The F Distribution and the F Ratio

Learning objectives

The distribution used for the hypothesis test is a new one. It is called the F distribution, named after Sir Ronald Fisher, an English statistician. The F statistic is a ratio (a fraction). There are two sets of degrees of freedom: one for the numerator and one for the denominator.

For example, if F follows an F distribution and the number of degrees of freedom for the numerator is 4, and the number of degrees of freedom for the denominator is 10, then F ~ F_4,10.

To calculate the F ratio, two estimates of the variance are made.

1. Variance between samples: an estimate of σ² that is the variance of the sample means multiplied by n, when the sample sizes are the same. If the samples are different sizes, the variance between samples is weighted to account for the different sample sizes. The variance is also called variation due to treatment or explained variation.
2. Variance within samples: an estimate of σ² that is the average of the sample variances, also known as a pooled variance. When the sample sizes are different, the variance within samples is weighted. The variance is also called the variation due to error or unexplained variation.

• SS_between = the sum of squares that represents the variation among the different samples
• SS_within = the sum of squares that represents the variation within samples that is due to chance

To find a sum of squares mean, add together squared quantities which, in some cases, may be weighted. We used sum of squares to calculate the sample variance and the sample standard deviation in Descriptive Statistics.

MS means mean square. MS_between is the variance between groups, and MS_within is the variance within groups.

Calculation of Sum of Squares and Mean Square

• k = the number of different groups
• n_j = the size of the j^th group
• s_j = the sum of the values in the j^th group
• n = total number of all the values combined (total sample size: ∑n_j)
• x = one value: ∑x = ∑s_j
• Sum of squares of all values from every group combined: ∑x²
• Between group variability: SS_total = ∑x² – $\frac{\left(\sum x^2\right)}{n}$
• Total sum of squares: ∑x² – $\frac{{\left(\sum x\right)}^2}{n}$
• Explained variation: sum of squares representing variation among the different samples ${SS}_{\left(\text{between}\right)}=\sum \left[\frac{{\left(s_j\right)}^2}{n_j}\right]-\frac{{\left(\sum s_j\right)}^2}{n }$
• Unexplained variation: sum of squares representing variation within samples due to chance $S{S}_{\text{within}}=S{S}_{\text{total}}–S{S}_{\text{between}}$
• dfs for different groups (dfs for the numerator): df = k – 1
• Equation for errors within samples (dfs for the denominator): df_within = n – k
• Mean square (variance estimate) explained by the different groups: MS_between = $\frac{S{S}_{\text{between}}}{d{f}_{\text{between}}}$
• Mean square (variance estimate) that is due to chance (unexplained): MS_within = $\frac{S{S}_{\text{within}}}{d{f}_{\text{within}}}$

MS_between and MS_within can be written as follows:

• $M{S}_{\text{between}}=\frac{S{S}_{\text{between}}}{d{f}_{\text{between}}}=\frac{S{S}_{\text{between}}}{k-1}$
• $M{S}_{within}=\frac{S{S}_{within}}{d{f}_{within}}=\frac{S{S}_{within}}{n-k}$

The one-way ANOVA test depends on the fact that MS_between can be influenced by population differences among means of the several groups. Since MS_within compares values of each group to its own group mean, the fact that group means might be different does not affect MS_within.

The null hypothesis says that all groups are samples from populations having the same normal distribution. The alternate hypothesis says that at least two of the sample groups come from populations with different normal distributions. If the null hypothesis is true, MS_between and MS_within should both estimate the same value.

F Ratio or F Statistic

If MS_between and MS_within estimate the same value, following the belief that H₀ is true, then the F ratio should be approximately equal to 1. Mostly, just sampling errors would contribute to variations away from 1. As it turns out, MS_between consists of the population variance plus a variance produced from the differences between the samples. MS_within is an estimate of the population variance. Since variances are always positive, if the null hypothesis is false, MS_between will generally be larger than MS_within. Then the F ratio will be larger than 1. However, if the population effect is small, it is not unlikely that MS_within will be larger in a given sample.

The previous calculations were done with groups of different sizes. If the groups are the same size, the calculations simplify somewhat and the F ratio can be written as follows:

F Ratio formula when the groups are the same size

where

• n = the sample size
• df_numerator = k – 1
• df_denominator = n – k
• s² pooled = the mean of the sample variances (pooled variance)
• $s_{\overline{x}}{}^2$ = the variance of the sample means

Data is typically put into a table for easy viewing. One-way ANOVA results are often displayed in this manner by computer software.

Three different diet plans are to be tested for mean weight loss. The entries in the table are the weight losses for the different plans. The one-way ANOVA results are shown in eip-id2754369.

s₁ = 16.5, s₂ = 15, s₃ = 15.5

Following are the calculations needed to fill in the one-way ANOVA table. The table is used to conduct a hypothesis test.

where n₁ = 4, n₂ = 3, n₃ = 3, and n = n₁ + n₂ + n₃ = 10

Notation

The notation for the F distribution is F ~ F_{df(num),df(denom)},

where df(num) = df_between and df(denom) = df_within.

The mean for the F distribution is $\mu =\frac{df\left(\text{denom}\right)}{df\left(\text{denom}\right)–2}.$

References

Marist College School of Science. (n.d.). Tomato data (Unpublished student research). Marist College School of Science, Poughkeepsie, NY.

Chapter Review

Analysis of variance compares the means of a response variable for several groups. ANOVA compares the variation within each group to the variation of the mean of each group. The ratio of these two is the F statistic from an F distribution with (number of groups – 1) as the numerator degrees of freedom and (number of observations – number of groups) as the denominator degrees of freedom. These statistics are summarized in the ANOVA table.

Formula Review

$S{S}_{\text{between}}={\sum }^{\text{}}\left[\frac{{\left(s_j\right)}^2}{n_j}\right]-\frac{{\left({\sum }^{\text{}}{s}_j\right)}^2}{n }$

$S{S}_{\text{total}}={\sum }^{\text{}}{x}^2-\frac{{\left({\sum }^{\text{}}x\right)}^2}{n}$

$S{S}_{\text{within}}=S{S}_{\text{total}}-S{S}_{\text{between}}$

df_between = df(num) = k – 1

df_within = df(denom) = n – k

MS_between = $\frac{S{S}_{\text{between}}}{d{f}_{\text{between}}}$

MS_within = $\frac{S{S}_{\text{within}}}{d{f}_{\text{within}}}$

F = $\frac{M{S}_{\text{between}}}{M{S}_{\text{within}}}$

F ratio when the groups are the same size: F = $\frac{n{s}_{\overline{x}}{}^2}{s^{\text{2}}{}_{pooled}}$

Mean of the F distribution: µ = $\frac{df\left(num\right)}{df\left(denom\right)-1}$

where

• k = the number of groups
• n_j = the size of the j^th group
• s_j = the sum of the values in the j^th group
• n = the total number of all values (observations) combined
• x = one value (one observation) from the data
• $s_{\overline{x}}{}^2$ = the variance of the sample means
• $s^2{}_{pooled}$ = the mean of the sample variances (pooled variance)

Use the following information to answer the next seven exercises. Groups of men from three different areas of the country are to be tested for mean weight. The entries in eip-556 are the weights for the different groups.

Use the following information to answer the next eight exercises. Girls from four different soccer teams are to be tested for mean goals scored per game. The entries in eip-461 are the goals per game for the different teams.

Homework

Use the following information to answer the next three exercises. Suppose a group is interested in determining whether teenagers obtain their drivers licenses at approximately the same average age across the country. Suppose that the following data are randomly collected from five teenagers in each region of the country. The numbers represent the age at which teenagers obtained their drivers licenses.

H₀: µ₁ = µ₂ = µ₃ = µ₄ = µ₅

Hα: At least any two of the group means µ₁, µ₂, …, µ₅ are not equal.