## LEARNING OBJECTIVES

- Find the area under an [latex]F[/latex]-distribution.
- Find the [latex]F[/latex]-score for a given area under the curve of an [latex]F[/latex]-distribution.

The [latex]F[/latex]-distribution is a continuous probability distribution. The graph of an [latex]F[/latex]-distribution is shown below. The [latex]F[/latex]-distribution is used in statistical inference to test the equality of population variances, test the difference in three or more population means, or to test the overall multiple regression model.

**Properties of the [latex]F[/latex]-distribution:**

- The graph of an [latex]F[/latex]-distribution is positively-skewed and asymmetrical with a minimum value of 0 and no maximum value.
- An [latex]F[/latex]-distribution is determined by two different degrees of freedom, [latex]df_1[/latex] and [latex]df_2[/latex]. [latex]df_1[/latex] is the degrees of freedom for the numerator of the [latex]F[/latex]-score and [latex]df_2[/latex] is the degrees of freedom for the denominator of the [latex]F[/latex]-score. The values of the degrees of freedom depends on how the [latex]F[/latex]-distribution is used. There is a different [latex]F[/latex]-distribution for every set of degrees of freedom. As the values of [latex]df_1[/latex] and [latex]df_2[/latex] get larger, the [latex]F[/latex]-distribution approaches a normal distribution.
- The total area under the graph of an [latex]F[/latex]-distribution is 1.
- Probabilities associated with an [latex]F[/latex]-distribution are given by the area under the curve of the [latex]F[/latex]-distribution.

## USING EXCEL TO CALCULATE THE AREA UNDER AN [latex]\textcolor{white}F[/latex]-DISTRIBUTION

**To find the area in the left tail:**

- To find the area under an [latex]F[/latex]-distribution to the left of a given [latex]F[/latex]-score, use the
**f.dist([latex]F[/latex], degrees of freedom 1, degrees of freedom 2, logic operator)**function.- For
**[latex]F[/latex]**, enter the [latex]F[/latex]-score. - For
**degrees of freedom 1**, enter the value of [latex]df_1[/latex] for the [latex]F[/latex]-distribution. - For
**degrees of freedom 2**, enter the value of [latex]df_2[/latex] for the [latex]F[/latex]-distribution. - For
**logic operator**, enter**true**.

- For
- The output from the
**f.dist**function is the area to the left of the entered [latex]F[/latex]-score. - Visit the Microsoft page for more information about the
**f.dist**function.

**To find the area in the right tail:**

- To find the area under an [latex]F[/latex]-distribution to the right of a given [latex]F[/latex]-score, use the
**f.dist.rt([latex]F[/latex], degrees of freedom 1, degrees of freedom 2)**function.- For
**[latex]F[/latex]**, enter the [latex]F[/latex]-score. - For
**degrees of freedom 1**, enter the value of [latex]df_1[/latex] for the [latex]F[/latex]-distribution. - For
**degrees of freedom 2**, enter the value of [latex]df_2[/latex] for the [latex]F[/latex]-distribution.

- For
- The output from the
**f.dist.rt**function is the area to the right of the entered [latex]F[/latex]-score. - Visit the Microsoft page for more information about the
**f.dist.rt**function.

## EXAMPLE

Consider an [latex]F[/latex]-distribution with [latex]df_1=12[/latex] and [latex]df_2=27[/latex].

- Find the area under the [latex]F[/latex]-distribution to the left of [latex]F=0.69[/latex].
- Find the area under the [latex]F[/latex]-distribution to the right of [latex]F=1.53[/latex].

**Solution:**

**Function**f.dist **Answer****Field 1**0.69 0.2535 **Field 2**12 **Field 3**27 **Field 4**true **Function**f.dist.rt **Answer****Field 1**1.53 0.1738 **Field 2**12 **Field 3**27

## USING EXCEL TO CALCULATE [latex]\textcolor{white}F[/latex]-SCORES

**To find the [latex]F[/latex]-score for the a given left-tail area:**

- To find the [latex]F[/latex]-score for a given area under an [latex]F[/latex]-distribution to the left of the [latex]F[/latex]-score, use the
**f.inv(area to the left, degrees of freedom 1, degrees freedom 2)**function.- For
**area to the left**, enter the area to the left of required [latex]F[/latex]-score. - For
**degrees of freedom 1**, enter the value of [latex]df_1[/latex] for the [latex]F[/latex]-distribution. - For
**degrees of freedom 2**, enter the value of [latex]df_2[/latex] for the [latex]F[/latex]-distribution.

- For
- The output from the
**f.inv**function is the value of [latex]F[/latex]-score so that the area to the left of the [latex]F[/latex]-score is the entered area. - Visit the Microsoft page for more information about the
**f.inv**function.

**To find the [latex]F[/latex]-score for the a given right-tail area:**

- To find the [latex]F[/latex]-score for a given area under an [latex]F[/latex]-distribution to the right of the [latex]F[/latex]-score, use the
**f.inv.rt(area to the right, degrees of freedom 1, degrees of freedom 2)**function.- For
**area to the right**, enter the area to the right of required [latex]F[/latex]-score. - For
**degrees of freedom 1**, enter the value of [latex]df_1[/latex] for the [latex]F[/latex]-distribution. - For
**degrees of freedom 2**, enter the value of [latex]df_2[/latex] for the [latex]F[/latex]-distribution.

- For
- The output from the
**f.inv.rt**function is the value of [latex]F[/latex]-score so that the area to the right of the [latex]F[/latex]-score is the entered area. - Visit the Microsoft page for more information about the
**f.inv.rt**function.

## EXAMPLE

Consider an [latex]F[/latex]-distribution with [latex]df_1=37[/latex] and [latex]df_2=15[/latex].

- Find the [latex]F[/latex]-score so that the area under the [latex]F[/latex]-distribution to the left of [latex]F[/latex] is 0.413.
- Find the [latex]F[/latex]-score so that the area under the [latex]F[/latex]-distribution to the right of [latex]F[/latex] is 0.148.

**Solution:**

**Function**f.inv **Answer****Field 1**0.413 0.934 **Field 2**37 **Field 3**15 **Function**f.dist.rt **Answer****Field 1**0.269 1.354 **Field 2**37 **Field 3**15

## Concept Review

The [latex]F[/latex]-distribution is a useful tool for assessment in a series of problem categories. These problem categories include: statistical inference for two population variances, testing the equality of three or more population means (one-way ANOVA), and testing the overall significance of the multiple regression model.

Important parameters in an [latex]F[/latex]-distribution are the degrees of freedom in a given problem. The [latex]F[/latex]-distribution curve is skewed to the right, and its shape depends on the degrees of freedom. As the degrees of freedom increase, the curve of an [latex]F[/latex]-distribution approaches a normal distribution.

## Attribution

“13.3Facts About the F Distribution“ in Introductory Statistics by OpenStaxis licensed under aCreative Commons Attribution 4.0 International License.