- CFA Exams
- 2024 Level I
- Topic 1. Quantitative Methods
- Learning Module 4. Common Probability Distributions
- Subject 11. Student's t-, Chi-Square, and F-Distributions

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##### Subject 11. Student's t-, Chi-Square, and F-Distributions PDF Download

#### Student's t-Distribution

The t-distribution describes the standardized distances of sample means to the population mean when the population standard deviation is not known, and the observations come from a normally distributed population.

The t-distribution is similar to a normal distribution. Like the normal distribution, the t-distribution is symmetric, and has a smooth shape. Like a standard normal distribution (or z-distribution), the t-distribution has a mean of zero.

Differences:

- The normal distribution assumes that the population standard deviation is known. The t-distribution does not make this assumption. It is based on the sample standard deviation.
- The t-distribution is defined by the degrees of freedom.
- The t-distribution is most useful for small sample sizes, when the population standard deviation is not known, or both. As the sample size increases, the t-distribution becomes more similar to a normal distribution.

In the above graph, all of the distributions have a smooth shape. All are symmetric. All have a mean of zero.

**Degrees of Freedom**refers to the maximum number of logically independent values, which are values that have the freedom to vary, in the data sample. The shape of the t-distribution depends on the degrees of freedom, which is n - 1. The curves with more degrees of freedom are taller and have thinner tails. All three t-distributions have "heavier tails" than the z-distribution.

The t-distribution plays a role in a number of widely used statistical analyses, including Student's t-test for assessing the statistical significance of the difference between two sample means, the construction of confidence intervals for the difference between two population means, and in linear regression analysis. These topics are covered in the reading of "Hypothesis Testing".

#### Chi-Square Distribution

The Chi-square distribution is a family of distributions. Each distribution is defined by the degrees of freedom.

The figure below shows three different Chi-square distributions with different degrees of freedom.

You can see that the blue curve with 8 degrees of freedom is somewhat similar to a normal curve (the familiar bell curve). But, it has a longer tail to the right than a normal distribution and is not symmetric. Compare the blue curve to the orange curve with 4 degrees of freedom. The orange curve is very different from a normal curve. The purple curve has 3 degrees of freedom and looks even less like a normal curve than the other two curves.

The higher the degrees of freedom for a Chi-square distribution, the more it looks like a normal distribution.

#### F Distribution

F distribution is used for one-way ANOVA and the test of two variances. Here are some facts about the F distribution.

- The curve is not symmetrical but skewed to the right.
- There is a different curve for each set of dfs.
- The F statistic is greater than or equal to zero.
- As the degrees of freedom for the numerator and for the denominator get larger, the curve approximates the normal.

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