Step-by-Step Process of Calculating One-way ANOVA (Analysis of Variance) with Example

ANOVA (Analysis of Variance) is a method used to compare the means of three or more groups to see if at least one mean is different from the others. Let’s go through a step-by-step guide to perform a one-way ANOVA with an example.

Step 1: State the Hypotheses for One-way ANOVA

• Null Hypothesis (H0): The means of all groups are equal.
• Alternative Hypothesis (Ha): At least one group mean is different from the others.

Step 2: Choose a Significance Level

• Commonly, a 5% level (0.05) is used.

Step 3: Gather Data

You need data from each group you’re comparing. For example, let’s say we’re comparing test scores from three different teaching methods, with scores for each method as follows:

Step 4: Calculate Group Means and Overall Mean

Mean of Method 1 =

ȳ_{1} =\dfrac{80+85+88}{3} = 84.33

Mean of Method 2 =

ȳ_{2} =\dfrac{78+82+80}{3} = 80.00

Mean of Method 3 =

ȳ_{3} =\dfrac{90+92+91}{3} = 91.00

Overall Mean =

ȳ =\dfrac{80+85+88+78+82+80+90+92+91}{9} = 85.22

Step 5: Calculate the Sum of Squares Between Groups (SSB)

First, the equation for SSB = Σ nᵢ(ȳᵢ – ȳ)²,

where:

nᵢ is the number of scores in each group (in this case is the method),

ȳᵢ is the mean of each group, and

ȳ is the overall mean.

Second, the calculation of SSB for our data =

3*(84.33-85.22)² + 3*(80.00-85.22)² + 3*(91.00-85.22)² = 183.22

Step 6: Calculate the Sum of Squares Within Groups (SSW)

SSW = Σ (yᵢj – ȳᵢ)²,

where:

yᵢj is each individual score and

ȳᵢ is the mean of that score’s group.

SSW for our data =

 (80 - 84.33)² + (85 - 84.33)² + (88 - 84.33)² + \\
(78 - 80)² + (82 - 80)² + (80 - 80)² + (90 - 91)² + \\
 (92 - 91)² + (91 - 91)² = 59.33

Step 7: Calculate Degrees of Freedom

DF Between = Number of groups – 1 =

3 - 1 = 2

DF Within = Total number of observations – Number of groups =

 9 - 3 = 6

Step 8: Calculate the Mean Square Values

Mean Square Between (MSB) =

 \dfrac{SSB}{ DF Between} =  \dfrac{183.22}{2} = 91.61

Mean Square Within (MSW) =

 \dfrac{SSW}{ DF Within} =   \dfrac{59.33}{6} = 9.89

Step 9: Calculate the F-Statistic

The F-statistic in our example is crucial for determining whether the differences in test scores across three teaching methods are statistically significant, guiding educational decisions and research.

F_{stats}= \dfrac{MSB}{MSW} =   \dfrac{91.61}{9.89} = 9.27

Step 10: Compare F-Statistic to Critical Value

From above calculation, the F-statistic was 9.27. The degrees of freedom for the numerator (between groups) is the number of groups minus one, which is 2 (since we have 3 groups), and for the denominator (within groups), it’s the total number of observations minus the number of groups, which is 6 (9 total observations – 3 groups).

Now, let’s find the critical value:

• Degrees of Freedom (numerator, between groups) = 3 – 1= 2
• Degrees of Freedom (denominator, within groups) = 9 – 3 = 6
• Significance level (commonly used) = 0.05

You would normally use an F-distribution table in the statistical book or in this link: (F-distribution table). Find the critical value for these degrees of freedom at a 0.05 significance level which is as following:

 F_{critical} =  5.1433

Comparison Process:

1. Look up the critical value for DF1 = 2, DF2 = 6 at the 0.05 significance level in an F-distribution table.
2. Compare the critical value to our calculated F-statistic (9.27).
3. If the F-statistic is greater than the critical value, you reject the null hypothesis, indicating a significant difference in group means.

5.1433 < 9.27 \\  F_{critical} <  F_{stats}

Conclusion

• In our example, since the critical value at a 0.05 significance level for DF Between = 2 and DF Within = 6 is less than 9.27, we can reject the null hypothesis, suggesting that the teaching methods have significantly different effects on test scores.