Analysis of Variance (ANOVA)

An analysis of variance (ANOVA) is a statistical test conducted to examine difference in a continuous variable by a categorical variable. Let’s talk about:

1. The variables in ANOVA,
   
2. The assumptions of ANOVA,
   
3. The logic of ANOVA, and
   
4. What the ANOVA results indicate.
   

I am going to limit the conversation to a one-way ANOVA (i.e., an ANOVA with just 1 independent variable).

Variables in ANOVA

The variables in ANOVA: there are two variables in an ANOVA—a dependent variable and an independent variable. For example, let’s imagine we want to examine differences in SAT scores by gender. SAT scores are the ANOVA dependent variable (i.e., the scores depend on the participants), and it’s a continuous variable because the scores range from 200 to 800. Gender is the ANOVA independent variable (i.e., the designation of male and female are independent of the participant). Further, the independent variable is categorical—you are either male or female. (For more on variables, look here.)

The Assumptions of ANOVA

The assumptions of ANOVA: when an ANOVA is conducted, there are three assumptions. The first ANOVA assumption is that of independence—in this example, males’ scores are unrelated or unaffected with the females’ scores. This ANOVA assumption cannot be violated; if it is, then a different test needs to be conducted. The second ANOVA assumption is normality—that is, the distribution of females’ scores are not dissimilar from a normal bell curve.

The third ANOVA assumption is homogeneity of variance. This ANOVA assumption essentially assesses whether the standard deviation of males and females scores are similar (or homogeneous); that is that the males’118.15 standard deviation is not dissimilar from females standard deviation of 101.03 (Table 1). The ANOVA assumption is homogeneity of variance can be assessed with the Levene test. Table 2 shows the resulting Levene test statistic, where a non-significant difference (i.e., sig > .05) indicates no difference between the standard deviations and the assumption is met.

Table 1.

Descriptives
sat
	N	Mean	Std. Deviation
male	13	608.5385	118.15288
female	13	506.7692	101.03148
Total	26	557.6538	119.55415

Table 2.

Test of Homogeneity of Variances
sat
Levene Statistic	df1	df2	Sig.
.062	1	24	.806

The Logic of ANOVA

The logic of ANOVA: the logic of ANOVA is to test whether the males mean score (M=608.53) differs from females mean score (M=506.76).

What the ANOVA Results Indicate

What the ANOVA results indicate: the ANOVA (Table 3) shows the resulting F-value (F=5.571) with a significance level of .027, indicating that the F-value would occur by chance less than 3 times in 100. We can than say there is a statistically significant difference between the male and female scores, with male achieving a higher average scores compared to females.

Table 3.

	df	F	Sig.
Between Groups	1	5.571	.027
Within Groups	24
Total	25

For a customized, confidential help with ANOVA and/or conducting your statistical analysis, please email us at James@StatisticsSolutions.com or call Statistics Solutions Inc. at (877) 437-8622 for a free 30-minute consultation.