Chi Square Distribution

Prerequisites

• Distributions, Standard Normal Distribution, Degrees of Freedom

Define the Chi Square distribution in terms of squared normal deviates

• 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 χ²(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 2².

Example

• 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) of being six or higher is 0.05

How does the shape of the Chi Square distribution change its degrees of freedom increase?

• 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 increase, the Chi Square Distribution approaches a normal distribution
• Notice how the skew decreases as the degrees of freedom increases.

Where can we use Chi Square distribution ?

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

Questions

1 Imagine that you sample 12 scores from a standard normal distribution, square each score, and sum the squares. How many degrees of freedom does the Chi Square distribution that corresponds to this sum have?

Answer >>

The degrees of freedom of the Chi Square distribution are equal to the number of standard normal deviates being summed (which is 12 in this case).

2 What is the mean of a Chi Square distribution with 8 degrees of freedom?

Answer >>

The mean of a Chi Square distribution is its degrees of freedom.

3 Which Chi Square distribution looks the most like a normal distribution?

	A Chi Square distribution with 0 df
	A Chi Square distribution with 1 df
	A Chi Square distribution with 2 df
	A Chi Square distribution with 10 df

Answer >>

As the degrees of freedom of a Chi Square distribution increase, the Chi Square distribution begins to look more and more like a normal distribution. Thus, out of these choices, a Chi Square distribution with 10 df would look the most similar to a normal distribution.

4 Imagine that you sample 3 scores from a standard normal distribution, square each score, and sum the squares. What is the probability that the sum of these 3 squares will be 9 or higher?

Answer >>

Because three scores are sampled, the answer can be found using the Chi Square distribution with three degrees of freedom. A Chi Square calculator can be used to find that the probability of a Chi Square (with 3 df) being 9 or higher is .0293.

Chi Square | One-Way Tables >