#set document(title: "17.1 Chi Square Distribution", author: "OpenStax")
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== 17.1#h(0.6em)Chi Square Distribution


    #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Prerequisites]


      Distributions,
                Standard Normal Distribution, Degrees of Freedom  #linebreak()  #linebreak()
]


    #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Learning Objectives]


      + Define the Chi Square distribution in terms of squared normal deviates
+ Describe how the shape of the Chi Square distribution changes as its  degrees
        of freedom increase
]


    A  standard normal deviate is a random sample from the standard 
    normal distribution. The Chi Square distribution is the distribution
               of the sum of squared standard normal deviates. The degrees
               of freedom of the distribution is equal to the number of standard
               normal deviates being summed. Therefore, Chi Square with one degree
               of freedom, written as χ#super[2] (1), is
               simply the distribution of a single normal deviate squared. The
               area of a Chi Square distribution below 4 is the same as the area
               of a standard normal distribution below 2, since 4 is 2#super[2].


    Consider the following problem: you sample two scores
               from a standard normal distribution, square each score, and sum
               the squares. What is the probability that the sum of these two
               squares will be six or higher? Since two scores are sampled, the
               answer can be found using the Chi Square distribution with two
               degrees of freedom. A Chi Square calculator can be used to find
               that the probability of a Chi Square (with 2 df)  being six
               or higher is 0.050.


    The mean of a Chi Square distribution is its degrees
               of freedom. Chi Square distributions are positively skewed, with
               the degree of skew decreasing with increasing degrees of freedom.
               As the degrees of freedom increases, the Chi Square distribution
               approaches a normal distribution. Figure 1 shows density functions
               for three Chi Square distributions. Notice how the skew decreases
               as the degrees of freedom increases.


    #figure(figph[Three stacked plots of Chi Square density against Chi Square (x-axis 0 to 16), labeled df = 2, df = 4 and df = 6. With df = 2 the curve starts at its maximum of 0.5 at zero and decays; with df = 4 it rises from 0 to a peak of about 0.18 near 2; with df = 6 it peaks at about 0.135 near 4. As the degrees of freedom grow the distribution shifts right and becomes less skewed.], alt: "Three stacked plots of Chi Square density against Chi Square (x-axis 0 to 16), labeled df = 2, df = 4 and df = 6. With df = 2 the curve starts at its maximum of 0.5 at zero and decays; with df = 4 it rises from 0 to a peak of about 0.18 near 2; with df = 6 it peaks at about 0.135 near 4. As the degrees of freedom grow the distribution shifts right and becomes less skewed.", caption: [Figure 1. Chi Square distributions with 2, 4, and 6 degrees of freedom.])


    The Chi Square distribution is very important because
               many test statistics are approximately distributed as Chi Square.
               Two of the more common tests using the Chi Square distribution
               are tests of deviations of differences between  theoretically
               expected and observed frequencies (one-way tables) and the relationship
               between categorical variables (contingency tables). Numerous other
               tests beyond the scope of this work are based on the Chi Square
               distribution.