Statistics: Ten facts about the chi-square distribution

Tutoring statistics, distributions are of constant interest. The tutor brings up ten points about the chi-square distribution.

The chi-square distribution may not be discussed much in a first-level stats course. It’s used to estimate or evaluate variance, rather than central tendency. Here are ten facts about the chi-square distribution:

  1. Typically, Χ2v is used to denote a chi-square variable or distribution with v degrees of freedom.
  2. Its parameter used for finding its values in tables is the degrees of freedom, typically referred to as v or just n-1, where n is the number of values in the sample.
  3. Its expected value is v.
  4. Its variance is 2v.
  5. It’s not symmetrical, but skewed right.
  6. Since a chi-square random variable is calculated from summing squares, it can’t be negative.
  7. (n-1)s22 has a chi-square distribution with n-1 degrees of freedom. In this context, s2 is the sample variance, while σ2 is the true population variance.
  8. Σ(observed-predicted)2/predicted, for predicted values using a model, follows a chi-square distribution.
  9. The chi-square distribution is used to estimate or test population variance.
  10. The chi-square distribution is used to test goodness-of-fit between a model and a sample.

I’ll be talking more about the uses of the chi-square distribution.


Harnett, Donald L. and James L. Murphy. Statistical Analysis for Business and Economics, third edition. Don Mills: Addison-Wesley, 1986.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

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