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The chi-square distribution is used for “interval estimation of population variance,” “goodness of fit test,” and “independence test.” Similar to the t-distribution, this distribution’s shape changes depending on the degrees of freedom.
In this study, we explain the chi-square distribution, which is often seen in research papers.
Chi-square distribution (χ2 distribution)
The chi-square distribution (χ2 distribution) is a type of probability distribution and one of the most widely used in statistical testing. It was discovered by Friedrich Robert Helmert (1843–1917, Germany) and was named Karl Pearson (1857–1936, UK).
The shape of the graph depends on the parameter value and degrees of freedom, f, and when f is large, it corresponds to the z-distribution (standard normal distribution). However, as f decreases, the peak of the graph shifts to the left compared with the z distribution.
A chi-square distribution with degrees of freedom f = 5, 10, and 20 is shown.
[Figure 1] Distribution of cases
Probabilities and percentiles of the chi-square distribution
The significance of the chi-square test is determined by calculating the upper probability and percentile of the chi-square distribution. (Figure 2)
Any value x on the horizontal axis of the chi-square distribution is called a “percentile.” The probability above this percentile is called the “upper probability.”
[Figure 2] How the chi-square test determines significant differences
The probability of a chi-square distribution can be easily calculated using Excel.
=CHIDIST(x, degrees of freedom)
We calculate the upper and lower probabilities in Figure 3.
[Figure 3]
Upper probability = CHIDIST(10, 5) = 0.0752
Lower probability = CHIDIST(2, 5) = 0.1509
Percentiles can also be easily calculated using Excel.
= CHIINV(upper probability and degrees of freedom)
We calculate the percentiles with an upper probability of 0.05 and a lower probability of 0.05.
[Figure 4] Percentiles with five degrees of freedom
Upper percentile = CHIINV(0.05, 5) = 11.07
Lower percentile = CHIINV(1-0.05, 5) = 1.15
A test that uses a chi-square distribution is called a “chi-square test.” The most representative tests are the “chi-square test of independence” and the “goodness-of-fit test.”
Next, we explain the F distribution, which is similar to the chi-square distribution widely used in statistical testing.
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