One-way analysis of variance

One-way analysis of variance (ANOVA) is a statistical technique used to compare means of three or more samples (using the F distribution) to understand if at least one of the sample means significantly differs from the others. It is particularly useful when dealing with complex data sets where variations within groups and between groups need to be isolated. The technique assumes that the various groups are sampled from populations with the same true variance.

Overview

One-way ANOVA tests the null hypothesis that two or more groups have the same population mean. The test is called "one-way" because it tests the effect of a single factor on the dependent variable, even though there can be multiple levels of this factor. For example, it can be used to test if the mean test scores differ between students in different classrooms. If significant, we conclude that not all group means are equal.

Assumptions

The application of one-way ANOVA comes with several assumptions:

• The populations from which the samples are drawn are normally distributed.
• Samples are independent of each other.
• The variances of the populations are equal (homogeneity of variances).

Procedure

The procedure involves several steps:

1. Calculate the overall mean.
2. Compute the sum of squares between groups (SSB) and within groups (SSW).
3. Calculate the mean square for between groups (MSB) by dividing SSB by the number of groups minus one.
4. Calculate the mean square for within groups (MSW) by dividing SSW by the total number of observations minus the number of groups.
5. Compute the F-ratio by dividing MSB by MSW.
6. Compare the computed F-ratio with the critical value from the F-distribution table at a predetermined significance level (alpha). If the F-ratio is larger, the null hypothesis is rejected.

Interpretation

A significant result indicates that at least one group mean is different from the others. However, it does not tell us which groups are different. Post-hoc tests are required to identify the specific group differences.

Applications

One-way ANOVA is widely used in various fields such as psychology, education, medicine, and agriculture to analyze the differences among group means in a sample.

Limitations

While one-way ANOVA is a powerful tool, it has limitations:

• It cannot be used to determine which specific means are different.
• It is sensitive to outliers.
• The assumption of homogeneity of variances can be violated in practice.

See Also

• F-test
• Analysis of covariance
• Two-way analysis of variance
• Post-hoc analysis