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^{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^{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 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.