One-Way ANOVA

Student’s Name

Institutional Affiliation

One-Way ANOVA

Section 1

Data Context 

The grade.sav data set provides a raw data for students’ details regarding their social demographics including gender, ethnicity, year and section of the study, as well as, their performance reflected through the score in quizzes, reviews, and grades for a given subject. For instance, the final grade which determines fail or pass is hypothesized to be influenced by some explanatory variables including the current GPA, a section of the study, the quiz scores, gender, ethnicity and review outcomes. For this analysis, the researcher focuses on n 3determining if the Quiz 3 scores vary across the section of the study of the students. 

In other words, the research would investigate if the mean scores for students in different sections are different from each other. In this case, the section that the student is currently in influences the hypothesized scores for the Quiz 3. Accordingly, learning process may be different across these sections hence resulting to differences in scores by the students. Therefore, the section is treated as an explanatory variable while Quiz 3 scores are treated as the dependent outcome or dependent variable. For the analysis, the sample size is depicted as n = 105. 

Section 2

Assumptions of a One-Way ANOVA

For a one-factor ANOVA analysis, there are basic assumptions that must be held. Ideally, the variances of the population groups are assumed to be equal, and each group sample is drawn from a population that is normally distributed – the residuals for response variables are approximately normally distributed (Girden, 2002). Again, the samples should be drawn independently indicating that the responses for each given population group are independent, as well as, identically distributed. Lastly, the distributions are assumed not to include any outliers to achieve an F-statistics that well behaves because it is robust to any violation of normality given equal sample sizes and symmetrical populations (Girden, 2002).

Histogram for Quiz 3

Fig. 1

Histogram for Quiz 3

From the above histogram representing the distribution, the mean is given as 8.05 with a standard deviation of 2.322 indicating that the distribution is approximately normal, given that n = 105.  Most of the scores are seen to lie within the group mean. However, if the distribution curve is not given and the group means, the data seems to skew to the left. However, such an aspect may be subjected to skewness and Kurtosis or plotting a histogram of the standardized scores for Quiz 3 – which confirms that the data is normal. 

Histogram of the standardized scores for Quiz 3

Quiz 3 Descriptive statistics – Skewness and Kurtosis 

Table 1

Skewness and Kurtosis for Quiz 3

Descriptive Statistics
	N	Mean	Std. Deviation	Skewness	Kurtosis
Statistic	Statistic	Statistic	Statistic	Std. Error	Statistic	Std. Error
Quiz3	105	8.05	2.322	-1.177	.236	.805	.467
Valid N (listwise)	105

The skewness appears to be – 1.177 which shows that the data is slightly skewed to the left. For a normally distributed data, it should be symmetrical and a skewness of -/+ 1. However, if we consider the Kurtosis, which assumes a Gaussian distribution that considers a -/+ 3 variation of the Kurtosis statistics, the data is normal as the statistics are given as 0.805 (lies within the accepted band). 

Shapiro-Wilk Test

Table 2

Showing Shapiro-Wilk test for Quiz 3

Tests of Normality
	Kolmogorov-Smirnova	Shapiro-Wilk
Statistic	df	Sig.	Statistic	df	Sig.
Quiz3	.221	105	.000	.818	105	.000
Lilliefors Significance Correction

Shapiro-Wilk test tests the idea that the null-hypothesis is drawn from normally distributed population. The test indicates that Shapiro-Wilk statistic is 0.818 and a P-value of 0.000 indicating that the null hypothesis is rejected and concluded that the sample is drawn from a population that is not normally distributed. 

Levene’s Test 

Table 3

Showing Levene’ Statistic for homogeneity 

Test of Homogeneity of Variances
Quiz3
Levene Statistic	df1	df2	Sig.
1.576	2	102	.212

The Levene’s statistic tests the homogeneity of variances – if population variances are equal. Given that the P-value = 0.212 > 0.05, the null hypothesis is rejected and it can be evidently claimed that there is a difference between group variances. The quiz 3 scores and section variables are independent. 

Summary of assumptions 

The normal distribution curve and the histogram, the Kurtosis analysis and the Levene’s test confirm that the assumptions of ANOVA are met except normality which from the Shapiro-Wilk test proves that the data is drawn from a distribution that is not normally distributed. However, looking at the assumptions of ANOVA, sample populations are just approximated to have normal residuals that are usually distributed. Hence the ANOVA test can be conducted. 

Section 3

Research Question 

Are the Quiz 3 cores different across the sections?

Hypothesis statement 

H0: The mean scores for quiz 3 are equal across the three sections 

H1: The mean scores for quiz 3 are different from the three sections 

Alpha 

The ANOVA test adopts an alpha of 0.05 indicating that the rejection region would be 0.05 and any P-value lying within this region would lead to rejection of the null hypothesis. Given 95% confidence, does quiz 3 mean scores vary across the three sections?

Section 4

Mean plot for Quiz 3

Fig. 3

Mean plots for Quiz 3 and section 

From the mean plots for the Quiz three scores, it is clear that all the group means are different. The group means decreases from section one towards sections 2 and 3. Such an aspect is interesting for running an ANOVA test to see if there is statistical evidence to claim that the group means are different across the three sections. 

Descriptive statistics 

Table 4

Means and standard deviations for Quiz 3

Descriptives
Quiz3
	N	Mean	Std. Deviation	Std. Error	95% Confidence Interval for Mean	Minimum	Maximum
Lower Bound	Upper Bound
1	33	9.00	2.107	.367	8.25	9.75	2	10
2	39	7.62	2.098	.336	6.94	8.30	2	10
3	33	7.61	2.549	.444	6.70	8.51	0	10
Total	105	8.05	2.322	.227	7.60	8.50	0	10

Table 5

ANOVA test analysis 

ANOVA
Quiz3
	Sum of Squares	df	Mean Square	F	Sig.
Between Groups	43.652	2	21.826	4.305	.016
Within Groups	517.110	102	5.070
Total	560.762	104

Given the ANOVA table above, the F-value is given as 4.305 with 2 degrees of freedom for between groups and 102 degrees of freedom for the within groups. Additionally, the P-value = 0.016 which is less than 0.05 (alpha) indicating that the P-value lies within the rejection region (Field, 2009). Therefore, the null hypothesis is rejected for the alternative hypothesis – indicating that the quiz 3 scores are different across the three sections. From the ANOVA table the Eta-Squared can give the effect size:

Effect size (Eta-Squared) = Sum of Squares (between)/ Sum of squares (Total)

= 43.652 / 560.762

= 0.0778 or 7.78%

The effect size indicates that 7.78% of the total variance is accounted by the treatment effect on the outcome variable (Quiz 3in this case) (Sawilowsky, 2009).

Post-hoc tests

Table 6

Showing Post-hoc tests 

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	1.385*	.533	.029	.12	2.65
3	1.394*	.554	.036	.08	2.71
2	1	-1.385*	.533	.029	-2.65	-.12
3	.009	.533	1.000	-1.26	1.28
3	1	-1.394*	.554	.036	-2.71	-.08
2	-.009	.533	1.000	-1.28	1.26
*. The mean difference is significant at the 0.05 level.

The post hoc tests are used to identify which groups differ from each other in case the null hypothesis is rejected. Since the data met the homogeneity of variance assumption, the Tukey’s HSD is used to identify which groups differ from each other. Section 1 is statistically different from 2 (P=0.029) and 3 (P=0.036), respectively. However, section 2 is not statistically different from 3 since P = 1.000 > 0.05 (Field, 2009). 

Section 5

In conclusion, the study answers the research question; do quiz 3 scores vary across the three sections? Evidently, the scores for quiz 3 are statistically different from the three sections. The study has rejected the null hypothesis that claimed that the scores for quiz 3 are equal. The one-way ANOVA is a strong statistical tool that identifies the mean group differences. Clearly, the post hoc tests indicate that the group means for section 1 is different from sections 2 and 3 while the group means for sections 3 and 2 are not different. Again, the ANOVA test is effective when the predictor variable is a categorical one and has more than two groups. However, one of the main limitation is what has been identified in this study – is assumes data is sampled from a normally distributed population which is not always the case (Donley, 2012). Another assumption that the standard deviation between the groups should be similar holds another limitation – in that the greater the difference in standard deviation, the higher the chance of having inaccurate tests. 

References

Field, A. (2009). Discovering statistics using SPSS (3rd ed.). Los Angeles [i.e. Thousand Oaks, Calif.]: SAGE Publications. 

Girden, E. R. (2002). ANOVA: Repeated measures. Newbury Park, Calif: Sage Publications.

Donley, A. (2012). Research Methods. New York : Infobase Pub.

Sawilowsky, S (2009). “New effect size rules of thumb.”. Journal of Modern Applied Statistical Methods. 8 (2): 467–474.