Statistics Minus the Maths

As noted in the main text, 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 \(\chi^2(1)\text{,}\) is simply the distribution of a single normal deviate squared. The area of the \(\chi^2(1)\) distribution below 4 is the same as the area of a standard normal distribution below 2, since 4 is \(2^2\text{.}\)

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.