Analysis and Interpretation Using the One-Way ANOVA

Analysis and Interpretation Using the One-Way ANOVA

The one-way analysis of variance (ANOVA) is a statistical tool that is used to compare the means between two or more groups of data. The one-way ANOVA aids researchers in determining whether there is a statistical significance when comparing the means of multiple variables, usually independent (Warner, 2013).  This report examines and analyses the data variables in the grades.sav data set (section and quiz3), and using SPSS, interprets the data with supporting statistical output using tables and graphs.

Section 1: Data File Description

The grades.sav data set may be described as follows. The data was presented as a fictional data set by a teacher, who recorded the demographics of his students, as well as their performances across quizzes and a final exam. The performance data was recorded across three sections of the course in question.  The total sample size N is 105, with each section consisting of 35 students. The variables in the data set are section and quiz3, where the variables may be interpreted as the predictor and outcome variables respectively. In terms of scales of measurement, section is a nominal scale while quiz3 may be interpreted as a ratio scale. Warner (2013) explained that nominal scales of measurement are non-numeric variables where numbers have no value, while ratio scales of measurement consist of variables that are measurable, possessing a true zero. In this regard, section is nominal while quiz3 is a variable in the ratio scale of measurement. In the data set, 1 denotes the first class, 2 denotes the second class, and 3 signifies the third class.

Section 2: Testing Assumptions

In order for the one-way ANOVA test to be applicable, the researcher assumes the following. First is that each sample is obtained from a data set that is evenly distributed (normality). In addition, the researcher assumes that each sample has been taken from groups that are unrelated or independent from each other. Moreover, before applying the one-way ANOVA, the researcher another assumption is that the variance in the data in the groups under comparison is the same (equality of variance). Finally, another assumption is that the dependent variable (in this case data in quiz3) can be put on a scale that can be divided into increments, or it is continuous.

The following is Figure 1: the SPSS histogram output for quiz3 and a discussion of the graphic analyses.

Figure 1: the SPSS histogram output for quiz3

Figure 1 represents the histogram output for quiz3 on the SPSS software. From the analysis, the scores of the students ranged from one to ten. The mean is 7.48, the standard deviation is 2.034, and a sample size is N = 105. From the representation, the data does not have anomalies, and the scores occur with few outliers, occurring within a small range. Therefore, the distribution appears to be normal.

The following Figure 2, which is the SPSS description and output showing skewness and kurtosis values for quiz3 and their subsequent interpretations:

Table 1: the SPSS description and output showing skewness and kurtosis values for quiz3

As is evident in Table 1, N = 105, a figure that represents the sample total. The minimum score in quiz 3 was 1, while the highest student scored 10. Kurtosis and skewness are statistical measures that help to describe skewness. Skewness differentiates extreme values in the two tails of data, while Kurtosis measures the extreme values in either of the tails (Stein, 2019). According to Stein (2019), a kurtosis value between ±1.0 is considered excellent, and a value between ±2.0 is considered acceptable. The Kurtosis value for quiz3 is -.168, which is below -1.0 depicting it as an excellent value. When it comes to skewness, Stein (2019) opined that values between ±1.0 are excellent, and ±2.0 is acceptable The skewness falls at -.559 which is below -1.0, proving a normal distribution of data, and thus, the data passes the assumption of normality.

Table 2: SPSS output for Tests of Normality

Table 2 represents the SPSS output for the Shapiro-Wilk test of quiz3. The data set has a sample size of 105 making it the Shapiro-Wilk test the most suitable for this case, due to its small sample size.  If the Sig. value of the Shapiro-Wilk test is over .05, the data is normal. Below .05 means the data significantly deviates from normal distribution. For this case, a value of .000 means that the data significantly deviates from the normal distribution.

Table 3: Test of homogeneity

Table 3 portrays the SPSS output for the Levene test of quiz3. The sig. value of .081 is greater than .05 and consequently, this figure specifies that the variability of the groups is equal.

Summary

The data meets all the assumptions except the normality assumption. The Shapiro-Wilk test of normality confirms that quiz 3 is not normally distributed because of for this case, a value of .000 means that the data significantly deviates from the normal distribution.

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Section 3: Research Question, Hypotheses, and Alpha Level

A suitable research question for this investigation could be:  Is there a significant difference between the scores of quiz 3, between sections 1, 2 and 3? The null hypothesis is that there is not a significant difference on quiz3 among the three sections.  The alternative hypothesis is there is a significant difference in mean scores on quiz3.  The alpha level is set at .05.

Section 4: Interpretation

Figure 2 represents the means plot for the sections on quiz3 scores. An analysis of the output informs the researcher that when it came to quiz3, section 1 depicted the lowest mean,

while section 2 depicted the highest mean.  The means across the sections were different.

Figure 2: Means across the three sections

Table 4: means and standard deviations

In table 4, the researcher can establish that the means of sections 1, 2 and 3 respectively were 6.21, 8.33 and 7.73, with standard deviations of 1.833, 1.528 and 2.169.

The following is the SPSS ANOVA output (Table 5) and a analysis of the results of the F test, including the degrees of freedom, F value, p value, and effect size. The one-way ANOVA test compares the mean scores across sections 1, 2 and 3.

Table 5: SPSS one-way ANOVA output

With an f value in the output is 12.277, while the sum of squares between groups stands at 83.463, while within groups is 346.727.  Because  the critical value or rejecting a null hypothesis is between 3.07 and 3.15 (Warner, 2013), the null hypothesis in this study would be rejected.

The Sig. value in the output is .000, which is lower than the acceptable value of .05.  The .000 value also backs the resolution to reject the null hypothesis.

The post-hoc (Tukey HSD) output Table 6 designates that section 2 had a significantly different mean of  2.121 from section one. Section 2 diverges by  .606 from the mean of section 3.  In this regard, the null hypothesis will be rejected.

Table 6. Multiple Comparisons

Section 5: Conclusion

The one-way ANOVA test was suitable for this study and research question, because there were more than two independent groups to be investigated. The results inform the researcher to reject the null hypothesis, and adopt the alternative hypothesis. The study found out that there is a statistically significant difference in the mean scores of quiz3 across sections 1, 2 and 3. The one-way ANOVA test is useful because it can measure mean differences more than two variables. A limitation, however, is that it assumes groups have very similar standard deviations. When groups differ in this regard, the results may be inaccurate.

References

Stein, R. (2019). Why Does Skewness Matter? Ask Kurtosis. Ask Kurtosis.(August 20, 2019).

Warner, R. M. (2013). Applied statistics: From bivariate through multivariate techniques (2nd       ed.). Thousand Oaks, CA: Sage Publications.