{"id":1907,"date":"2026-06-24T15:47:58","date_gmt":"2026-06-24T08:47:58","guid":{"rendered":"https:\/\/people.usd.ac.id\/~ydkristanto\/?page_id=1907"},"modified":"2026-06-24T16:42:34","modified_gmt":"2026-06-24T09:42:34","slug":"discovering-the-idea-behind-the-chi-squared-test","status":"publish","type":"page","link":"https:\/\/people.usd.ac.id\/~ydkristanto\/index.php\/learning-resources\/discovering-the-idea-behind-the-chi-squared-test\/","title":{"rendered":"Discovering the Idea Behind the Chi-Squared Test"},"content":{"rendered":"\n
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Imagine asking a group of children, “What makes you well liked by your classmates?” Would they say that good grades, sports, attractive appearance, or having lots of money are most important? Would girls and boys give different answers? Questions like these were investigated by a group of researchers at Michigan State University in the United States<\/a>. Part of their data is shown in Figure 1<\/a>.<\/p>\n\n\n\n

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Figure 1: Preferences of girls and boys regarding the factor they considered most important for being well liked by their classmates<\/figcaption><\/figure>\n\n\n\n

The data shown in Figure 1<\/a> are represented more clearly in Visualization 1<\/a>. In this visualization, each child is represented by a square, and the color of the square indicates that child’s preference. The 16 squares in the top row represent the preferences of girls, while the 16 squares in the bottom row represent the preferences of boys.<\/p>\n\n\n\n

Take another look at Figure 1<\/a>. Do the differences shown in the data suggest a relationship between gender and children’s preferences? Or could these differences have arisen by chance alone? In this interactive simulation, you will explore how the chi-squared test helps answer this question.<\/p>\n\n\n\n

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Could the Differences Be Due to Chance?<\/h3>\n\n\n\n

Let’s imagine that gender has no influence on children’s preferences. If that were true, what might the data look like? Click “Shuffle” to find out!<\/p>\n\n\n\n

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Visualization 1: Original data (adapted from Chase & Dummer (1992)<\/a>]<\/figcaption>\n<\/figure>\n<\/div>\n\n\n\n
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Visualization 2: Children’s preferences randomly reassigned (left), then sorted within each gender group (right)<\/figcaption>\n<\/figure>\n<\/div>\n<\/div>\n\n\n\n

The results of each shuffle in Visualization 2<\/a> are also summarized in Table 1<\/a>. The corresponding \u03c7<\/em>\u00b2 statistics from repeated shuffles are displayed in Visualization 3<\/a>.<\/p>\n\n\n\n\n \n \n \n \n \n
\n Table 1: Distribution of preferences by gender\n <\/caption>\n\n
Gender<\/th>\n Grades<\/th>\n Sports<\/th>\n Appearance<\/th>\n Money<\/th>\n <\/tr>\n <\/thead>\n\n
Girls<\/td>\n <\/td>\n <\/td>\n <\/td>\n <\/td>\n <\/tr>\n
Boys<\/td>\n <\/td>\n <\/td>\n <\/td>\n <\/td>\n <\/tr>\n <\/tbody>\n<\/table>\n<\/div><\/div>\n\n\n\n
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Note:<\/strong> In both Visualization 1<\/a> and Visualization 2<\/a>, you’ve seen the chi-squared statistic (\u03c7<\/em>\u00b2). Put simply, this statistic measures how different the observed pattern in the data is from the pattern we would expect if there were no relationship between the two variables.<\/p>\n<\/div><\/div>\n\n\n\n

Now examine the distribution of \u03c7<\/em>\u00b2 values in Visualization 3<\/a>. Remember that these \u03c7<\/em>\u00b2 values were generated under the assumption that gender has no influence on children’s preferences.<\/p>\n\n\n\n

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Visualization 3: Distribution of \u03c7<\/i>\u00b2 values from repeated shuffles<\/figcaption>\n<\/figure>\n\n