Chi-Square Distribution

Suppose we conduct the following statistical experiment. We select a random sample of size n from a normal population, having a standard deviation equal to σ. We find that the standard deviation in our sample is equal to s. Given these data, we can define a statistic, called chi-square, using the following equation:

Χ² = [ ( n - 1 ) * s² ] / σ²

If we repeated this experiment an infinite number of times, we could obtain a sampling distribution for the chi-square statistic. The chi-square distribution is defined by the following probability density function:

Y = Y₀ * ( Χ² ) ^{( v/2 - 1 )} * e^{-Χ2 / 2}

where Y₀ is a constant that depends on the number of degrees of freedom, Χ² is the chi-square statistic, v = n - 1 is the number of degrees of freedom, and e is a constant equal to the base of the natural logarithm system (approximately 2.71828). Y₀ is defined, so that the area under the chi-square curve is equal to one.

In the figure below, the red curve shows the distribution of chi-square values computed from all possible samples of size 3, where degrees of freedom is n - 1 = 3 - 1 = 2. Similarly, the the green curve shows the distribution for samples of size 5 (degrees of freedom equal to 4); and the blue curve, for samples of size 11 (degrees of freedom equal to 10).

The chi-square distribution has the following properties:

• The mean of the distribution is equal to the number of degrees of freedom: μ = v.
• The variance is equal to two times the number of degrees of freedom: σ² = 2 * v
• When the degrees of freedom are greater than or equal to 2, the maximum value for Y occurs when Χ² = v - 2.
• As the degrees of freedom increase, the chi-square curve approaches a normal distribution.

Cumulative Probability and the Chi-Square Distribution

The chi-square distribution is constructed so that the total area under the curve is equal to 1. The area under the curve between 0 and a particular chi-square value is a cumulative probability associated with that chi-square value. For example, in the figure below, the shaded area represents a cumulative probability associated with a chi-square statistic equal to A; that is, it is the probability that the value of a chi-square statistic will fall between 0 and A.

Chi-Square Distribution Calculator

Test Your Understanding of This Lesson

Problem 1

The Acme Battery Company has developed a new cell phone battery. On average, the battery lasts 60 minutes on a single charge. The standard deviation is 4 minutes.

Suppose the manufacturing department runs a quality control test. They randomly select 7 batteries. The standard deviation of the selected batteries is 6 minutes. What would be the chi-square statistic represented by this test?

Solution

We know the following:

• The standard deviation of the population is 4 minutes.
• The standard deviation of the sample is 6 minutes.
• The number of sample observations is 7.

To compute the chi-square statistic, we plug these data in the chi-square equation, as shown below.

Χ² = [ ( n - 1 ) * s² ] / σ²
Χ² = [ ( 7 - 1 ) * 6² ] / 4² = 13.5

where Χ² is the chi-square statistic, n is the sample size, s is the standard deviation of the sample, and σ is the standard deviation of the population.

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Problem 2
Let's revisit the problem presented above. The manufacturing department ran a quality control test, using 7 randomly selected batteries. In their test, the standard deviation was 6 minutes, which equated to a chi-square statistic of 13.5.

Suppose they repeated the test with a new random sample of 7 batteries. What is the probability that the standard deviation in the new test would be greater than 6 minutes?

Solution

We know the following:

• The sample size n is equal to 7.
• The degrees of freedom are equal to n - 1 = 7 - 1 = 6.
• The chi-square statistic is equal to 13.5 (see Example 1 above).

Given the degrees of freedom, we can determine the cumulative probability that the chi-square statistic will fall between 0 and any positive value. To find the cumulative probability that a chi-square statistic falls between 0 and 13.5, we enter the degrees of freedom (6) and the chi-square statistic (13.5) into the Chi-Square Distribution Calculator. The calculator displays the cumulative probability: 0.96.

This tells us that the probability that a standard deviation would be less than or equal to 6 minutes is 0.96. This means (by the subtraction rule) that the probability that the standard deviation would be greater than 6 minutes is 1 - 0.96 or .04.