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The F Distribution and the F Ratio

Module by: Susan Dean, Barbara Illowsky, Ph.D.. E-mail the authors

Summary: This module describes how to calculate the F Ratio and F Distribution based on the hypothesis test for the One-Way ANOVA.

The distribution used for the hypothesis test is a new one. It is called the FF distribution, named after Sir Ronald Fisher, an English statistician. The FF 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 F follows an F F distribution and the degrees of freedom for the numerator are 4 and the degrees of freedom for the denominator are 10, then FF ~ F 4 , 10 F 4 , 10 .

Note:

The FF distribution is derived from the Student's-t distribution. One-Way ANOVA expands the tt-test for comparing more than two groups. The scope of that derivation is beyond the level of this course.

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

  1. Variance between samples: An estimate of σ 2 σ 2 that is the variance of the sample means multiplied by n (when there is equal n). 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 σ 2 σ 2 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 = SS between = the sum of squares that represents the variation among the different samples.
  • SS within = SS within = the sum of squares that represents the variation within samples that is due to chance.

To find a "sum of squares" means to 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.

MSMS means "mean square." MS between MS between is the variance between groups and MS within MS within is the variance within groups.

Calculation of Sum of Squares and Mean Square

  • kk size 12{k} {} = the number of different groups
  • njnj size 12{n rSub { size 8{j} } } {} = the size of the jthjth size 12{ ital "jth"} {} group
  • sjsj size 12{s rSub { size 8{j} } } {}= the sum of the values in the jthjth size 12{ ital "jth"} {} group
  • nn size 12{n} {} = total number of all the values combined. (total sample size: njnj size 12{ Sum {n rSub { size 8{j} } } } {})
  • xx = one value: x=sjx=sj size 12{ Sum {x} = Sum {s rSub { size 8{j} } } } {}
  • Sum of squares of all values from every group combined: x2x2 size 12{ Sum {x rSup { size 8{2} } } ={}} {}
  • Between group variability: SS total = x 2 x 2 n SS total = x 2 x 2 n size 12{ ital "SS" rSub { size 8{ ital "total"} } = Sum {x rSup { size 8{2} } } - { { left ( Sum {x} right ) rSup { size 8{2} } } over {n} } } {}
  • Total sum of squares: x 2 ( x ) 2 n x 2 ( x ) 2 n size 12{ Sum {x rSup { size 8{2} } } - { { \( Sum {x} \) rSup { size 8{2} } } over {n} } } {}
  • Explained variation- sum of squares representing variation among the different samples SS between = [ ( sj ) 2 n j ] ( s j ) 2 n SS between = [ ( sj ) 2 n j ] ( s j ) 2 n size 12{ ital "SS" rSub { size 8{ ital "between"} } = Sum { \[ { { \( ital "sj" \) rSup { size 8{2} } } over {n rSub { size 8{j} } } } \] } - { { \( Sum {s rSub { size 8{j} } \) rSup { size 8{2} } } } over {n} } } {}
  • Unexplained variation- sum of squares representing variation within samples due to chance: SS within = SS total SS between SS within = SS total SS between size 12{ ital "SS" rSub { size 8{ ital "within"} } = ital "SS" rSub { size 8{ ital "total"} } - ital "SS" rSub { size 8{ ital "between"} } } {}
  • df's for different groups (df's for the numerator): df between = k 1 df between = k 1 size 12{ ital "df" rSub { size 8{ ital "between"} } =k - 1} {}
  • Equation for errors within samples (df's for the denominator): df within = n k df within = n k size 12{ ital "df" rSub { size 8{ ital "within"} } = n - k} {}
  • Mean square (variance estimate) explained by the different groups: MS between = SS between df between MS between = SS between df between size 12{ ital "MS" rSub { size 8{ ital "between"} } = { { ital "SS" rSub { size 8{ ital "between"} } } over { ital "df" rSub { size 8{ ital "between"} } } } } {}
  • Mean square (variance estimate) that is due to chance (unexplained): MS within = SS within df within MS within = SS within df within size 12{ ital "MS" rSub { size 8{ ital "within"} } = { { ital "SS" rSub { size 8{ ital "within"} } } over { ital "df" rSub { size 8{ ital "within"} } } } } {}

MS between MS between and MS within MS within can be written as follows:

  • MS between = SS between df between = SS between k 1 MS between = SS between df between = SS between k 1
  • MS within = SS within df within = SS within n k MS within = SS within df within = SS within n k

The One-Way ANOVA test depends on the fact that MS between MS between can be influenced by population differences among means of the several groups. Since MS within 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 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 MS between and MS within MS within should both estimate the same value.

Note:

The null hypothesis says that all the group population means are equal. The hypothesis of equal means implies that the populations have the same normal distribution because it is assumed that the populations are normal and that they have equal variances.

F-Ratio or F Statistic

F = MS between MS within F= MS between MS within
(1)

If MS between MS between and MS within MS within estimate the same value (following the belief that H o H o 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 MS between consists of the population variance plus a variance produced from the differences between the samples. MS within MS within is an estimate of the population variance. Since variances are always positive, if the null hypothesis is false, MS between MS between will generally be larger than MS within MS within . Then the F-ratio will be larger than 1. However, if the population effect size is small it is not unlikely that MS within MS within will be larger in a give sample.

The above 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:

F-Ratio Formula when the groups are the same size

F = n s x_ 2 s 2 pooled F = n s x_ 2 s 2 pooled
(2)

where ...

  • n=n= size 12{n={}} {}the sample size
  • df numerator = k 1 df numerator = k 1 size 12{ ital "df" rSub { size 8{ ital "numerator"} } =k - 1} {}
  • df denominator = n k df denominator = n k size 12{ ital "df" rSub { size 8{ ital "denominator"={}} } k \( n - 1 \) =N - k} {}
  • s 2 pooled = s 2 pooled = the mean of the sample variances (pooled variance)
  • sx¯ 2 = s x 2 = the variance of the sample means

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

Table 1
Source of Variation Sum of Squares (SS) Degrees of Freedom (df) Mean Square (MS) F
Factor
(Between)
SS(Factor) k - 1 MS(Factor) = SS(Factor)/(k-1) F = MS(Factor)/MS(Error)
Error
(Within)
SS(Error) n - k MS(Error) = SS(Error)/(n-k)  
Total SS(Total) n - 1    

Example 1

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 table is shown below.

Table 2
Plan 1 Plan 2 Plan 3
5 3.5 8
4.5 7 4
4   3.5
3 4.5  

One-Way ANOVA Table: The formulas for SS(Total), SS(Factor) = SS(Between) and SS(Error) = SS(Within) are shown above. This same information is provided by the TI calculator hypothesis test function ANOVA in STAT TESTS (syntax is ANOVA(L1, L2, L3) where L1, L2, L3 have the data from Plan 1, Plan 2, Plan 3 respectively).

Table 3
Source of Variation Sum of Squares (SS) Degrees of Freedom (df) Mean Square (MS) F
Factor
(Between)
SS(Factor)
= SS(Between)
=2.2458
k - 1
= 3 groups - 1
= 2
MS(Factor)
= SS(Factor)/(k-1)
= 2.2458/2
= 1.1229
F =
MS(Factor)/MS(Error)
= 1.1229/2.9792
= 0.3769
Error
(Within)
SS(Error)
= SS(Within)
= 20.8542
n - k
= 10 total data - 3 groups
= 7
MS(Error)
= SS(Error)/(n-k)
= 20.8542/7
= 2.9792
 
Total SS(Total)
= 2.9792 + 20.8542
=23.1
n - 1
= 10 total data - 1
= 9
   

The One-Way ANOVA hypothesis test is always right-tailed because larger F-values are way out in the right tail of the F-distribution curve and tend to make us reject H o H o .

Notation

The notation for the F distribution is FF ~ F df(num) , df(denom) F df(num) , df(denom)

where df(num) = df between df(num)= df between and df(denom) = df within df(denom) = df within

The mean for the F distribution is μ = df(num) df(denom) 1 μ= df(num) df(denom) 1

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