{"id":805,"date":"2025-12-23T10:35:39","date_gmt":"2025-12-23T10:35:39","guid":{"rendered":"https:\/\/alandix.com\/statistics\/?page_id=805"},"modified":"2025-12-23T10:41:28","modified_gmt":"2025-12-23T10:41:28","slug":"chapter-4","status":"publish","type":"page","link":"https:\/\/alandix.com\/statistics\/second-edition\/toc\/chapter-4\/","title":{"rendered":"Chapter 4: Characterising randomness through probability distributions"},"content":{"rendered":"<div class=\"embedurl\" data-url=\"https:\/\/alandix.com\/books\/hcistats\/content2e\/chaps\/chap-4.html\" ><!--  Chapter 4 Characterising randomness through probability distributions  -->\n\n<script>\nvar chapnos = 4;\nvar json_url = \"https:\\\/\\\/alandix.com\\\/books\\\/hcistats\\\/content2e\\\/chaps\\\/chap-4.json\";\n<\/script>\n\n\n\n\n\n\n<h3> Contents <\/h3>\n<div class=\"toc\">\n<dl>\n<dt>4.1&nbsp;&nbsp;Types of probability distribution<\/dt><dd><dl>\n<dt>4.1.1&nbsp;&nbsp;Continuous or discrete?<\/dt>\n<dt>4.1.2&nbsp;&nbsp;Finite or unbounded<\/dt>\n<dt>4.1.3&nbsp;&nbsp;UK income distribution &ndash; a long tail<\/dt>\n<dt>4.1.4&nbsp;&nbsp;One tail or two<\/dt>\n<\/dl><\/dd>\n<dt>4.2&nbsp;&nbsp;Normal or not<\/dt><dd><dl>\n<dt>4.2.1&nbsp;&nbsp;Approximations<\/dt>\n<dt>4.2.2&nbsp;&nbsp;The central limit theorem &ndash; (nearly) everything is Normal<\/dt>\n<dt>4.2.3&nbsp;&nbsp;Non-Normal &ndash; what can go wrong?<\/dt>\n<\/dl><\/dd>\n<dt>4.3&nbsp;&nbsp;Power law distributions<\/dt>\n<dt>4.4&nbsp;&nbsp;Parametric and non-parametric<\/dt><dd><dl>\n<dt>4.4.1&nbsp;&nbsp;Transforming data<\/dt>\n<dt>4.4.2&nbsp;&nbsp;Contingency table<\/dt>\n<dt>4.4.3&nbsp;&nbsp;Order statistics<\/dt>\n<\/dl><\/dd>\n<\/dl><\/div>\n\n\n<h3> Glossary items referenced in this chapter <\/h3>\n<div class=\"toc\">\n<a href=\"https:\/\/alandix.com\/glossary\/hcistats\/anova\">ANOVA (Analysis of Variance)<\/a>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/applied%20mathematics\">applied mathematics<\/a>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/approximate%20distribution\">approximate distribution<\/a>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/approximately%20normal\">approximately Normal<\/a>, <strong><a href=\"https:\/\/alandix.com\/glossary\/hcistats\/arithmetical%20data\">arithmetical data<\/a><\/strong>, <strong><a href=\"https:\/\/alandix.com\/glossary\/hcistats\/asymmetric%20distribution\">asymmetric distribution<\/a><\/strong>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/bias\">bias<\/a>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/bimodal%20distribution\">bimodal distribution<\/a>, <strong><a href=\"https:\/\/alandix.com\/glossary\/hcistats\/binary%20data\">binary data<\/a><\/strong>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/binomial%20distribution\">Binomial distribution<\/a>, <strong><a href=\"https:\/\/alandix.com\/glossary\/hcistats\/bounded%20below\">bounded below<\/a><\/strong>, <strong><a href=\"https:\/\/alandix.com\/glossary\/hcistats\/bounded%20data\">bounded data<\/a><\/strong>, <strong><a href=\"https:\/\/alandix.com\/glossary\/hcistats\/categorical%20data\">categorical data<\/a><\/strong>, <strong><a href=\"https:\/\/alandix.com\/glossary\/hcistats\/central%20limit%20theorem\">Central Limit Theorem<\/a><\/strong>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/chi%20squared\">chi squared<\/a>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/citation%20data\">citation data<\/a>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/coin%20tossing\">coin tossing<\/a>, <strong><a href=\"https:\/\/alandix.com\/glossary\/hcistats\/contingency%20table\">contingency table<\/a><\/strong>, <strong><a href=\"https:\/\/alandix.com\/glossary\/hcistats\/continuous%20data\">continuous data<\/a><\/strong>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/continuous%20distribution\">continuous distribution<\/a>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/count%20data\">count data<\/a>, <strong><a href=\"https:\/\/alandix.com\/glossary\/hcistats\/discrete%20data\">discrete data<\/a><\/strong>, <strong><a href=\"https:\/\/alandix.com\/glossary\/hcistats\/empirical%20distribution\">empirical distribution<\/a><\/strong>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/error%20rate\">error rate<\/a>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/fair%20coin\">fair coin<\/a>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/false%20negative\">false negative<\/a>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/feedback%20effects\">feedback effects<\/a>, <strong><a href=\"https:\/\/alandix.com\/glossary\/hcistats\/finite%20data\">finite data<\/a><\/strong>, <strong><a href=\"https:\/\/alandix.com\/glossary\/hcistats\/finite%20variance\">finite variance<\/a><\/strong>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/income%20distribution\">income distribution<\/a>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/independence\">independence<\/a>, <strong><a href=\"https:\/\/alandix.com\/glossary\/hcistats\/independence%20condition\">independence condition<\/a><\/strong>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/kruskal-wallis%20test\">Kruskal-Wallis test<\/a>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/likert%20scale\">Likert scale<\/a>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/linear%20regression\">linear regression<\/a>, <strong><a href=\"https:\/\/alandix.com\/glossary\/hcistats\/linearity%20condition\">linearity condition<\/a><\/strong>, <strong><a href=\"https:\/\/alandix.com\/glossary\/hcistats\/log-normal%20distribution\">Log-Normal distribution<\/a><\/strong>, <strong><a href=\"https:\/\/alandix.com\/glossary\/hcistats\/long-tail%20distribution\">long-tail distribution<\/a><\/strong>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/mathematics\">mathematics<\/a>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/mean\">mean (&amp;mu;)<\/a>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/negative%20binomial\">negative binomial<\/a>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/network%20data\">network data<\/a>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/non-independence\">non-independence<\/a>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/nonlinearity\">nonlinearity<\/a>, <strong><a href=\"https:\/\/alandix.com\/glossary\/hcistats\/nonparametric%20statistics\">nonparametric statistics<\/a><\/strong>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/normal%20approximation\">Normal approximation<\/a>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/normal%20distribution\">Normal distribution<\/a>, <strong><a href=\"https:\/\/alandix.com\/glossary\/hcistats\/one-tailed%20test\">one-tailed test<\/a><\/strong>, <strong><a href=\"https:\/\/alandix.com\/glossary\/hcistats\/order%20statistics\">order statistics<\/a><\/strong>, <strong><a href=\"https:\/\/alandix.com\/glossary\/hcistats\/ordinal%20data\">ordinal data<\/a><\/strong>, <strong><a href=\"https:\/\/alandix.com\/glossary\/hcistats\/parametric%20statistics\">parametric statistics<\/a><\/strong>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/pearson%20correlation%20coefficient\">Pearson correlation coefficient<\/a>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/poisson%20distribution\">Poisson distribution<\/a>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/positive%20feedback\">positive feedback<\/a>, <strong><a href=\"https:\/\/alandix.com\/glossary\/hcistats\/power-law%20distribution\">power-law distribution<\/a><\/strong>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/probability%20distribution\">probability distribution<\/a>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/quartile\">quartile<\/a>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/residuals\">residuals<\/a>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/sample%20size\">sample size<\/a>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/scale%20free%20distribution\">scale free distribution<\/a>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/social%20network%20data\">social network data<\/a>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/spearman%27s%20rank%20correlation%20coefficient\">Spearman&#039;s rank correlation coefficient<\/a>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/standard%20deviation\">standard deviation (s.d., &amp;sigma;)<\/a>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/statistical%20analysis\">statistical analysis<\/a>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/statistical%20power\">statistical power<\/a>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/student%27s%20t-test\">Student&#039;s t-test<\/a>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/survey%20data\">survey data<\/a>, <strong><a href=\"https:\/\/alandix.com\/glossary\/hcistats\/tail\">tail<\/a><\/strong>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/tail%20heavy\">tail heavy<\/a>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/task%20completion%20time\">task completion time<\/a>, <strong><a href=\"https:\/\/alandix.com\/glossary\/hcistats\/theoretical%20distribution\">theoretical distribution<\/a><\/strong>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/threshold%20effect\">threshold effect<\/a>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/transforming%20data\">transforming data<\/a>, <strong><a href=\"https:\/\/alandix.com\/glossary\/hcistats\/two-tailed%20test\">two-tailed test<\/a><\/strong>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/uk%20income%20distribution\">UK income distribution<\/a>, <strong><a href=\"https:\/\/alandix.com\/glossary\/hcistats\/unbounded%20data\">unbounded data<\/a><\/strong>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/unbounded%20tail\">unbounded tail<\/a>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/usability%20evaluation\">usability evaluation<\/a>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/user%20experience%20studies\">user experience studies<\/a>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/user%20studies\">user studies<\/a>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/variability\">variability<\/a>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/variance\">variance<\/a>, <a href=\"https:\/\/alandix.com\/glossary\/hcistats\/wilcoxon%20rank%20sum%20test\">Wilcoxon rank sum test<\/a><\/div>\n\n\n<!-- ?php\n\tif ( @$page['prolog_examples'] ):\n?>\n<h3> Prolog examples (from 1st ed.) <\/h3>\n<!-- ?php\n\techo $page['prolog_examples'];\n?>\n<!-- ?php\n\tendif; \/\/ @$page['prolog_examples']\n? -->\n\n<\/div>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":2,"featured_media":0,"parent":801,"menu_order":4,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_themeisle_gutenberg_block_has_review":false,"footnotes":""},"class_list":["post-805","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/alandix.com\/statistics\/wp-json\/wp\/v2\/pages\/805","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/alandix.com\/statistics\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/alandix.com\/statistics\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/alandix.com\/statistics\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/alandix.com\/statistics\/wp-json\/wp\/v2\/comments?post=805"}],"version-history":[{"count":2,"href":"https:\/\/alandix.com\/statistics\/wp-json\/wp\/v2\/pages\/805\/revisions"}],"predecessor-version":[{"id":837,"href":"https:\/\/alandix.com\/statistics\/wp-json\/wp\/v2\/pages\/805\/revisions\/837"}],"up":[{"embeddable":true,"href":"https:\/\/alandix.com\/statistics\/wp-json\/wp\/v2\/pages\/801"}],"wp:attachment":[{"href":"https:\/\/alandix.com\/statistics\/wp-json\/wp\/v2\/media?parent=805"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}