{"id":450,"date":"2023-03-23T10:20:40","date_gmt":"2023-03-23T10:20:40","guid":{"rendered":"https:\/\/www.editage.com\/blog\/?p=450"},"modified":"2026-06-04T09:25:32","modified_gmt":"2026-06-04T09:25:32","slug":"chi-square-test-types-explained-for-biomedical-researchers","status":"publish","type":"post","link":"https:\/\/www.editage.com\/blog\/chi-square-test-types-explained-for-biomedical-researchers\/","title":{"rendered":"What is a Chi-Square Test? Types, Formula &#038; Examples"},"content":{"rendered":"\n<p>The <strong>chi-square test<\/strong> (\u03c7\u00b2, pronounced &#8220;kai-square&#8221;) is specifically designed for <strong>categorical data<\/strong>: data that can be sorted into distinct groups or categories rather than measured on a numerical scale. Whether you want to check if your sample matches an expected distribution or investigate whether two categorical variables are related, the chi-square test provides a straightforward and robust method to answer these questions.<\/p>\n\n\n\n<p>In this article, we cover the definition, types, formula, assumptions, step-by-step examples, and reporting guidelines for chi-square tests.<\/p>\n\n\n\n<p><strong>Contents<\/strong><\/p>\n\n\n\n<ol type=\"1\"><li><a href=\"#_Toc231476712\">What is a Chi-Square Test?<\/a><\/li><li><a href=\"#_Toc231476713\">Types of Chi-Square Tests<\/a><\/li><li><a href=\"#_Toc231476714\">The Chi-Square Formula<\/a><\/li><li><a href=\"#_Toc231476715\">Assumptions of the Chi-Square Test<\/a><\/li><li><a href=\"#_Toc231476716\">Step-by-Step Examples<\/a><\/li><li><a href=\"#_Toc231476717\">The Chi-Square Distribution<\/a><\/li><li><a href=\"#_Toc231476718\">Chi-Square Critical Value Table (Selected Values)<\/a><\/li><li><a href=\"#_Toc231476719\">How to Perform a Chi-Square Test: Step-by-Step<\/a><\/li><li><a href=\"#_Toc231476720\">Chi-Square Test vs. Other Statistical Tests<\/a><\/li><li><a href=\"#_Toc231476721\">The Null Hypothesis in Chi-Square Tests<\/a><\/li><li><a href=\"#_Toc231476722\">Interpreting Chi-Square Results<\/a><\/li><li><a href=\"#_Toc231476723\">Chi-Square Test in Medical and Biomedical Research<\/a><\/li><li><a href=\"#_Toc231476724\">Yates&#8217; Correction for Continuity<\/a><\/li><li><a href=\"#_Toc231476725\">Limitations of the Chi-Square Test<\/a><\/li><li><a href=\"#_Toc231476726\">How to Report Chi-Square Results<\/a><\/li><li><a href=\"#_Toc231476727\">Frequently Asked Questions<\/a><\/li><li><a href=\"#_Toc231476728\">Summary<\/a><\/li><\/ol>\n\n\n\n<h2><a id=\"_Toc231476712\">What is a Chi-Square Test?<\/a><\/h2>\n\n\n\n<p>A chi-square test is a <strong><a href=\"https:\/\/www.editage.com\/insights\/an-introduction-to-non-parametric-tests-for-biomedical-researchers\">non-parametric<\/a> hypothesis test<\/strong> that uses the chi-square (\u03c7\u00b2) statistic to evaluate whether observed frequencies in categorical data differ significantly from expected frequencies. In other words, it tells you whether any differences between your groups are due to a real association or simply due to chance.<\/p>\n\n\n\n<p>The symbol \u03c7\u00b2 is the Greek letter chi squared. The test belongs to the family of <a href=\"https:\/\/www.editage.com\/insights\/an-introduction-to-non-parametric-tests-for-biomedical-researchers\">parametric and non-parametric tests<\/a>, but unlike <a href=\"https:\/\/www.editage.com\/insights\/what-biomedical-researchers-need-to-know-about-t-tests\">t-tests<\/a> or <a href=\"https:\/\/www.editage.com\/insights\/anova-testing-in-statistics\">ANOVA<\/a>, it does not require normally distributed data and works with count or frequency data.<\/p>\n\n\n\n<h3>Key Characteristics<\/h3>\n\n\n\n<ul><li>Works with categorical (nominal or ordinal) variables<\/li><li>Based on the comparison of observed versus expected frequencies<\/li><li>Does not require the assumption of normality<\/li><li>Results in a test statistic (\u03c7\u00b2) that follows a chi-square distribution<\/li><li>The p-value derived from this statistic is compared against a chosen significance level (e.g., \u03b1 = 0.05)<\/li><\/ul>\n\n\n\n<h2><a id=\"_Toc231476713\">Types of Chi-Square Tests<\/a><\/h2>\n\n\n\n<p>There are three main types of chi-square tests used in research. The appropriate test depends on your <a href=\"https:\/\/www.editage.com\/insights\/qualitative-quantitative-or-mixed-methods-a-quick-guide-to-choose-the-right-design-for-your-research\">study design<\/a> and the number of variables you are examining.<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td><strong>Test Type<\/strong><\/td><td><strong>Number of Variables<\/strong><\/td><td><strong>Purpose<\/strong><\/td><td><strong>Example<\/strong><\/td><\/tr><tr><td><strong>Goodness of Fit<\/strong><\/td><td>One categorical variable<\/td><td>Does observed distribution match expected distribution?<\/td><td>Are M&amp;M colors equally distributed in a bag?<\/td><\/tr><tr><td><strong>Test of Independence<\/strong><\/td><td>Two categorical variables<\/td><td>Are two variables related\/associated in a population?<\/td><td>Is smoking status related to lung disease diagnosis?<\/td><\/tr><tr><td><strong>Test of Homogeneity<\/strong><\/td><td>Two or more groups<\/td><td>Do two or more populations have the same distribution?<\/td><td>Do patients in two hospitals have the same blood type distribution?<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h3>1. Chi-Square Goodness of Fit Test<\/h3>\n\n\n\n<p>The goodness of fit test is used when you have <strong>one categorical variable<\/strong> and want to determine whether the observed distribution of that variable matches a hypothesized (expected) distribution. You are essentially asking: &#8220;Does my sample reflect what I would expect?&#8221;<\/p>\n\n\n\n<h4>When to use:<\/h4>\n\n\n\n<p>When comparing a single categorical variable against a theoretical or known distribution.<\/p>\n\n\n\n<h4>Example:<\/h4>\n\n\n\n<p>A die is rolled 60 times. You want to test whether each face appears equally (expected: 10 times each). The goodness of fit test tells you if the observed outcomes significantly deviate from the expected equal frequencies.<\/p>\n\n\n\n<h3>2. Chi-Square Test of Independence<\/h3>\n\n\n\n<p>This is the most commonly used chi-square test. It examines whether <strong>two categorical variables are independent<\/strong> of each other (i.e., not associated) in a single population. Data is typically arranged in a contingency table (cross-tabulation).<\/p>\n\n\n\n<h4>When to use:<\/h4>\n\n\n\n<p>When examining the relationship between two categorical variables measured on the same subjects.<\/p>\n\n\n\n<h4>Example:<\/h4>\n\n\n\n<p>Is there an association between gender (male\/female) and preference for a political party (Party A\/Party B\/Party C)? This test determines if gender and party preference are independent or related.<\/p>\n\n\n\n<h3>3. Chi-Square Test of Homogeneity<\/h3>\n\n\n\n<p>The test of homogeneity is used to compare the distribution of a categorical variable across <strong>two or more independent groups<\/strong>. While it uses the same formula as the test of independence, the key difference lies in the <a href=\"https:\/\/www.editage.com\/insights\/qualitative-quantitative-or-mixed-methods-a-quick-guide-to-choose-the-right-design-for-your-research\">study design<\/a>: in the test of homogeneity, you sample separately from each group.<\/p>\n\n\n\n<h4>When to use:<\/h4>\n\n\n\n<p>When comparing the distribution of a categorical outcome across multiple predefined groups or populations.<\/p>\n\n\n\n<h4>Example:<\/h4>\n\n\n\n<p>Do patients from three different hospitals have the same distribution of blood types (A, B, AB, O)?<\/p>\n\n\n\n<h3>Independence vs. Homogeneity: Key Difference<\/h3>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td><strong>Feature<\/strong><\/td><td><strong>Test of Independence<\/strong><\/td><td><strong>Test of Homogeneity<\/strong><\/td><\/tr><tr><td><strong>Sampling<\/strong><\/td><td>One sample from one population<\/td><td>Separate samples from multiple populations<\/td><\/tr><tr><td><strong>Question<\/strong><\/td><td>Are two variables associated?<\/td><td>Do groups have the same distribution?<\/td><\/tr><tr><td><strong>Formula<\/strong><\/td><td>Same \u03c7\u00b2 formula<\/td><td>Same \u03c7\u00b2 formula<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h2><a id=\"_Toc231476714\">The Chi-Square Formula<\/a><\/h2>\n\n\n\n<p>The chi-square statistic is calculated using the following formula:<\/p>\n\n\n\n<p><strong>\u03c7\u00b2 = \u03a3 [ (O \u2212 E)\u00b2 \/ E ]<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td><strong>Symbol<\/strong><\/td><td><strong>Meaning<\/strong><\/td><\/tr><tr><td><strong>\u03c7\u00b2<\/strong><\/td><td>Chi-square test statistic<\/td><\/tr><tr><td><strong>\u03a3<\/strong><\/td><td>Summation across all categories or cells<\/td><\/tr><tr><td><strong>O<\/strong><\/td><td>Observed frequency, the actual count recorded in each category<\/td><\/tr><tr><td><strong>E<\/strong><\/td><td>Expected frequency, the count expected under the <a href=\"https:\/\/www.editage.com\/insights\/the-null-hypothesis-what-researchers-often-get-wrong\">null hypothesis<\/a><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>The larger the difference between observed and expected frequencies, the larger the \u03c7\u00b2 statistic, and the more likely the result is statistically significant. This value is then compared to a critical value from the <strong>chi-square distribution<\/strong> based on degrees of freedom (df) and the chosen significance level (\u03b1).<\/p>\n\n\n\n<h3>Degrees of Freedom<\/h3>\n\n\n\n<p>Degrees of freedom determine which chi-square distribution to use when looking up critical values.<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td><strong>Test Type<\/strong><\/td><td><strong>Degrees of Freedom Formula<\/strong><\/td><td><strong>Notation<\/strong><\/td><\/tr><tr><td><strong>Goodness of Fit<\/strong><\/td><td>Number of categories \u2212 1<\/td><td>df = k \u2212 1<\/td><\/tr><tr><td><strong>Independence \/ Homogeneity<\/strong><\/td><td>(Rows \u2212 1) \u00d7 (Columns \u2212 1)<\/td><td>df = (r\u22121)(c\u22121)<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h3>Calculating Expected Frequencies<\/h3>\n\n\n\n<p>For the test of independence, expected frequencies for each cell in a contingency table are calculated as:<\/p>\n\n\n\n<p><strong>E = (Row Total \u00d7 Column Total) \/ Grand Total<\/strong><\/p>\n\n\n\n<p>For the goodness of fit test, expected frequencies are derived from the hypothesized proportions multiplied by the total sample size.<\/p>\n\n\n\n<h2><a id=\"_Toc231476715\">Assumptions of the Chi-Square Test<\/a><\/h2>\n\n\n\n<p>To ensure valid results, the following assumptions must be met before applying a chi-square test. Violations can lead to misleading conclusions, so checking these in your <a href=\"https:\/\/www.editage.com\/insights\/how-to-write-the-methods-section-of-a-research-paper\">methods section<\/a> is important:<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td><strong>Assumption<\/strong><\/td><td><strong>Details<\/strong><\/td><\/tr><tr><td><strong>Categorical data<\/strong><\/td><td>Both variables must be categorical (nominal or ordinal). The chi-square test is not appropriate for continuous numerical data.<\/td><\/tr><tr><td><strong>Independent observations<\/strong><\/td><td>Each observation must be independent, one subject should contribute to only one cell in the table.<\/td><\/tr><tr><td><strong>Mutually exclusive categories<\/strong><\/td><td>Each observation must belong to only one category. Categories cannot overlap.<\/td><\/tr><tr><td><strong>Expected frequency \u2265 5<\/strong><\/td><td>At least 80% of cells should have expected frequencies of 5 or more. If this is violated, consider combining categories or using Fisher&#8217;s Exact Test.<\/td><\/tr><tr><td><strong>Adequate sample size<\/strong><\/td><td>A sufficiently large sample is needed. Very small samples make the test unreliable. See guidance on sample size for details.<\/td><\/tr><tr><td><strong>Random sampling<\/strong><\/td><td>Data should be collected through random or representative sampling to allow valid inference to the population.<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p><strong>Note: <\/strong>When expected cell counts are less than 5, Yates&#8217; continuity correction (for 2\u00d72 tables) or Fisher&#8217;s Exact Test may be more appropriate.<\/p>\n\n\n\n<h2><a id=\"_Toc231476716\">Step-by-Step Examples<\/a><\/h2>\n\n\n\n<h3>Example 1: Chi-Square Goodness of Fit Test<\/h3>\n\n\n\n<p><strong>Research question: <\/strong>Is a six-sided die fair? After 60 rolls, do the observed frequencies match the expected equal distribution?<\/p>\n\n\n\n<p><strong>Null hypothesis (H\u2080): <\/strong>Each face has an equal probability of appearing (p = 1\/6 for each face).<\/p>\n\n\n\n<p><strong>Alternative hypothesis (H\u2081): <\/strong>The die is not fair; at least one face appears with a different frequency.<\/p>\n\n\n\n<p><strong>Significance level: <\/strong>\u03b1 = 0.05<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td><strong>Face<\/strong><\/td><td><strong>Observed (O)<\/strong><\/td><td><strong>Expected (E)<\/strong><\/td><td><strong>(O\u2212E)\u00b2<\/strong><\/td><td><strong>(O\u2212E)\u00b2\/E<\/strong><\/td><\/tr><tr><td>1<\/td><td>8<\/td><td>10<\/td><td>4<\/td><td>0.40<\/td><\/tr><tr><td>2<\/td><td>9<\/td><td>10<\/td><td>1<\/td><td>0.10<\/td><\/tr><tr><td>3<\/td><td>11<\/td><td>10<\/td><td>1<\/td><td>0.10<\/td><\/tr><tr><td>4<\/td><td>14<\/td><td>10<\/td><td>16<\/td><td>1.60<\/td><\/tr><tr><td>5<\/td><td>9<\/td><td>10<\/td><td>1<\/td><td>0.10<\/td><\/tr><tr><td>6<\/td><td>9<\/td><td>10<\/td><td>1<\/td><td>0.10<\/td><\/tr><tr><td><strong>Total<\/strong><\/td><td><strong>60<\/strong><\/td><td><strong>60<\/strong><\/td><td>&nbsp;<\/td><td><strong>\u03c7\u00b2 = 2.40<\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p><strong>Degrees of freedom: <\/strong>df = 6 \u2212 1 = 5<\/p>\n\n\n\n<p><strong>Critical value (\u03c7\u00b2 at df=5, \u03b1=0.05): <\/strong>11.07<\/p>\n\n\n\n<p><strong>Result: <\/strong>\u03c7\u00b2 = 2.40 &lt; 11.07 \u2192 Fail to reject H\u2080. There is no significant evidence that the die is unfair.<\/p>\n\n\n\n<h3>Example 2: Chi-Square Test of Independence<\/h3>\n\n\n\n<p><strong>Research question: <\/strong>Is there an association between a new drug treatment and recovery outcome in 200 patients?<\/p>\n\n\n\n<p><strong>Null hypothesis (H\u2080): <\/strong>Treatment type and recovery outcome are independent.<\/p>\n\n\n\n<p><strong>Alternative hypothesis (H\u2081): <\/strong>Treatment type and recovery outcome are associated.<\/p>\n\n\n\n<p><strong>Observed Contingency Table:<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td>&nbsp;<\/td><td><strong>Recovered<\/strong><\/td><td><strong>Not Recovered<\/strong><\/td><td><strong>Row Total<\/strong><\/td><\/tr><tr><td><strong>Drug Treatment<\/strong><\/td><td>90<\/td><td>10<\/td><td><strong>100<\/strong><\/td><\/tr><tr><td><strong>Placebo<\/strong><\/td><td>60<\/td><td>40<\/td><td><strong>100<\/strong><\/td><\/tr><tr><td><strong>Column Total<\/strong><\/td><td><strong>150<\/strong><\/td><td><strong>50<\/strong><\/td><td><strong>200<\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p><strong>Expected Frequencies (E = Row Total \u00d7 Column Total \/ Grand Total):<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td>&nbsp;<\/td><td><strong>Recovered<\/strong><\/td><td><strong>Not Recovered<\/strong><\/td><td><strong>(O\u2212E)\u00b2\/E<\/strong><\/td><\/tr><tr><td><strong>Drug Treatment<\/strong><\/td><td>E = 75<\/td><td>E = 25<\/td><td>3.0 + 9.0 = 12.0<\/td><\/tr><tr><td><strong>Placebo<\/strong><\/td><td>E = 75<\/td><td>E = 25<\/td><td>3.0 + 9.0 = 12.0<\/td><\/tr><tr><td><strong>\u03c7\u00b2 Total<\/strong><\/td><td>&nbsp;<\/td><td>&nbsp;<\/td><td><strong>\u03c7\u00b2 = 24.0<\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p><strong>Degrees of freedom: <\/strong>df = (2\u22121)(2\u22121) = 1<\/p>\n\n\n\n<p><strong>Critical value (\u03c7\u00b2 at df=1, \u03b1=0.05): <\/strong>3.84<\/p>\n\n\n\n<p><strong>Result: <\/strong>\u03c7\u00b2 = 24.0 &gt;&gt; 3.84 \u2192 Reject H\u2080. There is a significant association between treatment type and recovery outcome (p &lt; 0.05).<\/p>\n\n\n\n<h2><a id=\"_Toc231476717\">The Chi-Square Distribution<\/a><\/h2>\n\n\n\n<p>The chi-square distribution is a continuous probability distribution that arises when independent standard normal variables are squared and summed. It is <strong>right-skewed<\/strong> and takes only non-negative values. Key properties:<\/p>\n\n\n\n<ul><li>The shape depends on the degrees of freedom (df)<\/li><li>With small df, the distribution is very right-skewed; it becomes more symmetric as df increases<\/li><li>The mean of a chi-square distribution equals its degrees of freedom<\/li><li>As df increases, the distribution approaches a normal distribution<\/li><\/ul>\n\n\n\n<p>A larger \u03c7\u00b2 statistic indicates a greater discrepancy between observed and expected values. To determine statistical significance, the computed \u03c7\u00b2 value is compared against the critical value from a chi-square table at the relevant df and \u03b1 level.<\/p>\n\n\n\n<h2><a id=\"_Toc231476718\">Chi-Square Critical Value Table (Selected Values)<\/a><\/h2>\n\n\n\n<p>The table below shows commonly used critical values for the chi-square distribution at \u03b1 = 0.05 and \u03b1 = 0.01:<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td><strong>Degrees of Freedom (df)<\/strong><\/td><td><strong>\u03b1 = 0.10<\/strong><\/td><td><strong>\u03b1 = 0.05<\/strong><\/td><td><strong>\u03b1 = 0.01<\/strong><\/td><\/tr><tr><td>1<\/td><td>2.706<\/td><td>3.841<\/td><td>6.635<\/td><\/tr><tr><td>2<\/td><td>4.605<\/td><td>5.991<\/td><td>9.210<\/td><\/tr><tr><td>3<\/td><td>6.251<\/td><td>7.815<\/td><td>11.345<\/td><\/tr><tr><td>4<\/td><td>7.779<\/td><td>9.488<\/td><td>13.277<\/td><\/tr><tr><td>5<\/td><td>9.236<\/td><td>11.070<\/td><td>15.086<\/td><\/tr><tr><td>6<\/td><td>10.645<\/td><td>12.592<\/td><td>16.812<\/td><\/tr><tr><td>8<\/td><td>13.362<\/td><td>15.507<\/td><td>20.090<\/td><\/tr><tr><td>10<\/td><td>15.987<\/td><td>18.307<\/td><td>23.209<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h2><a id=\"_Toc231476719\">How to Perform a Chi-Square Test: Step-by-Step<\/a><\/h2>\n\n\n\n<p>Whether you are using a chi-square goodness of fit test or a test of independence, the general process follows these steps. It is advisable to <a href=\"https:\/\/www.editage.com\/services\/publishing-services-packs\/statistical-analysis\">consult a biostatistician<\/a> if you are unsure which test applies to your data:<\/p>\n\n\n\n<ol type=\"1\"><li><strong>Define your hypotheses: <\/strong>State the null hypothesis (H\u2080) and alternative hypothesis (H\u2081) clearly before collecting data.<\/li><li><strong>Choose your significance level (\u03b1): <\/strong>Typically \u03b1 = 0.05, though \u03b1 = 0.01 or 0.10 may be used in specific contexts.<\/li><li><strong>Collect and organize your data: <\/strong>Create a frequency table or contingency table for your categorical variables.<\/li><li><strong>Calculate expected frequencies: <\/strong>Use the formula E = (Row Total \u00d7 Column Total) \/ Grand Total for the test of independence, or E = n \u00d7 p for goodness of fit.<\/li><li><strong>Check assumptions: <\/strong>Confirm all expected cell frequencies are \u2265 5 and observations are independent.<\/li><li><strong>Compute the \u03c7\u00b2 statistic: <\/strong>Apply \u03c7\u00b2 = \u03a3 [(O \u2212 E)\u00b2 \/ E] across all categories or cells.<\/li><li><strong>Determine degrees of freedom: <\/strong>df = k\u22121 (goodness of fit) or df = (r\u22121)(c\u22121) (independence\/homogeneity).<\/li><li><strong>Find the critical value or p-value: <\/strong>Compare your \u03c7\u00b2 statistic to the critical value from the chi-square table, or obtain a <a href=\"https:\/\/www.editage.com\/insights\/correct-way-report-p-values\">p-value<\/a> using statistical software.<\/li><li><strong>Draw your conclusion: <\/strong>If \u03c7\u00b2 &gt; critical value (or p &lt; \u03b1), reject H\u2080. Otherwise, fail to reject H\u2080.<\/li><\/ol>\n\n\n\n<h2><a id=\"_Toc231476720\">Chi-Square Test vs. Other Statistical Tests<\/a><\/h2>\n\n\n\n<p>Choosing the right test is important. The following comparison helps distinguish the chi-square test from other commonly used tests. For further guidance, see the overview of <a href=\"https:\/\/www.editage.com\/insights\/3-simple-steps-to-help-you-pick-the-right-statistical-test\">choosing the right statistical test<\/a>:<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td><strong>Test<\/strong><\/td><td><strong>Data Type<\/strong><\/td><td><strong>Groups<\/strong><\/td><td><strong>Purpose<\/strong><\/td><td><strong>Key Assumption<\/strong><\/td><\/tr><tr><td><strong>Chi-square<\/strong><\/td><td>Categorical<\/td><td>2+<\/td><td>Association \/ distribution<\/td><td>Expected freq \u2265 5<\/td><\/tr><tr><td><strong>T-test<\/strong><\/td><td>Continuous<\/td><td>1 or 2<\/td><td>Compare means<\/td><td>Normality, equal variance<\/td><\/tr><tr><td><strong>ANOVA<\/strong><\/td><td>Continuous<\/td><td>3+<\/td><td>Compare means<\/td><td>Normality, homogeneity of variance<\/td><\/tr><tr><td><strong>Fisher&#8217;s Exact<\/strong><\/td><td>Categorical<\/td><td>2<\/td><td>Association (small samples)<\/td><td>Small samples (2\u00d72 table)<\/td><\/tr><tr><td><strong>McNemar&#8217;s<\/strong><\/td><td>Categorical<\/td><td>2<\/td><td>Paired categorical data<\/td><td>Matched pairs<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>For continuous data with two groups, consider a <a href=\"https:\/\/www.editage.com\/insights\/what-biomedical-researchers-need-to-know-about-t-tests\">t-test<\/a>. For three or more groups, <a href=\"https:\/\/www.editage.com\/insights\/anova-testing-in-statistics\">ANOVA<\/a> is preferred. For correlation between continuous variables, see <a href=\"https:\/\/www.editage.com\/insights\/5-things-biomedical-researchers-need-to-know-about-correlation-analysis\">correlation analysis<\/a> or <a href=\"https:\/\/www.editage.com\/insights\/choosing-the-right-regression-method-a-handy-guide-for-biomedical-researchers\">regression<\/a>.<\/p>\n\n\n\n<h2><a id=\"_Toc231476721\">The Null Hypothesis in Chi-Square Tests<\/a><\/h2>\n\n\n\n<p>Every chi-square test begins with a clearly stated <a href=\"https:\/\/www.editage.com\/insights\/the-null-hypothesis-what-researchers-often-get-wrong\">null hypothesis<\/a>. Understanding what the null hypothesis means is crucial to interpreting your results correctly.<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td><strong>Test Type<\/strong><\/td><td><strong>Null Hypothesis (H\u2080)<\/strong><\/td><td><strong>Alternative Hypothesis (H\u2081)<\/strong><\/td><\/tr><tr><td><strong>Goodness of Fit<\/strong><\/td><td>The observed distribution matches the expected distribution<\/td><td>The observed distribution does not match the expected distribution<\/td><\/tr><tr><td><strong>Test of Independence<\/strong><\/td><td>The two variables are independent (no association)<\/td><td>The two variables are not independent (there is an association)<\/td><\/tr><tr><td><strong>Test of Homogeneity<\/strong><\/td><td>The distribution is the same across all groups<\/td><td>The distribution differs across at least one group<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>Rejecting H\u2080 does not tell you the magnitude or direction of the association \u2014 only that a statistically significant difference or relationship exists. Consider calculating effect sizes (such as Cram\u00e9r&#8217;s V or phi) to quantify the practical significance of your findings.<\/p>\n\n\n\n<h2><a id=\"_Toc231476722\">Interpreting Chi-Square Results<\/a><\/h2>\n\n\n\n<h3>The p-Value<\/h3>\n\n\n\n<p>The <a href=\"https:\/\/www.editage.com\/insights\/correct-way-report-p-values\">p-value<\/a> tells you the probability of obtaining a test statistic as extreme as yours, assuming the null hypothesis is true.<\/p>\n\n\n\n<ul><li>p &lt; \u03b1 (typically 0.05): Reject H\u2080 &nbsp;(statistically significant result)<\/li><li>p \u2265 \u03b1: Fail to reject H\u2080 (not statistically significant)<\/li><\/ul>\n\n\n\n<h3>Effect Size: Cram\u00e9r&#8217;s V<\/h3>\n\n\n\n<p>Statistical significance alone does not convey the strength of an association. Cram\u00e9r&#8217;s V is the most commonly used effect size measure for chi-square tests:<\/p>\n\n\n\n<p><strong>V = \u221a[ \u03c7\u00b2 \/ (n \u00d7 (min(r,c) \u2212 1)) ]<\/strong><\/p>\n\n\n\n<p>Where n is the total sample size, r is the number of rows, and c is the number of columns.<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td><strong>Cram\u00e9r&#8217;s V Value<\/strong><\/td><td><strong>Interpretation<\/strong><\/td><\/tr><tr><td>0.10 \u2013 0.19<\/td><td>Small effect<\/td><\/tr><tr><td>0.20 \u2013 0.29<\/td><td>Medium effect<\/td><\/tr><tr><td>\u2265 0.30<\/td><td>Large effect<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h3>Confidence Intervals<\/h3>\n\n\n\n<p>While the chi-square test provides a p-value, reporting a <a href=\"https:\/\/www.editage.com\/blog\/what-is-confidence-intervals-and-why-is-it-important\/\">confidence interval<\/a> (e.g., for proportions or odds ratios derived from the contingency table) provides additional context about the precision of the estimate.<\/p>\n\n\n\n<h2><a id=\"_Toc231476723\">Chi-Square Test in Medical and Biomedical Research<\/a><\/h2>\n\n\n\n<p>The chi-square test is among the most frequently reported statistical tests in <a href=\"https:\/\/www.editage.com\/insights\/a-young-researchers-guide-to-a-clinical-trial\">clinical trials<\/a> and biomedical publications. Common applications include:<\/p>\n\n\n\n<ul><li>Comparing treatment outcomes (recovered vs. not recovered) across two patient groups<\/li><li>Assessing whether adverse events are equally distributed among drug dosage groups<\/li><li>Examining the association between a risk factor (e.g., smoking) and a disease outcome (e.g., lung cancer)<\/li><li>Analyzing baseline characteristics in randomized controlled trials to confirm comparability of groups<\/li><li>Evaluating screening test performance in terms of sensitivity and specificity categories<\/li><\/ul>\n\n\n\n<p><strong>&nbsp;Important note: <\/strong>In <a href=\"https:\/\/www.editage.com\/insights\/cross-sectional-studies-overview-applications-advantages-and-challenges\">cross-sectional studies<\/a> and case-control studies, chi-square tests are commonly used to examine associations. However, they cannot establish causality, only association.<\/p>\n\n\n\n<h2><a id=\"_Toc231476724\">Yates&#8217; Correction for Continuity<\/a><\/h2>\n\n\n\n<p>For 2\u00d72 contingency tables, particularly when sample sizes are small, Yates&#8217; continuity correction is applied to reduce the overestimation of statistical significance:<\/p>\n\n\n\n<p><strong>\u03c7\u00b2(Yates) = \u03a3 [ (|O \u2212 E| \u2212 0.5)\u00b2 \/ E ]<\/strong><\/p>\n\n\n\n<p>Yates&#8217; correction makes the test more conservative, reducing the risk of Type I errors (false positives). When the expected cell count in any cell of a 2\u00d72 table is less than 5, Fisher&#8217;s Exact Test is generally preferred over chi-square with or without correction.<\/p>\n\n\n\n<h2><a id=\"_Toc231476725\">Limitations of the Chi-Square Test<\/a><\/h2>\n\n\n\n<p>While the chi-square test is widely applicable, it has limitations researchers should be aware of:<\/p>\n\n\n\n<ul><li>Cannot establish causality; it only tests association or distribution fit<\/li><li>Sensitive to sample size: very large samples can produce statistically significant chi-square values even for trivial associations<\/li><li>Requires adequate expected cell frequencies: cells with E &lt; 5 can produce unreliable results<\/li><li>Does not indicate the direction or strength of an association without supplementary measures (e.g., Cram\u00e9r&#8217;s V, odds ratio)<\/li><li>Not suitable for continuous data but only categorical variables<\/li><li>Affected by outliers in the sense that unusual distributions in one cell can dominate the overall chi-square value<\/li><\/ul>\n\n\n\n<p>&nbsp;When small expected frequencies are a concern, consider Fisher&#8217;s Exact Test (2\u00d72 tables) or collecting more data. For count data more generally, reviewing <a href=\"https:\/\/www.editage.com\/insights\/5-popular-statistical-tests-for-count-data\">statistical tests for count data<\/a> may be helpful.<\/p>\n\n\n\n<h2><a id=\"_Toc231476726\">How to Report Chi-Square Results<\/a><\/h2>\n\n\n\n<p>In academic manuscripts, chi-square results should be reported in the <a href=\"https:\/\/www.editage.com\/insights\/how-to-write-the-results-section\">results section<\/a> with all relevant statistics. Standard reporting format:<\/p>\n\n\n\n<p><strong>\u03c7\u00b2(df) = [value], p = [value], N = [sample size]<\/strong><\/p>\n\n\n\n<p><strong>Example: <\/strong>&#8220;There was a significant association between treatment type and recovery outcome, \u03c7\u00b2(1) = 24.0, p &lt; .001, N = 200, Cram\u00e9r&#8217;s V = 0.35.&#8221;<\/p>\n\n\n\n<p>Elements to include in your report:<\/p>\n\n\n\n<ul><li>The chi-square statistic (\u03c7\u00b2)<\/li><li>Degrees of freedom in parentheses<\/li><li>The exact p-value (use p &lt; .001 when appropriate)<\/li><li>Sample size (N)<\/li><li>Effect size measure (e.g., Cram\u00e9r&#8217;s V, phi coefficient)<\/li><li>A frequency or contingency table in the results<\/li><\/ul>\n\n\n\n<p>&nbsp;When reporting, follow the guidelines of your target journal (APA, Vancouver, etc.). For guidance on <a href=\"https:\/\/www.editage.com\/insights\/correct-way-report-p-values\">reporting p-values correctly<\/a> and using <a href=\"https:\/\/www.editage.com\/blog\/what-are-descriptive-statistics-types-choosing-reporting\/\">descriptive statistics<\/a> to contextualize your findings, refer to established resources.<\/p>\n\n\n\n<h2><a id=\"_Toc231476727\">Frequently Asked Questions<\/a><\/h2>\n\n\n\n<h3>When should I use a chi-square test instead of a t-test?<\/h3>\n\n\n\n<p>Use a chi-square test when your outcome variable is categorical (e.g., yes\/no, blood type, disease status). Use a <a href=\"https:\/\/www.editage.com\/insights\/what-biomedical-researchers-need-to-know-about-t-tests\">t-test<\/a> when your outcome variable is continuous (e.g., height, blood pressure) and you are comparing means between two groups.<\/p>\n\n\n\n<h3>Can I use a chi-square test for ordinal data?<\/h3>\n\n\n\n<p>Technically yes, but the chi-square test ignores the ordering of ordinal categories. Other tests (such as the Cochran-Armitage trend test or Spearman&#8217;s rank correlation) may be more appropriate for ordered categorical data.<\/p>\n\n\n\n<h3>What if my expected cell count is less than 5?<\/h3>\n\n\n\n<p>If any expected frequency falls below 5, the chi-square approximation may not be valid. Consider: (1) merging categories to increase expected counts, (2) using Fisher&#8217;s Exact Test for 2\u00d72 tables, or (3) collecting more data.<\/p>\n\n\n\n<h3>What is the difference between the chi-square test and a z-test for proportions?<\/h3>\n\n\n\n<p>For a 2\u00d72 contingency table, the chi-square test of independence and the two-proportion z-test are mathematically equivalent (\u03c7\u00b2 = z\u00b2). The chi-square test is more general and applicable to tables larger than 2\u00d72.<\/p>\n\n\n\n<h3>Does a significant chi-square result prove causation?<\/h3>\n\n\n\n<p>No. A chi-square test can identify a statistically significant association but cannot establish causality. Causal inference requires careful <a href=\"https:\/\/www.editage.com\/insights\/qualitative-quantitative-or-mixed-methods-a-quick-guide-to-choose-the-right-design-for-your-research\">study design<\/a>, control for confounders, and often prospective or experimental methods.<\/p>\n\n\n\n<h3>How do I report chi-square in APA style?<\/h3>\n\n\n\n<p>In <a href=\"https:\/\/www.editage.com\/insights\/cheat-sheet-american-psychological-association-manual-of-style\">APA style<\/a>, report as: \u03c7\u00b2(df, N = sample size) = chi-square value, p = p-value. Example: \u03c7\u00b2(2, N = 150) = 8.45, p = .015.<\/p>\n\n\n\n<h2><a id=\"_Toc231476728\">Summary<\/a><\/h2>\n\n\n\n<p>The chi-square test is an essential tool for researchers working with categorical data. Here is a quick reference summary:<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td><strong>Feature<\/strong><\/td><td><strong>Details<\/strong><\/td><\/tr><tr><td><strong>Type of data<\/strong><\/td><td>Categorical (nominal or ordinal)<\/td><\/tr><tr><td><strong>Main tests<\/strong><\/td><td>Goodness of fit, Test of independence, Test of homogeneity<\/td><\/tr><tr><td><strong>Formula<\/strong><\/td><td>\u03c7\u00b2 = \u03a3 [(O \u2212 E)\u00b2 \/ E]<\/td><\/tr><tr><td><strong>Degrees of freedom<\/strong><\/td><td>k\u22121 (goodness of fit); (r\u22121)(c\u22121) (independence\/homogeneity)<\/td><\/tr><tr><td><strong>Key assumption<\/strong><\/td><td>Expected frequency \u2265 5 in each cell<\/td><\/tr><tr><td><strong>Effect size<\/strong><\/td><td>Cram\u00e9r&#8217;s V, phi coefficient<\/td><\/tr><tr><td><strong>Alternative (small samples)<\/strong><\/td><td>Fisher&#8217;s Exact Test (2\u00d72 tables)<\/td><\/tr><tr><td><strong>Software<\/strong><\/td><td>SPSS, R, SAS, Stata, Python (scipy.stats.chi2_contingency)<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>For additional support with your statistical analysis or manuscript preparation, consider working with <a href=\"https:\/\/www.editage.com\/services\/publishing-services-packs\/statistical-analysis\">expert statistical consultants<\/a> who can help you choose the right test, interpret results accurately, and present findings clearly in your paper.<\/p>\n\n\n\n<p><em>This article was originally published on March 3, 2023, and updated on June 4, 2026.<\/em><\/p>\n","protected":false},"excerpt":{"rendered":"Biomedical researchers often handle categorical or nominal data in their analyses (i.e., data with labels rather than numerical values, such as race, smoking status, and sex). Since categorical data cannot be summarized using means or medians, the measure of central tendency of categorical variables is the mode. Because categorical variables can only have a few specific values, such data does not follow a normal distribution. Hence, categorical data cannot be analyzed with commonly used tests like ANOVA or Pearson's correlation, because these tests require continuous data. Instead, categorical data is often analyzed using the chi-square test.","protected":false},"author":2,"featured_media":451,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":[],"categories":[14],"tags":[23,24],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v20.6 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>What Is Chi-Square Test: Definition and Types of Chi-Square Tests | Editage<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.editage.com\/blog\/chi-square-test-types-explained-for-biomedical-researchers\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"What Is Chi-Square Test: Definition and Types of Chi-Square Tests | Editage\" \/>\n<meta property=\"og:description\" content=\"Biomedical researchers often handle categorical or nominal data in their analyses (i.e., data with labels rather than numerical values, such as race, smoking status, and sex). 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