Analysis of Variance (ANOVA) is a statistical test used to determine if more than two population means are equal. 

The test uses the F-distribution (probability distribution) function and information about the variances of each population (within) 

and grouping of populations (between) to help decide if variability between and within each populations are significantly different.

So the method of ANOVA test the hypotheses that: 

 

H0:   or

Ha: Not all the means are equal

1. Know the purpose of the analysis of variance test.

The analysis of variance (ANOVA) test statistics is used to test if more than 2 population means are equal.

2. Know the difference between the within-sample estimate of the variance and the between-sample estimate of the variance 

and how to calculate them.

When comparing two or more populations there are several ways to estimate the variance.

The within-sample or treatment variance or variation is the average of the all the variances for each population and is an estimate of 

 whether the null hypothesis, H0 is true or not.

, for j = 1 to k, where k is the number of samples or populations.

Example: Given 5 populations with sample variances: 2.0, 2.2, 2.35, 2.41 and 2.45, the within standard deviation is:

The within-sample variance is often called the unexplained variation.

The between-sample variance or error is the average of the square variations of each population mean from the mean 

or all the data (Grand Mean,  ) and is a estimate of   only if the null hypothesis, H0 is true. 

When the null hypothesis is false this variance is relatively large and by comparing it with the within-sample variance 

we can tell statistically whether H0 is true or not.

The between-sample variance is associated with the explained variation of our experiment.

Example: Given the means of 5 samples: 210, 212, 213, 214 and 214,

Then another estimate of the sample variance of the means is:

for j = 1 to k, where k is the number of samples or populations. 

 

Sample Sample mean  

(1) 

( )

Mean - Grand

Mean (2) 

( )

Sum of

Square  

( )2 

(3)

1 210 1 6.76

2 212 2 0.36

3 213 3 0.16

4 214 4 1.96

5 214 5 1.96

Average (1)

=212.6

Total (3)

=11.2

Grand Mean,   = 212.6, k = 5, and s2B (between) = ( )2 / (k-1) = 11.2 / 4 = 2.8

When the null hypothesis, H0 is true the within-sample variance and the between-sample variance will be about the same; 

however, if the between-sample variance is much larger than the within, we would reject H0.

If the data from both examples above are from the same 5 samples or populations then a ratio of both estimates of the variance 

would give the following:

This ratio has a F-distribution.

3. Know the properties of an F Distribution.

There is an infinite number of F-Distribution based on the combination of alpha significance level, 

the degree of freedom (df1) of the within-sample variance and the degree of freedom (df1) of the between-sample variance.

The F-Distribution is the ratio of the between-sample estimate of   and the within-sample estimate:

If there are k number of population and n number of data values of the all the sample, 

then the degree of freedom of the within-sample variance, df1 = k -1 and 

the degrees of freedom of the between-sample variance is given has df2 = n - k.

The graph of an F probability distribution starts a 0 and extends indefinitely to the right. 

It is skewed to the right similar to the graph shown below. 

 

F-Distribution Graphs:

4. Know how sum of squares relate to Analysis of Variance.

Remember back in Chapter 3 (Regression) we introduced the concept that the total sum of squares 

is equal to the sum of the explained and unexplained variation; this section is an extension of that discussion. 

 

Total Variation = Explained Variation + Unexplained

Variation.

Sum of Squares

The sum of squares for the between-sample variation is either given by the symbol SSB (sum of squares between) 

or SSTR (sum of squares for treatments) and is the explained variation.

To calculate SSB or SSTR, we sum the squared deviations of the sample treatment means from the grand mean 

and multiply by the number of observations for each sample.

The sum of squares for the within-samplevariation is either given by the symbol SSW (sum of square within) 

or SSE (sum of square for error).

To calculate the SSW we first obtained the sum of squares for each sample and then sum them.

The Total Sum of Squares, SSTO = SSB + SSW

The between-sample variance,  , where k is the number of samples or treatment and is often 

called the Mean Square Between,  ,

The within-sample variance,  , where n is the total number of observations in all the samples is often 

called the Mean Square Within or Mean Square Error,  ,

Examples Compute the SSB, SSE and SSTO for the following samples: 

 

Rows, i Treatment of samples j ( 1 to k), k = 3

Sample 1 Sample 2 Sample 3

1 77 83 80

2 79 91 82

3 87 94 86

4 85 88 85

5 78 85 80

Mean 81.2 88.2 82.6

Grand

Mean, 

84

Sum of Square Worksheet 

 

Rows, i Treatment of samples j ( 1 to k), k = 3

SS (Column 1) SS (Column 2) SS (Column 3)

1 (77-81.2)2 = 17.6 (83-88.2)2 = 27.0 (80-82.6)2 = 6.76

4 4

2 (79-81.2)2 = 4.84 (91-88.2)2 = 7.84 (82-82.6)2 = 0.36

3 (87-81.2)2 = 33.6

4

(94-88.2)2 = 33.6

4

(86-82.6)2 = 11.5

6

4 (85-81.2)2 = 14.4

4

(88-88.2)2 = 0.04 (85-82.6)2 = 5.76

5 (78-81.2)2 = 10.2

4

(85-88.2)2 = 10.2

4

(80-82.6)2 = 6.76

SSj (Col

)

80.8 78.8 31.2

Sample

size, nj

5 5 5

SSW = SSE =   = 80.8 + 78.8 + 31.2 = 190.8

SSB = SSTR =   = 5[(81.2 - 84)2 + (88.2 - 84)2 + (82.6 - 84)2 ] =137.2

MSB =   and MSW = MSE = 

F = 

5. Know how to construct an ANOVA Table.

The various statistics computed from the analysis of variance above can be summarized in an ANOVA Table 

as shown below: These summaries are then used to draw inference about the various samples or treatments of which we are studying.

ANOVA Table - Basic layout: 

 

Source Sum of

Squares

(SS)

Degree of

Freedom

(df)

Mean

Square

F-Statistics P-value

Between

Samples

(Explained)

SSB k-1MSB= F=

Value from

Table

Within

Samples

(Unexplained

)

SSE n-kMSE=

Total SSTO n-1

The ANOVA table is easily constructed from the ANOVA program by entering each observations for each sample in 

appropriate columns and deleting any division by 0 in selected regions of the program.

For the data above the ANOVA table is: 

 

6. Know how to interpret the data in the ANOVA table against the null hypothesis.

The ANOVA table program computes the necessary statistics for evaluating the null hypothesis that 

the means are equal: H0:  .

Use the degrees of freedom and an alpha significance level to obtain the expected F-Distribution statistics 

from the lookup table or from the ANOVA program.

Acceptance Criteria for the Null Hypothesis:

If the F-statistics computed in the ANOVA table is less than the F-table statistics or 

the P-value if greater than the alpha level of significance, then there is not reason to reject the null hypothesis 

that all the means are the same: 

 

That is accept H0 if: F-Statistics < F-table or P-value >

alpha.

For the example above we would reject the null hypothesis at the 5% significance level and conclude 

that the means are not equal since F-Stat > F-Table.

7. Know the procedure for testing the null hypothesis that the mean for more than two populations are equal.

Step 1 - Formulate Hypotheses:

H0:   and

Ha: Not all the means are equal

Step 2. Select the F-Statistics Test for equality of more than two means

Step 3. Obtain or decide on a significance level for alpha, say 

Step 4. Compute the test statistics from the ANOVA table .

Step 5. Identify the critical Region: The region of rejection of H0 is obtained from the F-table with alpha and degrees of freedom (k-1, n-k).

Step 6. Make a decision:

That is accept H0 if: F-Statistics < F-table or P-value > alpha.

Worksheet for Computing Analysis of Variance Statistics: 

 

Rows, i Treatment of samples j ( 1 to k

Sum of

Square 

j=1 

Sum of

Square 

j=2 

Sum of

Square 

j=3 

Sum of

Square 

j=4 

Sum of

Square 

j=5 

1

2

3

4

5

6

7

8

9

10

11

12

13

14

SSj (Colu

mn Total)

Sample

size, nj

SSW = SSE =  =

SSB = SSTR =   =

ANOVA Table 

 

Source Sum of

Squares

(SS)

Degree of

Freedom

(df)

Mean

Square

F-Statistics P-value /

F-Table

Between

Samples

(Explained)

SSB= k-1=MSB= = F= =

Within

Samples

(Unexplained)

SSE= n-k=MSE= =

Total SSTO= n-1=