>Open the <strong>Input mode<\/strong> dropdown and choose <em>Mean &amp; standard deviation known<\/em> if you already have \\(\\mu\\) and \\(\\sigma\\), or <em>From a data set<\/em> if you want the calculator to compute them for you.<\/li>\n<li>In <strong>Raw value (x)<\/strong>, enter the value you want to convert into a z-score.<\/li>\n<li>In direct mode, fill in <strong>Population mean (\u03bc)<\/strong> and <strong>Standard deviation (\u03c3)<\/strong>. The standard deviation must be greater than 0.<\/li>\n<li>In data-set mode, paste your numbers into <strong>Data set<\/strong>, separated by commas, spaces, semicolons or line breaks. You need at least 2 numbers, and they cannot all be identical. The calculator computes the mean and the sample standard deviation (using \\(n - 1\\)) automatically.<\/li>\n<li>Read the <strong>Z-score<\/strong>, the <strong>percentile<\/strong> \\(P(X \\le x)\\), the probability of a higher value \\(P(X &gt; x)\\), and a plain-language <strong>Interpretation<\/strong> with a summary table.<\/li>\n<\/ol>\n<h2>What a z-score means<\/h2>\n<p>A z-score (also called a standard score) answers a simple question: how unusual is this value? It converts any measurement into a common scale by asking how many standard deviations it sits away from the mean. Because every distribution's raw units are stripped away, z-scores let you compare values from completely different measurements \u2014 for example, whether a height of 190 cm or an exam score of 130 is more exceptional relative to its own population.<\/p>\n<p>The sign carries the direction. A z-score of +2 means the value is two standard deviations above the mean; a z-score of \u22121.5 means it is one and a half standard deviations below. A z of exactly 0 means the value equals the mean.<\/p>\n<p>The calculator also converts your z-score into probabilities, assuming the data follow a normal (Gaussian) distribution. The percentile \\(\\Phi(z)\\) is the share of values expected at or below yours; \\(1 - \\Phi(z)\\) is the share expected above. For the example above, \\(z = 0.6667\\) places the value at about the 75th percentile \u2014 a good but not exceptional result.<\/p>\n<h2>Interpreting how extreme a z-score is<\/h2>\n<p>The familiar 68\u201395\u201399.7 rule for normal distributions gives quick benchmarks:<\/p>\n<table>\n<thead>\n<tr>\n<th>|z| range<\/th>\n<th>What it means<\/th>\n<th>Share of values at least this extreme<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>under 1<\/td>\n<td>Typical \u2014 within 1 standard deviation of the mean<\/td>\n<td>\u2248 32%<\/td>\n<\/tr>\n<tr>\n<td>1 to 2<\/td>\n<td>Somewhat uncommon<\/td>\n<td>\u2248 27%<\/td>\n<\/tr>\n<tr>\n<td>2 to 3<\/td>\n<td>Unusual<\/td>\n<td>\u2248 4.6%<\/td>\n<\/tr>\n<tr>\n<td>3 or more<\/td>\n<td>Very unusual<\/td>\n<td>under 0.3%<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Z-scores beyond \u00b13 are often treated as outliers or as signals worth investigating, though what counts as \"extreme\" depends on your field \u2014 in physics, 5-sigma (\\(|z| \\ge 5\\)) is a common threshold for a discovery.<\/p>\n<h2>Assumptions, limitations and tips<\/h2>\n<ul>\n<li><strong>Percentiles assume normality.<\/strong> The z-score itself is valid for any distribution with a defined mean and standard deviation, but the percentile and tail probabilities \\(\\Phi(z)\\) are only accurate when the data are approximately normally distributed. For skewed data, the reported probabilities will be misleading even though the z-score is correct.<\/li>\n<li><strong>Sample vs. population standard deviation.<\/strong> In data-set mode the calculator uses the sample standard deviation with Bessel's correction (dividing by \\(n - 1\\)), which is the right choice when your numbers are a sample from a larger population. If your numbers are the entire population, enter \\(\\sigma\\) manually in direct mode instead.<\/li>\n<li><strong>Small samples are noisy.<\/strong> With only a handful of values, the sample mean and standard deviation are unstable estimates, so z-scores computed from them should be treated with caution.<\/li>\n<li><strong>Watch for outliers at the input stage.<\/strong> A single mistyped value in the data set can inflate the standard deviation and flatten everyone else's z-scores. Double-check the summary table in the interpretation block \u2014 it shows the mean, standard deviation and sample size that were actually used.<\/li>\n<li><strong>Use the raw value, not a z-score, in the x field.<\/strong> The calculator expects an original measurement; it computes the standardization for you.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Convert any raw value into a z-score using the mean and standard deviation, or let the calculator derive them from your own data set. You also get the percentile and tail probabilities.<\/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":[77],"class_list":["post-65464","post","type-post","status-publish","format-standard","hentry","category-statistics","tag-distributions"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/65464","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=65464"}],"version-history":[{"count":3,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/65464\/revisions"}],"predecessor-version":[{"id":65736,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/65464\/revisions\/65736"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=65464"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=65464"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=65464"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}