Analysis for Variance

The analysis of variance is essentially a procedure for testing the difference between different groups of data for homogeneity.

The basic principle of ANOVA is to test for differences among the means of the populations by examining the amount of variation within each of these samples, relative to the amount of variation between samples.

Assumptions of Analysis of Variance

1. Population is normally distributed
2. Population has same or equal variances
3. Independent random samples are drawn

Types of ANOVA:

One-Way ANOVA:

• Compares means across a single factor (independent variable) with multiple levels.
• In a one- way classification we take into account the effect of only one variable.

Two-Way ANOVA:

• Examines the impact of two factors and their interaction on the dependent variable.
• If there is a two- way classification the effect of two variables or two factors can be studied.

Steps in Conducting ANOVA

1. Obtain the mean of each sample
2. Work out the mean of the sample means
3. Calculate sum of squares for variance between the samples (or SS between)
4. Obtain variance or mean squares (MS) between samples
5. Calculate sum of squares for variance within the samples (or SS within).
6. Obtain the variance or mean square (MS) within samples
7. Find sum of squares of deviations for total variance
8. Find F-ratio

Null and Alternative Hypothesis:

Null Hypothesis:

H0: µ1=µ2 =…=µk =µ or

H0: a1=a2=…=ak=0; or

H0: b1=b2=…=bh=0

Alternate Hypothesis:

Hit: At least two of the µi .’s are different or

Hit: At least one of the ai ’s is not zero or

Hit : At least one of bj ’s is not zero.

Example:

To study the performance of three detergents and three different water temperature, the following whiteness readings were obtained with specially designed equipment.