Answer to Question #7763 Submitted to "Ask the Experts"

Category: Instrumentation and Measurements — Surveys and Measurements (SM)

The following question was answered by an expert in the appropriate field:

What is the difference between using the chi-square value and the reduced chi-square value? I realize that the reduced value is simply the chi-square value divided by the degrees of freedom, but so what? Why would I want to know this instead of, or in addition to, the regular value?

You may very well not care about knowing the reduced chi-square value, and you won't necessarily be missing anything since you are familiar with the data being evaluated, and you know the number of degrees of freedom involved. One notable advantage of using the reduced value is in communicating results to others, because the listener or reader does not need to know how many degrees of freedom apply in order to interpret the value. If the value is close to unity, then the data, assuming a reasonable number of data points, likely exhibit acceptable variance. The reduced chi-square value is equal to the ratio of the observed experimental variance divided by the theoretical variance.

For example, if you are using a chi-square test to check the operation of a counting system by taking multiple counts (say 25) of a standard for equal counting durations, and the average count is 2,575, the theoretical variance is also 2,575 (Poisson statistics, and the variance is equal to the count). If you evaluate the experimental variance using normal statistics and find it to be 2,446, then the reduced chi-square value is 0.95, and it is likely that the counting system is behaving properly in that it is not showing excessive variance—or, said differently, the experimental variance is close to the expected theoretical variance. The actual value of chi-square for 24 degrees of freedom for this case would be 22.8.

Hope you enjoy continued good counting.

George Chabot, PhD, CHP

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