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Explain the chi-square, F and Student's t distributions

CFA® Level I · Quantitative Methods · Statistical Distributions for Financial Asset Prices and Returns · QM.6.d

The idea

Student's t distribution is symmetric like the normal but with fatter tails, governed by its degrees of freedom; it converges on the normal as those grow, and is the right distribution for inference about a mean when the population standard deviation is unknown. The chi-square distribution is asymmetric and bounded at zero and is used for inference about a variance. The F distribution is the ratio of two independent chi-square variables, each divided by its own degrees of freedom, and is used to compare two variances, which is what makes it the distribution behind the overall significance test in a regression.

A practice question

From the question bank, with the reason each wrong option is wrong.

How does Student's t distribution differ from the normal?

  1. It is skewed to the rightWhy not: It is symmetric, like the normal. What differs is the weight in both tails, not the balance between them.
  2. It has fatter tails and depends on degrees of freedom — correct
  3. It is bounded at zeroWhy not: It extends indefinitely in both directions. The chi-square is the one bounded at zero.

The t distribution's heavier tails reflect the extra uncertainty from estimating the standard deviation. As degrees of freedom rise it converges on the normal.

A harder one

When should a t distribution be used instead of a normal distribution for inference about a mean?

  1. When the population standard deviation is unknown and must be estimated — correct
  2. When the sample is larger than 100Why not: The t is used for *small* samples in practice, and by 100 observations it is nearly indistinguishable from the normal anyway. Sample size is not the criterion.
  3. When the data are skewedWhy not: Neither distribution handles skew. The t exists to account for an estimated standard deviation, not an asymmetric population.

Estimating the standard deviation from the sample adds uncertainty that the t distribution's fatter tails account for. With a known population standard deviation the normal applies.

The hardest tier

Which distribution underlies the test of whether two variances are equal?

  1. Student's tWhy not: That serves inference about a mean. Comparing two variances calls for a ratio, which t is not built from.
  2. Chi-squareWhy not: That is for inference about a *single* variance. A ratio of two chi-square variables, scaled, gives the F.
  3. The F distribution — correct

The F statistic is a ratio of two chi-square variables scaled by their degrees of freedom, which is exactly the form a ratio of sample variances takes. The same distribution gives a regression's overall significance test.

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