Statistics/Distributions/Chi-square

Chi-square distribution is related to normal distribution. A chi-square statistic is the sum of a number of independent and standard normal random variables.

Assume that we have n number of random variables Z, that are normally distributed. Therefore, we can write $Z\in N(0,1)$. If we square Z such that $Z^2$, then we get the chi-square distribution $Z^2\in {\chi }_1^2$. If we sum n number of ${\chi }_1^2$, we can write

$Y=Z_1^2+Z_2^2+...+Z_n^2\in {\chi }_n^2$.

One example could be that we want to know whether the weight of a set of eight apples is normally distributed. Chi-square distribution can be used to test for this. Assume that the apples weigh 88, 93, 110, 76, 78, 121, 92 and 86 grams. We obtain the normally distributed Z values by subtracting the average weight (93) and divide by the standard deviation (15.41). For example, the first apple has got $Z_1=\frac{88-93}{15.41}=-0.3245$ using four decimal points. Summing up for all values gives us Y = 7.0000.

Now when we have the value of the chi-square statistic Y, we compare it to the critical value of the chi-square distribution at 7 degrees of freedom and 95% level of significance, which is 14.1. The null hypothesis is that the sample of apples is normally distributed. It is rejected if the value of the test statistic is higher than the critical value. Because 7.0 is lower than 14.1, we do not reject the null hypothesis and conclude that the weights of the apples are normally distributed.