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Chi-square and F-Distributions

Chi-square and F-Distributions

Chi-square Distribution

A chi-square distribution is an asymmetrical family of distributions. A chi-square distribution with \(v\) degrees of freedom is the distribution of the sum of the squares of  \(v\) independent standard normally distributed random variables. Intuitively, chi-square distributions take only non-negative random variables.

A chi-square distribution is used to test the variance of a population that is distributed normally.

In a summary, the following are the properties of a chi-square distribution:

  • A chi-square distribution is a non-symmetrical distribution (skewed to the right).
  • A chi-square distribution is defined by one parameter: degrees of freedom (df), \(v = n – 1\).
  • A chi-square distribution is the sum of the squares of \(k\) independent standard normally distributed random variables. Hence, it is a non-negative distribution.
  • For each degree of freedom, there are different chi-square distributions.
  • The shape of a chi-square distribution changes with the change in the degrees of freedom. The more the degrees of free increase, the more the distribution assumes the shape of a standard normal distribution.

Chi-square Distribution


An F-distribution is used to test the equality of variances of two normally distributed populations from two independent random samples.

The following are the properties of an F-distribution:   

  • An F-distribution is an asymmetrical distribution (skewed to the right).
  • An F-distribution is defined by two parameters, i.e., degrees of freedom of the numerator ( \(m\)) and degrees of freedom of the denominator ( \(n\)).
  • Like a chi-square distribution, an F-distribution can only have positive values.
  • As the degrees of freedom for the numerator and the denominator increase, the F-distribution approximates the normal distribution.

Relationship Between the Chi-square and F-distributions

The F-distribution is the ratio of two chi-square distributions with degrees of freedom \(m\) and  \(n\), respectively, where each chi-square has first been divided by its degrees of freedom, i.e.,


Where  \(m\) is the numerator degrees of freedom and  \(n\) is the denominator degrees of freedom.


Which of the following are most likely common characteristics of F-distribution and chi-square distribution?

  1. Both can take only positive value.
  2. Both are defined by two parameters.
  3. Both are negatively skewed distribution. 


The correct answer is A.

Both F-distribution and chi-square distribution can only take non-negative values.

B is incorrect. A chi-square distribution is defined by one parameter (i.e., n-1 degrees of freedom) while an F-distribution is defined by parameters, i.e., degrees of freedom of the numerator (m) and degrees of freedom of the denominator (n).

C is incorrect. Both the F-distribution and the chi-square distribution are positively skewed distributions.

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