{"id":3309,"date":"2021-09-22T15:22:00","date_gmt":"2021-09-22T14:22:00","guid":{"rendered":"https:\/\/www.gironi.it\/blog\/?p=3309"},"modified":"2026-07-16T16:12:36","modified_gmt":"2026-07-16T15:12:36","slug":"statistical-parametric-and-non-parametric-tests","status":"publish","type":"post","link":"https:\/\/www.gironi.it\/blog\/en\/statistical-parametric-and-non-parametric-tests\/","title":{"rendered":"Statistical Parametric and Non-Parametric Tests"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Have you ever run an <a href=\"https:\/\/www.gironi.it\/blog\/en\/ab-testing-statistically-valid-experiments\/\" target=\"_blank\" rel=\"noopener noreferrer\">A\/B test<\/a> and wondered whether you should use a t-test or a Wilcoxon test? The choice matters: using the wrong test can cause you to miss a significant result \u2014 or, worse, make you think a result is significant when it\u2019s not.<\/p>\n\n\n<p class=\"wp-block-paragraph\"><a href=\"https:\/\/www.gironi.it\/blog\/en\/hypothesis-testing-a-step-by-step-guide\/\" target=\"_blank\" rel=\"noopener noreferrer\">Statistical tests<\/a> fall into two broad families: <strong>parametric<\/strong> and <strong>non-parametric<\/strong>. The fundamental difference? Parametric tests assume the data follow a known distribution (usually the normal); non-parametric tests make no such assumption. This is not a technical detail \u2014 it determines which tests you can use, how powerful they are, and how much you can trust the results.<\/p>\n\n\n<p class=\"wp-block-paragraph\">In this article we\u2019ll look at the differences, when to use each, and how the choice impacts the analysis of your site\u2019s data.<\/p>\n\n\n<!--more-->\n\n\n<p class=\"wp-block-paragraph\"><strong>What we\u2019ll cover<\/strong><\/p>\n<ul>\n<li><a href=\"#parametric\">Parametric tests: the power of normality<\/a><\/li>\n<li><a href=\"#non-parametric\">Non-parametric tests: flexibility and robustness<\/a><\/li>\n<li><a href=\"#comparison\">Comparison table<\/a><\/li>\n<li><a href=\"#try-it\">Try it yourself<\/a><\/li>\n<li><a href=\"#choosing\">Choosing the right test<\/a><\/li>\n<\/ul><\/p>\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n<h2 class=\"wp-block-heading\" id=\"parametric\">Parametric tests: the power of normality<\/h2>\n\n\n<p class=\"wp-block-paragraph\">Many <strong>parametric tests<\/strong> assume the data are an <strong>approximately normally distributed<\/strong>.<\/p>\n\n\n<p class=\"wp-block-paragraph\">The typical conditions for using them are:<\/p>\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Continuous or interval\/ratio<\/strong> data.\n<\/li>\n\n\n<li><strong>Approximately normal distribution<\/strong> of the data (or of sample means, thanks to the <a href=\"https:\/\/www.gironi.it\/blog\/en\/central-limit-theorem\/\" target=\"_blank\" rel=\"noopener noreferrer\">central limit theorem<\/a> for large samples).\n<\/li>\n\n\n<li><strong>Homogeneity of variances<\/strong> (<em>homoscedasticity<\/em>) across groups.\n<\/li>\n\n\n<li>A <strong>sufficiently large sample<\/strong>, when the model assumptions are reasonably met.\n<\/li>\n<\/ul>\n\n\n<p class=\"wp-block-paragraph\">The main advantage of parametric tests is <strong>higher statistical power<\/strong>: for the same effect size and sample size, a parametric test is more likely to detect a real difference. This is because they leverage the information about the data\u2019s distribution \u2014 provided the assumption holds.<\/p>\n\n\n<p class=\"wp-block-paragraph\"><strong>Examples of parametric tests<\/strong> we have covered include the <a href=\"https:\/\/www.gironi.it\/blog\/en\/the-normal-distribution\/#zscore\" target=\"_blank\" rel=\"noopener noreferrer\">Z-test<\/a>, <a href=\"https:\/\/www.gironi.it\/blog\/en\/the-t-distribution-and-hypothesis-testing\/\" target=\"_blank\" rel=\"noopener noreferrer\">Student\u2019s t-test<\/a>, <a href=\"https:\/\/www.gironi.it\/blog\/en\/analysis-of-variance-anova-explained-simply\/\" target=\"_blank\" rel=\"noopener noreferrer\">ANOVA<\/a> (Analysis of Variance), the <a href=\"https:\/\/www.gironi.it\/blog\/en\/correlation-and-regression-analysis-linear-regression\/#il-coefficiente-di-correlazione-r-di-pearson\" target=\"_blank\" rel=\"noopener noreferrer\">Pearson correlation coefficient r<\/a>, and <a href=\"https:\/\/www.gironi.it\/blog\/en\/correlation-and-regression-analysis-linear-regression\/\" target=\"_blank\" rel=\"noopener noreferrer\">linear regression<\/a>.<\/p>\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n<h2 class=\"wp-block-heading\" id=\"non-parametric\">Non-parametric tests: flexibility and robustness<\/h2>\n\n\n<p class=\"wp-block-paragraph\"><strong>Non-parametric tests<\/strong> (also called <em>distribution-free<\/em> tests) do not require assuming a particular distribution of the data nor estimating the parameters of such a distribution. They often work on <strong>ranks<\/strong> (the ordering of values) rather than on the values themselves.<\/p>\n\n\n<p class=\"wp-block-paragraph\">They fall into two broad categories:<\/p>\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Goodness-of-fit tests<\/strong>: compare observed frequencies with those expected under a hypothesis. The prime example is the <a href=\"https:\/\/www.gironi.it\/blog\/il-test-del-chi-quadrato-bonta-di-adattamento-e-test-di-indipendenza\/\" target=\"_blank\" rel=\"noopener noreferrer\">chi-square test<\/a>.\n<\/li>\n\n\n<li><strong>Non-parametric alternatives to parametric tests<\/strong>: tests that answer the same question without assuming normality. For example, the <a href=\"https:\/\/www.gironi.it\/blog\/en\/non-parametric-tests-the-wilcoxon-test-for-non-normal-data\/\" target=\"_blank\" rel=\"noopener noreferrer\">Wilcoxon test<\/a> is the alternative to the t-test for two samples, and <a href=\"https:\/\/www.gironi.it\/blog\/en\/correlation-and-regression-analysis-linear-regression\/#il-coefficiente-di-correlazione-per-ranghi-rho-di-spearman-e-un-accenno-al-tau-di-kendall\" target=\"_blank\" rel=\"noopener noreferrer\">Spearman\u2019s rho<\/a> is the non-parametric alternative to Pearson\u2019s correlation.\n<\/li>\n<\/ul>\n\n\n<p class=\"wp-block-paragraph\">Other examples include the Mann-Whitney U test (alternative to the t-test for two independent samples), the Kruskal-Wallis test (alternative to ANOVA), and Kendall\u2019s tau (alternative to Spearman for ties).<\/p>\n\n\n<p class=\"wp-block-paragraph\">The price of this versatility? Non-parametric tests generally have <strong>lower statistical power<\/strong> when the assumptions of parametric tests are met. They have lower <em>efficiency<\/em>: to achieve the same power, you need larger samples.<\/p>\n\n\n<div class=\"wp-block-group has-background\" style=\"background-color:#f5f7f9;margin-top:2.5rem;margin-bottom:2.5rem;padding-top:1.5rem;padding-right:1.5rem;padding-bottom:1rem;padding-left:1.5rem\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-container-core-group-is-layout-eed7543b wp-block-group-is-layout-constrained\">\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1050\" height=\"525\" src=\"https:\/\/www.gironi.it\/blog\/wp-content\/uploads\/2026\/07\/parametric-distribuzioni.png\" alt=\"Normal distribution (left) and skewed distribution (right): the shape of the data determines the choice between parametric and non-parametric tests.\" class=\"wp-image-4365\" srcset=\"https:\/\/www.gironi.it\/blog\/wp-content\/uploads\/2026\/07\/parametric-distribuzioni.png 1050w, https:\/\/www.gironi.it\/blog\/wp-content\/uploads\/2026\/07\/parametric-distribuzioni-300x150.png 300w, https:\/\/www.gironi.it\/blog\/wp-content\/uploads\/2026\/07\/parametric-distribuzioni-1024x512.png 1024w\" sizes=\"auto, (max-width: 709px) 85vw, (max-width: 909px) 67vw, (max-width: 1362px) 62vw, 840px\" \/><figcaption class=\"wp-element-caption\">Normal distribution (left) and skewed distribution (right): the shape of the data determines the choice between parametric and non-parametric tests.<\/figcaption><\/figure>\n\n<\/div><\/div>\n\n\n<div class=\"wp-block-group has-background\" style=\"background-color:#f5f7f9;margin-top:2.5rem;margin-bottom:2.5rem;padding-top:1.5rem;padding-right:1.5rem;padding-bottom:1rem;padding-left:1.5rem\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-container-core-group-is-layout-eed7543b wp-block-group-is-layout-constrained\">\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"825\" height=\"600\" src=\"https:\/\/www.gironi.it\/blog\/wp-content\/uploads\/2026\/07\/parametric-potenza.png\" alt=\"Statistical power of the t-test and the Wilcoxon test on normal and skewed data. The t-test loses less power when data deviate from normality.\" class=\"wp-image-4366\" srcset=\"https:\/\/www.gironi.it\/blog\/wp-content\/uploads\/2026\/07\/parametric-potenza.png 825w, https:\/\/www.gironi.it\/blog\/wp-content\/uploads\/2026\/07\/parametric-potenza-300x218.png 300w\" sizes=\"auto, (max-width: 709px) 85vw, (max-width: 909px) 67vw, (max-width: 984px) 61vw, (max-width: 1362px) 45vw, 600px\" \/><figcaption class=\"wp-element-caption\">Statistical power of the t-test and the Wilcoxon test on normal and skewed data. The t-test loses less power when data deviate from normality.<\/figcaption><\/figure>\n\n<\/div><\/div>\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n<h2 class=\"wp-block-heading\" id=\"comparison\">Comparison table<\/h2>\n\n\n<p class=\"wp-block-paragraph\">When to use each? Here is a summary:<\/p>\n\n\n<figure class=\"wp-block-table\"><table>\n<thead><tr><th>Feature<\/th><th>Parametric tests<\/th><th>Non-parametric tests<\/th><\/tr><\/thead>\n<tbody>\n<tr><td><strong>Assumed distribution<\/strong><\/td><td>Normal (approx.)<\/td><td>None<\/td><\/tr>\n<tr><td><strong>Data type<\/strong><\/td><td>Continuous, interval\/ratio<\/td><td>Any (including ordinal, ranks)<\/td><\/tr>\n<tr><td><strong>Statistical power<\/strong><\/td><td>Higher (if assumptions hold)<\/td><td>Lower (same sample size)<\/td><\/tr>\n<tr><td><strong>Robustness<\/strong><\/td><td>Sensitive to outliers, non-normality<\/td><td>Robust to outliers and skewness<\/td><\/tr>\n<tr><td><strong>Minimum sample<\/strong><\/td><td>Depends on model assumptions<\/td><td>Works even with small n<\/td><\/tr>\n<tr><td><strong>Output<\/strong><\/td><td>Parameter estimates + p-value<\/td><td>Rank-based test statistic<\/td><\/tr>\n<tr><td><strong>Examples<\/strong><\/td><td>t-test, ANOVA, Pearson, Z-test<\/td><td>Wilcoxon, Mann-Whitney, Kruskal-Wallis, Spearman<\/td><\/tr>\n<\/tbody><\/table>\n<figcaption class=\"wp-element-caption\">Quick comparison between parametric and non-parametric tests.<\/figcaption>\n<\/figure>\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n<h3 class=\"wp-block-heading\" id=\"try-it\">Try it yourself<\/h3>\n\n\n<p class=\"wp-block-paragraph\">An ecommerce site is testing two versions of a product page (A and B). Session times in seconds for 10 visitors each are:<\/p>\n\n\n<pre class=\"wp-block-code\"><code>version_A <- c(45, 52, 48, 61, 55, 49, 53, 47, 58, 51)\nversion_B <- c(42, 65, 44, 70, 50, 68, 55, 71, 49, 63)<\/code><\/pre>\n\n\n<p class=\"wp-block-paragraph\">Let\u2019s run both a t-test (parametric) and a Wilcoxon test (non-parametric) to see if the difference is significant:<\/p>\n\n\n<pre class=\"wp-block-code\"><code># Parametric test (assumes normality)\nt.test(version_A, version_B)\n\n# Non-parametric test (rank-based)\nwilcox.test(version_A, version_B)<\/code><\/pre>\n\n\n<p class=\"wp-block-paragraph\">Questions:<\/p>\n\n\n<p class=\"wp-block-paragraph\">1. <strong>Do the two tests give the same verdict (p &lt; 0.05)?<\/strong><\/p>\n\n\n<p class=\"wp-block-paragraph\">2. <strong>Which test makes more sense given only 10 data points per group?<\/strong><\/p>\n\n\n<p class=\"wp-block-paragraph\">3. <strong>How would the choice change with 100 visitors per version?<\/strong><\/p>\n\n\n<p class=\"wp-block-paragraph\"><strong>Answers:<\/strong><\/p>\n\n\n<p class=\"wp-block-paragraph\">1. With this particular data, the t-test gives p \u2248 0.12 (not significant), while Wilcoxon gives p \u2248 0.16 (not significant). They agree \u2014 but that\u2019s not always the case.<\/p>\n\n\n<p class=\"wp-block-paragraph\">2. With n = 10 it\u2019s hard to verify normality. Wilcoxon is the safer choice: it makes no distributional assumptions and works well with small samples.<\/p>\n\n\n<p class=\"wp-block-paragraph\">3. With n = 100, the <a href=\"https:\/\/www.gironi.it\/blog\/en\/central-limit-theorem\/\" target=\"_blank\" rel=\"noopener noreferrer\">central limit theorem<\/a> guarantees the sample mean is approximately normal even if the original data aren\u2019t. The t-test becomes more justified \u2014 and more powerful.<\/p>\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n<h3 class=\"wp-block-heading\" id=\"choosing\">Choosing the right test<\/h3>\n\n\n<p class=\"wp-block-paragraph\">The choice between a parametric and a non-parametric test does not depend on which is \"best\" in absolute terms, but on the <strong>nature of the data<\/strong> and the <strong>assumptions we are reasonably willing to make<\/strong>.<\/p>\n\n\n<p class=\"wp-block-paragraph\">The rule of thumb is:<\/p>\n\n\n<ul class=\"wp-block-list\">\n<li>If the data are <strong>normal<\/strong> (or the sample is large) and the variances are <strong>homogeneous<\/strong> \u2192 parametric test (more powerful).\n<\/li>\n\n\n<li>If the data are not normal, the sample is small, or there are <strong>outliers<\/strong> \u2192 non-parametric test (more robust).\n<\/li>\n\n\n<li>When in doubt, run <strong>both<\/strong>: if they agree, the answer is robust. If they disagree, investigate why.\n<\/li>\n<\/ul>\n\n\n<div class=\"wp-block-group has-background\" style=\"background-color:#f5f7f9;margin-top:2.5rem;margin-bottom:2.5rem;padding-top:1.5rem;padding-right:1.5rem;padding-bottom:1rem;padding-left:1.5rem\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-container-core-group-is-layout-eed7543b wp-block-group-is-layout-constrained\">\n<p><strong>Caution: power isn\u2019t everything.<\/strong><\/p>\n<p>A more powerful parametric test is not always the best choice. If the data violate the assumptions (normality, homoscedasticity), the parametric test\u2019s p-value can be unreliable \u2014 exposing you to a false positive or false negative. In such cases, a non-parametric test, though less powerful, gives a more honest result.<\/p>\n<p>For a practical guide to choosing between tests, see the <a href=\"https:\/\/www.gironi.it\/blog\/en\/guide-to-statistical-tests-for-a-b-analysis\/\" target=\"_blank\" rel=\"noopener noreferrer\">guide to statistical tests for A\/B analysis<\/a>, and for a systematic treatment of assumptions, <a href=\"https:\/\/www.gironi.it\/blog\/en\/effect-size-and-power-analysis-for-seo\/\" target=\"_blank\" rel=\"noopener noreferrer\">effect size and power analysis<\/a>.<\/p>\n<\/div><\/div>\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n<h3 class=\"wp-block-heading\">Further Reading<\/h3>\n\n\n<p class=\"wp-block-paragraph\">For a comprehensive treatment of the choice between parametric and non-parametric tests, with the applicability conditions of each, <a href=\"https:\/\/www.amazon.it\/dp\/8891910651?tag=consulenzeinf-21&#038;ascsubtag=statistical-parametric-and-non-parametric-tests\" rel=\"nofollow sponsored noopener\" target=\"_blank\"><em>Statistica<\/em><\/a> by Newbold, Carlson and Thorne offers the most systematic coverage available in Italian.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Have you ever run an A\/B test and wondered whether you should use a t-test or a Wilcoxon test? The choice matters: using the wrong test can cause you to miss a significant result \u2014 or, worse, make you think a result is significant when it\u2019s not. Statistical tests fall into two broad families: parametric &hellip; <a href=\"https:\/\/www.gironi.it\/blog\/en\/statistical-parametric-and-non-parametric-tests\/\" class=\"more-link\">Leggi tutto<span class=\"screen-reader-text\"> &#8220;Statistical Parametric and Non-Parametric Tests&#8221;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_uag_custom_page_level_css":"","footnotes":""},"categories":[161],"tags":[1232,1230],"class_list":["post-3309","post","type-post","status-publish","format-standard","hentry","category-statistics","tag-non-parametric-tests","tag-parametric-tests"],"lang":"en","translations":{"en":3309,"it":2306},"uagb_featured_image_src":{"full":false,"thumbnail":false,"medium":false,"medium_large":false,"large":false,"1536x1536":false,"2048x2048":false,"post-thumbnail":false},"uagb_author_info":{"display_name":"paolo","author_link":"https:\/\/www.gironi.it\/blog\/author\/paolo\/"},"uagb_comment_info":4,"uagb_excerpt":"Have you ever run an A\/B test and wondered whether you should use a t-test or a Wilcoxon test? The choice matters: using the wrong test can cause you to miss a significant result \u2014 or, worse, make you think a result is significant when it\u2019s not. Statistical tests fall into two broad families: parametric&hellip;","_links":{"self":[{"href":"https:\/\/www.gironi.it\/blog\/wp-json\/wp\/v2\/posts\/3309","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.gironi.it\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.gironi.it\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.gironi.it\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.gironi.it\/blog\/wp-json\/wp\/v2\/comments?post=3309"}],"version-history":[{"count":5,"href":"https:\/\/www.gironi.it\/blog\/wp-json\/wp\/v2\/posts\/3309\/revisions"}],"predecessor-version":[{"id":4370,"href":"https:\/\/www.gironi.it\/blog\/wp-json\/wp\/v2\/posts\/3309\/revisions\/4370"}],"wp:attachment":[{"href":"https:\/\/www.gironi.it\/blog\/wp-json\/wp\/v2\/media?parent=3309"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.gironi.it\/blog\/wp-json\/wp\/v2\/categories?post=3309"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.gironi.it\/blog\/wp-json\/wp\/v2\/tags?post=3309"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}