{"id":6421,"date":"2026-07-07T14:51:56","date_gmt":"2026-07-07T19:51:56","guid":{"rendered":"https:\/\/www.eastcentral.edu\/learning-center\/?page_id=6421"},"modified":"2026-07-07T15:23:41","modified_gmt":"2026-07-07T20:23:41","slug":"central-limit-theorem","status":"publish","type":"page","link":"https:\/\/www.eastcentral.edu\/learning-center\/central-limit-theorem\/","title":{"rendered":"Central Limit Theorem"},"content":{"rendered":"\n<ul class=\"wp-block-list\">\n<li>What are you being asked about?<\/li>\n<\/ul>\n\n\n\n<table style=\"width:100%; border-collapse:collapse; border:1px solid #000; table-layout:fixed;\">\n\n<tr>\n    <th style=\"border:1px solid #000; padding:8px;\"><\/th>\n    <th style=\"border:1px solid #000; padding:8px; font-size:24px;\">Individual<\/th>\n    <th style=\"border:1px solid #000; padding:8px; font-size:24px;\">Mean<\/th>\n    <th style=\"border:1px solid #000; padding:8px; font-size:24px;\">Proportion<\/th>\n<\/tr>\n\n<tr>\n    <th style=\"border:1px solid #000; padding:8px; font-size:18px;\">\n        Mean\n    <\/th>\n\n    <td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n        \u03bc\n    <\/td>\n\n    <td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n        \u03bc\n    <\/td>\n\n    <td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n        p\n    <\/td>\n<\/tr>\n\n<tr>\n    <th style=\"border:1px solid #000; padding:8px; font-size:18px;\">\n        Standard Deviation\n    <\/th>\n\n    <td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n        \u03c3\n    <\/td>\n\n    <td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n\n        <span style=\"display:inline-block; text-align:center;\">\n            <span style=\"display:block; border-bottom:1px solid #000;\">\n                \u03c3\n            <\/span>\n            <span style=\"display:block;\">\n                \u221an\n            <\/span>\n        <\/span>\n\n    <\/td>\n\n    <td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n\n        <span style=\"display:inline-block; white-space:nowrap;\">\n            <span style=\"font-size:36px;\">\u221a<\/span>\n            <span style=\"display:inline-block; border-top:1px solid #000; padding-top:2px;\">\n\n                <span style=\"display:inline-block; text-align:center;\">\n                    <span style=\"display:block; border-bottom:1px solid #000;\">\n                        p(1 \u2212 p)\n                    <\/span>\n                    <span style=\"display:block;\">\n                        n\n                    <\/span>\n                <\/span>\n\n            <\/span>\n        <\/span>\n\n    <\/td>\n<\/tr>\n\n<tr>\n    <th style=\"border:1px solid #000; padding:8px; font-size:18px;\">\n        Conditions\n    <\/th>\n\n    <td style=\"border:1px solid #000; padding:8px;\">\n        \u2022 Normal\n    <\/td>\n\n    <td style=\"border:1px solid #000; padding:8px;\">\n        \u2022 n \u2265 30<br>\n        \u2022 Normal\n    <\/td>\n\n    <td style=\"border:1px solid #000; padding:8px;\">\n        \u2022 np \u2265 10<br>\n        \u2022 n(1 \u2212 p) \u2265 10<br>\n        \u2022 Normal\n    <\/td>\n<\/tr>\n\n<tr>\n    <th style=\"border:1px solid #000; padding:8px; font-size:18px;\">\n        z-score\n    <\/th>\n\n    <td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:16px;\">\n\n        z =\n\n        <span style=\"display:inline-block; text-align:center; vertical-align:middle;\">\n            <span style=\"display:block; border-bottom:1px solid #000; padding:0 8px;\">\n                x \u2212 \u03bc\n            <\/span>\n            <span style=\"display:block;\">\n                \u03c3\n            <\/span>\n        <\/span>\n\n    <\/td>\n\n    <td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:16px;\">\n\n        z =\n\n        <span style=\"display:inline-block; text-align:center; vertical-align:middle;\">\n            <span style=\"display:block; border-bottom:1px solid #000; padding:0 8px;\">\n                x\u0304 \u2212 \u03bc\n            <\/span>\n\n            <span style=\"display:block;\">\n\n                <span style=\"display:inline-block; text-align:center;\">\n                    <span style=\"display:block; border-bottom:1px solid #000;\">\n                        \u03c3\n                    <\/span>\n\n                    <span style=\"display:block;\">\n                        \u221an\n                    <\/span>\n                <\/span>\n\n            <\/span>\n        <\/span>\n\n    <\/td>\n\n   <td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n\n<span style=\"display:inline-flex; align-items:center;\">\n\nz =\n\n<span style=\"display:inline-block; text-align:center; vertical-align:middle; margin-left:4px;\">\n\n    <span style=\"display:block; border-bottom:1px solid #000; padding:0 8px;\">\n        p\u0302 \u2212 p\n    <\/span>\n\n    <span style=\"display:block;\">\n\n        <span style=\"display:inline-block; white-space:nowrap;\">\n            <span style=\"font-size:36px;\">\u221a<\/span>\n\n            <span style=\"display:inline-block; border-top:1px solid #000; padding-top:2px;\">\n\n                <span style=\"display:inline-block; text-align:center;\">\n                    <span style=\"display:block; border-bottom:1px solid #000;\">\n                        p(1 \u2212 p)\n                    <\/span>\n\n                    <span style=\"display:block;\">\n                        n\n                    <\/span>\n                <\/span>\n\n            <\/span>\n\n        <\/span>\n\n    <\/span>\n\n<\/span>\n\n<\/span>\n\n<\/td>\n<tr>\n    <th style=\"border:1px solid #000; padding:8px; font-size:18px;\">\n        Other Equations\n    <\/th>\n\n    <td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n        N\/A\n    <\/td>\n\n    <td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n        N\/A\n    <\/td>\n\n    <td style=\"border:1px solid #000; padding:8px;\">\n\n        \u2022 p\u0302 =\n\n        <span style=\"display:inline-block; text-align:center; vertical-align:middle;\">\n            <span style=\"display:block; border-bottom:1px solid #000;\">\n                x\n            <\/span>\n            <span style=\"display:block;\">\n                n\n            <\/span>\n        <\/span>\n\n        <br>\n\n        \u2022 x = p\u0302 &times; n\n\n    <\/td>\n<\/tr>\n<tr>\n    <th style=\"border:1px solid #000; padding:8px; font-size:18px;\">\n        Key Phrases:<br>\n        Probability\n    <\/th>\n\n    <td style=\"border:1px solid #000; padding:8px;\">\n        \u2022 Individual<br>\n        \u2022 Random ____ Chosen\n    <\/td>\n\n    <td style=\"border:1px solid #000; padding:8px;\">\n        \u2022 Mean\n    <\/td>\n\n    <td style=\"border:1px solid #000; padding:8px;\">\n        \u2022 Proportion<br>\n        \u2022 Percent<br>\n        \u2022 x of n\n    <\/td>\n<\/tr>\n<\/table>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Does it say: <strong>&#8220;differs by&#8221;<\/strong>?<\/li>\n<\/ul>\n\n\n\n<table style=\"width:100%; border-collapse:collapse; border:1px solid #000; table-layout:fixed;\">\n\n<tr>\n    <th rowspan=\"2\" style=\"border:1px solid #000; padding:8px;\"><\/th>\n\n    <th rowspan=\"2\" style=\"border:1px solid #000; padding:8px; font-size:26px;\">\n        No\n    <\/th>\n\n    <th colspan=\"3\" style=\"border:1px solid #000; padding:8px; font-size:26px;\">\n        Differs By Less\n    <\/th>\n\n    <th rowspan=\"2\" style=\"border:1px solid #000; padding:8px; font-size:26px;\">\n        Differs By More\n    <\/th>\n<\/tr>\n\n<tr>\n\n    <th style=\"border:1px solid #000; padding:8px; font-size:18px;\">\n        Individual\n    <\/th>\n\n    <th style=\"border:1px solid #000; padding:8px; font-size:18px;\">\n        Mean\n    <\/th>\n\n    <th style=\"border:1px solid #000; padding:8px; font-size:18px;\">\n        Proportion\n    <\/th>\n\n<\/tr>\n\n<tr>\n\n    <th style=\"border:1px solid #000; padding:8px; font-size:18px;\">\n        Calculator<br>Functions\n    <\/th>\n\n    <td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:16px;\">\n        Follow<br>\n        normalcdf<br>\n        Rules\n    <\/td>\n\n    <td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n        \u2022 Lower: \u03bc \u2212 x<br>\n        \u2022 Upper: \u03bc + x<br>\n        \u2022 Mean: \u03bc<br>\n        \u2022 SD: \u03c3\n    <\/td>\n\n    <td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n        \u2022 Lower: \u03bc \u2212 x<br>\n        \u2022 Upper: \u03bc + x<br>\n        \u2022 Mean: \u03bc<br>\n\n        \u2022 SD:\n\n        <span style=\"display:inline-block; text-align:center; vertical-align:middle;\">\n            <span style=\"display:block; border-bottom:1px solid #000;\">\n                \u03c3\n            <\/span>\n            <span style=\"display:block;\">\n                \u221an\n            <\/span>\n        <\/span>\n    <\/td>\n\n    <td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n        \u2022 Lower: p \u2212 %<br>\n        \u2022 Upper: p + %<br>\n        \u2022 Mean: p<br>\n\n        \u2022 SD:\n\n    <span style=\"font-size:32px;\">\u221a<\/span>\n\n    <span style=\"display:inline-block; border-top:1px solid #000; padding-top:2px;\">\n\n        <span style=\"display:inline-block; text-align:center;\">\n\n            <span style=\"display:block; border-bottom:1px solid #000;\">\n                p(1 \u2212 p)\n            <\/span>\n\n            <span style=\"display:block;\">\n                n\n            <\/span>\n\n        <\/span>\n\n   <\/span>\n<td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n\n\u2022 1 \u2212 &#8220;Differs by Less&#8221;\n\n<br><br>\n\n\u2022 2 &#038;times normalcdf (one of the sides)\n<\/span>\n<\/td>\n<\/tr>\n<tr>\n\n    <th style=\"border:1px solid #000; padding:8px; font-size:18px;\">\n        Diagram\n    <\/th>\n\n    <td style=\"border:1px solid #000; padding:8px; text-align:center; vertical-align:middle;\">\n        N\/A\n    <\/td>\n\n    <td colspan=\"3\" style=\"border:1px solid #000; padding:8px; text-align:center; vertical-align:middle;\">\n        <img decoding=\"async\" src=\"https:\/\/www.eastcentral.edu\/learning-center\/wp-content\/uploads\/sites\/27\/2026\/07\/bell-curve-graph-between.png\"\n             alt=\"Differs By Less\"\n             style=\"max-width:200px; height:auto;\">\n    <\/td>\n\n    <td style=\"border:1px solid #000; padding:8px; text-align:center; vertical-align:middle;\">\n        <img decoding=\"async\" src=\"https:\/\/www.eastcentral.edu\/learning-center\/wp-content\/uploads\/sites\/27\/2026\/07\/bell-curve-tails.png\"\n             alt=\"Differs By More\"\n             style=\"max-width:auto; height:auto;\">\n    <\/td>\n\n<\/tr>\n<\/table>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><a href=\"https:\/\/www.eastcentral.edu\/learning-center\/wp-content\/uploads\/sites\/27\/2025\/05\/image.jpeg\"><img loading=\"lazy\" decoding=\"async\" width=\"602\" height=\"455\" src=\"https:\/\/www.eastcentral.edu\/learning-center\/wp-content\/uploads\/sites\/27\/2025\/05\/image.jpeg\" alt=\"91ԭƬ Learning Center Logo\" class=\"wp-image-4893\" style=\"width:200px\" srcset=\"https:\/\/www.eastcentral.edu\/learning-center\/wp-content\/uploads\/sites\/27\/2025\/05\/image.jpeg 602w, https:\/\/www.eastcentral.edu\/learning-center\/wp-content\/uploads\/sites\/27\/2025\/05\/image-300x227.jpeg 300w\" sizes=\"auto, (max-width: 602px) 100vw, 602px\" \/><\/a><figcaption class=\"wp-element-caption\">For more help, please contact The Learning Center<\/figcaption><\/figure>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>Individual Mean Proportion Mean \u03bc \u03bc p Standard Deviation \u03c3 \u03c3 \u221an \u221a p(1 \u2212 p) n Conditions \u2022 Normal \u2022 n \u2265 30 \u2022 Normal \u2022 np \u2265 10 \u2022 n(1 \u2212 p) \u2265 10 \u2022 Normal z-score z = x \u2212 \u03bc \u03c3 z = x\u0304 \u2212 \u03bc \u03c3 \u221an z = p\u0302 [&hellip;]<\/p>\n","protected":false},"author":12,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_eb_attr":"","footnotes":""},"yst_prominent_words":[],"class_list":["post-6421","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/pages\/6421","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/users\/12"}],"replies":[{"embeddable":true,"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/comments?post=6421"}],"version-history":[{"count":9,"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/pages\/6421\/revisions"}],"predecessor-version":[{"id":6434,"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/pages\/6421\/revisions\/6434"}],"wp:attachment":[{"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/media?parent=6421"}],"wp:term":[{"taxonomy":"yst_prominent_words","embeddable":true,"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/yst_prominent_words?post=6421"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}