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<\/figure>\n\n\n\nWe can see that the distribution is “right skew”: it has a “tail” off to the right. The Shapiro-Wilk test (q.v.) is highly statistically significant showing that it’s vanishingly unlikely that this quite large sample arose from sampling from a population in which the values have a Gaussian distribution.<\/p>\n\n\n\n
In this situation, before bootstrapping (q.v.) rendered the preoccupation with Gaussian distributions and parametric versus<\/em> non-parametric tests less important, a researcher might have tried to find a transformation of the data that pulled them closer to a Gaussian distribution. She might have tried taking the log of the values (as logs shrink large numbers much more than they shrink small numbers so they reduce right skew). Here’s that log (to base e) transform of the raw data. I have added a best fit linear regression line in blue to emphasise that the transformation has reduced the higher values and increased the smaller values relative what a linear transform would have done.<\/p>\n\n\n\n How has that affected the distribution?<\/p>\n\n\n\n That has overdone it and now the distribution is slightly left skew and though the misfit according to the Shapiro-Wilk test has been reduced, this is still not nearly Gaussian. What about using a square root transformation (square roots, only, for our purposes applicable to positive numbers, also reduce large numbers more than smaller ones). Here is the transformation mapping this time mapping the square root transform rather than the log transform.<\/p>\n\n\n\n You can see that that has again, relatively, shrunk the larger values and increased teh smaller values but less so than the log transform did. So how has that changed the distribution?<\/p>\n\n\n\n Bingo! The distribution is now, very narrowly, not statistically significantly implausible to have arisen by sampling from a Gaussian population as the Shapiro-Wilk test p value is .0562, above the conventional “< .05” criterion.<\/p>\n\n\n\n Bootstrapping<\/a> A bit esoteric these days, not covered in the book.<\/p>\n\n\n\n None yet, low on my priorities!<\/p>\n\n\n\n First created 1.iii.24, tweaked links 21.vi.24.<\/p>\n","protected":false},"excerpt":{"rendered":" Technically this could be any transforming of a variable or multiple variables so it could include dichotomising a continuous variable into (say) high\/low or standardising where the aim is to make a lot of different variables all have the same mean and standard deviation. However, it is generally used for transformations that bring a variable … Continue reading Transforming data\/variable(s)<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","template":"","meta":{"footnotes":""},"doc_category":[18],"glossaries":[],"doc_tag":[],"knowledge_base":[],"class_list":["post-3931","docs","type-docs","status-publish","hentry","doc_category-om-book"],"year_month":"2026-04","word_count":506,"total_views":"952","reactions":{"happy":"0","normal":"0","sad":"0"},"author_info":{"name":"chris","author_nicename":"chris","author_url":"https:\/\/www.psyctc.org\/psyctc\/author\/chris\/"},"doc_category_info":[{"term_name":"All OM book glossary entries","term_url":"https:\/\/www.psyctc.org\/psyctc\/glossary\/non-knowledgebase\/om-book\/"}],"doc_tag_info":[],"knowledge_base_info":[],"knowledge_base_slug":[],"_links":{"self":[{"href":"https:\/\/www.psyctc.org\/psyctc\/wp-json\/wp\/v2\/docs\/3931","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.psyctc.org\/psyctc\/wp-json\/wp\/v2\/docs"}],"about":[{"href":"https:\/\/www.psyctc.org\/psyctc\/wp-json\/wp\/v2\/types\/docs"}],"author":[{"embeddable":true,"href":"https:\/\/www.psyctc.org\/psyctc\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.psyctc.org\/psyctc\/wp-json\/wp\/v2\/comments?post=3931"}],"version-history":[{"count":4,"href":"https:\/\/www.psyctc.org\/psyctc\/wp-json\/wp\/v2\/docs\/3931\/revisions"}],"predecessor-version":[{"id":4318,"href":"https:\/\/www.psyctc.org\/psyctc\/wp-json\/wp\/v2\/docs\/3931\/revisions\/4318"}],"wp:attachment":[{"href":"https:\/\/www.psyctc.org\/psyctc\/wp-json\/wp\/v2\/media?parent=3931"}],"wp:term":[{"taxonomy":"doc_category","embeddable":true,"href":"https:\/\/www.psyctc.org\/psyctc\/wp-json\/wp\/v2\/doc_category?post=3931"},{"taxonomy":"glossaries","embeddable":true,"href":"https:\/\/www.psyctc.org\/psyctc\/wp-json\/wp\/v2\/glossaries?post=3931"},{"taxonomy":"doc_tag","embeddable":true,"href":"https:\/\/www.psyctc.org\/psyctc\/wp-json\/wp\/v2\/doc_tag?post=3931"},{"taxonomy":"knowledge_base","embeddable":true,"href":"https:\/\/www.psyctc.org\/psyctc\/wp-json\/wp\/v2\/knowledge_base?post=3931"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}![\"\"](\"https:\/\/www.psyctc.org\/psyctc\/wp-content\/uploads\/2024\/03\/log_transform_plot-1024x1024.png\"){kind=spacer}
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<\/figure>\n\n\n\nTry also<\/h4>\n\n\n\n
Dichotomising
Distribution shape<\/a>
Gaussian distribution<\/a>
Null hypothesis testing (NHST)<\/a>
Parametric tests<\/a>
Shapiro-Wilk test<\/a>
Skewness<\/a>
Standardising\/normalising<\/a><\/p>\n\n\n\nChapters<\/h4>\n\n\n\n
Online resources<\/h4>\n\n\n\n
Dates<\/h4>\n\n\n\n