One-Way ANOVA: Definition, Types, and Examples

Section 1 

The grades.sav data set is a data set that has a total of 21 variables. These variables constitute student information that include; names, gender, ethnicity, year of study, quiz test results, gpa among other variables. The dataset is considered the sample in this scenario and therefore the sample size is the 105 student records it has. The variables chosen for this one-way ANOVA test are the section variable and the quiz3 variable. The section variable’s scale of measurement is nominal and it is the independent variable placed in the factor section in the one-way ANOVA SPSS dialog box. The quiz 3 variable is the dependent variable and consequently it is placed in the dependent list in the one-way ANOVA SPSS dialog box. The reason the section variable is the independent variable is because there are 3 sections in outlined in it and each section has several quiz3 variables. Therefore, there is need to analyze the means of the 3 section and determine whether there is any statistically significant difference between the 3 groups.

Section 2 

For one-way ANOVA to be done, there are several assumptions that are made. The first assumption is that the sample is taken from a normally distributed population. Secondly, each sample needs to drawn independent of the other samples. The variance of each of the groups to be tested need to equal to each other and lastly, the dependent variable needs to be a continuous variable meaning that it can be measured on a scale.

Figure 1 Histogram output of quiz3 

Figure 1 shows the histogram output of quiz3. From histogram, it is possible to see that the data from quiz3 is unimodal with the mode being 7. Moreover, the shape of the histogram depicts a normally distributed sample.

Descriptive Statistics
	N	Skewness		Kurtosis
	Statistic	Statistic	Std. Error	Statistic	Std. Error
quiz3	105	-.256	.236	.039	.467
Valid N (listwise)	105

Table 1 Table showing the skewness and kurtosis of quiz3 variable. 

From table 1 it is possible to see that the distribution is negatively skewed meaning that the mean is less than the median and both the mean and the median are less than the mode. Moreover, the kurtosis value indicates that the distribution is platykurtic since its value is less than 3.

Tests of Normality
	Kolmogorov-Smirnov a			Shapiro-Wilk
	Statistic	df	Sig.	Statistic	df	Sig.
quiz3	.171	105	.000	.938	105	.000
a. Lilliefors Significance Correction

Table 2 Table showing the Shapiro-Wilk test 

From the Shapiro-Wilk test, the p-value is less than 0.05. Therefore, we reject Shapiro-Wilk null hypothesis that there is no statistically significant difference between the sample and a normal distribution.

Test of Homogeneity of Variances
		Levene Statistic	df1	df2	Sig.
quiz3	Based on Mean	1.291	2	102	.280
	Based on Median	.657	2	102	.520
	Based on Median and with adjusted df	.657	2	90.451	.521
	Based on trimmed mean	1.211	2	102	.302

Table 3 Table showing the Levene's statistic and its significance 

The Levene’s test for homogeneity of variance that is considered is the one based on the means. The significance value is at 0.280 and this is greater than 0.05 indicating that the variances for the groups do not differ significantly and are therefore homogenous.

From the above tests, most of the assumptions of the one-way ANOVA are met. For instance, the dependent variable is a continuous variable and the Levene’s test proves that the variances of the groups are similar. Moreover, the samples of the groups are independent on each other. However, the Shapiro-Wilk test and the measurement of skewness and kurtosis point to a distribution that is not normally distributed. Nonetheless, with 3 out of 4 check off, we can continue with the one-way ANOVA.

Section 3 

The research question basically is: Is there a statistically significant difference between the means of the quiz3 variables in the 3 sections available?

From the research question, the null hypothesis therefore is that there is no statistically significant difference between the means of the quiz3 variables in the 3 sections. The alternate hypothesis therefore becomes that there is a statistically significant difference between at least one of the means of the quiz 3 variables in the 3 sections. The alpha level that will be used in this one-way ANOVA test is 0.05.

Section 4 

Table 4 SPSS output of Means plot 

The means plot show that the mean of quiz3 marks in section 1 is lower than the means in section 2 and section 3. Moreover, section 2 has the highest quiz3 marks mean as compared to the rest of the sections.

Descriptives
quiz3
	N	Mean	Std. Deviation	Std. Error	95% Confidence Interval for Mean		Minimum	Maximum
					Lower Bound	Upper Bound
1	33	6.21	1.833	.319	5.56	6.86	1	9
2	39	8.33	1.528	.245	7.84	8.83	5	10
3	33	6.70	1.468	.256	6.18	7.22	4	10
Total	105	7.15	1.849	.180	6.79	7.51	1	10

Table 5 Table showing the descriptive statistics of quiz3 in the three different questions 

The section variable as mentioned earlier has 3 levels: 1, 2, 3. Section 1 has a mean of 6.21 with a standard deviation of 1.833, section 2, a mean of 8.33 with standard deviation of 1.528 and lastly, section 3 has a mean of 6.7 and a standard deviation of 1.468.

ANOVA
quiz3
	Sum of Squares	df	Mean Square	F	Sig.
Between Groups	90.410	2	45.205	17.390	.000
Within Groups	265.152	102	2.600
Total	355.562	104

Table 6 Table showing one-way ANOVA results 

The degrees of freedom between groups is 2 and the one within groups is 102. The F-value is 17.390 and the corresponding p-value is 0.000. Since the p-value of 0.000 is less than 0.05 indicating that the f value is significant.

Multiple Comparisons
Dependent Variable: quiz3
Tukey HSD
(I) section	(J) section	Mean Difference (I-J)	Std. Error	Sig.	95% Confidence Interval
					Lower Bound	Upper Bound
1	2	-2.121 *	.381	.000	-3.03	-1.21
	3	-.485	.397	.443	-1.43	.46
2	1	2.121 *	.381	.000	1.21	3.03
	3	1.636 *	.381	.000	.73	2.54
3	1	.485	.397	.443	-.46	1.43
	2	-1.636 *	.381	.000	-2.54	-.73
*. The mean difference is significant at the 0.05 level.

From the Post hoc tests it is evident that section 2 had a higher mean compared to section 1 and section 3. But section 1 and section 3 had a mean that is close to each other therefore the marginal mean difference is as a result of the difference in the means of section 2 and the rest.

Section 5 

From the results of the ANOVA, we see that the p-value is 0.000 which is less than 0.05 and therefore, we reject the null hypothesis and accept the alternate hypothesis that there is a statistically significant difference in quiz3 marks between at least one of the section groups. One strength of one-way ANOVA is that one can find out whether there is a difference between the means of two or more groups as compared to t-test which can only compare to groups. However, in the one-way ANOVA, one cannot identify which group means actually differ and therefore have to run additional tests to identify them.

References

George, D., & Mallery, P. (2016). IBM SPSS statistics 23 step by step: A simple guide and reference. Routledge. 

Stoline, M. R. (1981). The status of multiple comparisons: simultaneous estimation of all pairwise comparisons in one-way ANOVA designs.  The American Statistician ,  35 (3), 134-141. 

Gamage, J., & Weerahandi, S. (1998). Size performance of some tests in one-way ANOVA.  Communications in Statistics-Simulation and Computation ,  27 (3), 625-640.