{"id":65474,"date":"2026-09-29T12:27:03","date_gmt":"2026-09-29T09:27:03","guid":{"rendered":"https:\/\/wpcalc.com\/en\/?p=65474"},"modified":"2026-09-29T15:19:19","modified_gmt":"2026-09-29T12:19:19","slug":"poisson-distribution-calculator","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/statistics\/poisson-distribution-calculator\/","title":{"rendered":"Poisson Distribution 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 Poisson Probabilities<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"65474\" data-favorite-title=\"Poisson Distribution\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/statistics\/poisson-distribution-calculator\/\" data-favorite-excerpt=\"Calculate Poisson probabilities from the average rate \u03bb: the chance of exactly k events, a range, or cumulative at-most \/ at-least probabilities, plus mean and variance.\" 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\/poisson-distribution-calculator\/\" method=\"POST\" class=\"calculator\" id=\"calculator-65474\" data-post=\"65474\"><fieldset class=\"fieldset-input is-1\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field\" id=\"input-1\"><label for=\"field-1\">Probability type<\/label><div class=\"grid grid-cols-1\"><select id=\"field-1\" name=\"probability_type\"><option value=\"exact\" selected>Exactly k events \u2014 P(X = k)<\/option><option value=\"range\">Between k\u2081 and k\u2082 events \u2014 P(k\u2081 \u2264 X \u2264 k\u2082)<\/option><option value=\"atmost\">At most k events \u2014 P(X \u2264 k)<\/option><option value=\"atleast\">At least k events \u2014 P(X \u2265 k)<\/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\">Average rate (\u03bb)<\/label><div class=\"term\">\u03bb<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"average_rate_\" id=\"field-2\" aria-describedby=\"calc-info-2\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 3\" min=\"1.0E-6\" value=\"3\"\/><small id=\"calc-info-2\">The mean number of events per interval (must be positive).<\/small><\/div><div class=\"field has-term\" id=\"input-3\"><label for=\"field-3\">Number of events (k)<\/label><div class=\"term\">k<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"number_of_events_k\" id=\"field-3\" aria-describedby=\"calc-info-3\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 5\" min=\"0\" value=\"5\"\/><small id=\"calc-info-3\">A whole number of events (0 or larger).<\/small><\/div><div class=\"field has-term\" id=\"input-4\" hidden><label for=\"field-4\">Upper event count (k\u2082)<\/label><div class=\"term\">k\u2082<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"upper_event_count_k\" id=\"field-4\" aria-describedby=\"calc-info-4\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 8\" min=\"0\" value=\"8\"\/><small id=\"calc-info-4\">The upper bound of the range (only used for the range mode). Must be \u2265 k.<\/small><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"65474\">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>Requested probability<\/span><span class=\"result-term\">P<\/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>P(X = k)<\/span><span class=\"result-term\">P(X = k)<\/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>Cumulative P(X \u2264 k)<\/span><span class=\"result-term\">P(X \u2264 k)<\/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>Mean<\/span><span class=\"result-term\">\u03bc<\/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>Variance<\/span><span class=\"result-term\">\u03c3\u00b2<\/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>Interpretation &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=\"65474\" 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>Poisson Distribution 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\\[ P(X = k) = \\frac{e^{-\\lambda}\\,\\lambda^k}{k!} \\\\\n\\[ P(X \\le k) = \\sum_{j=0}^{k} \\frac{e^{-\\lambda}\\,\\lambda^j}{j!} \\\\\n\\[ P(X \\ge k) = 1 - \\sum_{j=0}^{k-1} \\frac{e^{-\\lambda}\\,\\lambda^j}{j!} \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(\\lambda\\) \u2014 the average (mean) number of events per interval; must be positive<\/li>\n<li>\\(k\\) \u2014 the number of events; a whole number, 0 or larger<\/li>\n<li>\\(e \\approx 2.71828\\) \u2014 Euler's number<\/li>\n<li>\\(k!\\) \u2014 the factorial of \\(k\\)<\/li>\n<\/ul>\n<p>The first formula gives the probability of observing exactly \\(k\\) events; the two sums give the cumulative probabilities \\(P(X \\le k)\\) and \\(P(X \\ge k)\\).<\/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>Poisson Distribution \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> a call center receives an average of \\(\\lambda = 3\\) calls per hour. What is the probability of receiving exactly \\(k = 5\\) calls in the next hour?<\/p>\n<p>\\[ P(X = 5) = \\frac{e^{-3} \\cdot 3^5}{5!} = \\frac{0.0497871 \\cdot 243}{120} = \\frac{12.0981}{120} \\approx 0.100819 \\]<\/p>\n<p><strong>Result:<\/strong> \\(P(X = 5) \\approx 0.1008\\), or about <strong>10.08%<\/strong>. With \\(\\lambda = 3\\), the mean and variance are both 3 and the standard deviation is \\(\\sqrt{3} \\approx 1.732\\).<\/p>\n<p>\n\r\n<\/section>\n<h2>How to use the calculator<\/h2>\n<ol>\n<li>Select the <strong>Probability type<\/strong>: exactly k events, a range between two counts, at most k events, or at least k events.<\/li>\n<li>Enter the <strong>Average rate (\u03bb)<\/strong> \u2014 the mean number of events per interval (per hour, per page, per batch, etc.). It must be greater than 0.<\/li>\n<li>Enter the <strong>Number of events (k)<\/strong> \u2014 a whole number, 0 or larger. In range mode this value acts as the lower bound k\u2081.<\/li>\n<li>If you chose the range mode, also fill in the <strong>Upper event count (k\u2082)<\/strong>, which must be greater than or equal to k.<\/li>\n<li>Read the <strong>Requested probability<\/strong> at the top of the results, then check \\(P(X = k)\\), the cumulative \\(P(X \\le k)\\), the mean, and the variance below it.<\/li>\n<\/ol>\n<h2>What is the Poisson distribution?<\/h2>\n<p>The Poisson distribution models the number of times a random event occurs within a fixed interval of time, space, or volume, when events happen independently at a constant average rate. It is named after French mathematician Sim\u00e9on Denis Poisson and is one of the most widely used discrete probability distributions in statistics.<\/p>\n<p>A single parameter, \\(\\lambda\\), fully describes the distribution: it is both the mean and the variance, so \\(\\mu = \\sigma^2 = \\lambda\\). Classic applications include the number of customer arrivals per hour, typos per page, radioactive decays per second, website visits per minute, or insurance claims per month. If you know the historical average count for the interval, the Poisson formula lets you estimate the probability of any specific outcome.<\/p>\n<p>The most likely outcome is near \\(\\lambda\\) itself, and probabilities taper off on both sides. For example, with \\(\\lambda = 3\\), getting 2, 3, or 4 events is fairly likely, while 0 or 8 events is uncommon but not impossible. The calculator's neighbouring-probabilities table shows this spread around your chosen k, which helps you judge whether an observed count is typical or unusual. Cumulative answers (at most \/ at least) are often more useful than exact probabilities: they tell you the chance of hitting a threshold, such as \"more than 10 arrivals\" or \"no more than 1 defect\".<\/p>\n<h2>Assumptions, limitations, and tips<\/h2>\n<ul>\n<li><strong>Independence:<\/strong> one event must not influence the probability of another. Arrivals triggered by the same cause (e.g. a viral post spiking visits) violate this assumption.<\/li>\n<li><strong>Constant rate:<\/strong> \\(\\lambda\\) must stay the same across the interval. If the rate varies by hour or season, compute each period separately or use a different model.<\/li>\n<li><strong>Count data only:<\/strong> k counts whole events in a fixed interval. The Poisson distribution does not describe continuous quantities like durations or distances.<\/li>\n<li><strong>Scaling \u03bb:<\/strong> if you double the interval, double \\(\\lambda\\). A rate of 3 calls per hour becomes \\(\\lambda = 6\\) for a two-hour window or \\(\\lambda = 0.05\\) per minute.<\/li>\n<li><strong>Large \u03bb:<\/strong> as \\(\\lambda\\) grows (roughly above 20), the Poisson distribution becomes nearly symmetric and closely approximates a normal distribution with mean \\(\\lambda\\) and variance \\(\\lambda\\).<\/li>\n<li><strong>Complements:<\/strong> remember that \\(P(X \\ge k) = 1 - P(X \\le k-1)\\). The \"at least\" answer already accounts for this, so \\(P(X \\ge 0)\\) is always 100%.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Calculate Poisson probabilities from the average rate \u03bb: the chance of exactly k events, a range, or cumulative at-most \/ at-least probabilities, plus mean and variance.<\/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-65474","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\/65474","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=65474"}],"version-history":[{"count":3,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/65474\/revisions"}],"predecessor-version":[{"id":65713,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/65474\/revisions\/65713"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=65474"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=65474"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=65474"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}