Analysis of the Group Statistics

The group statistics (N, Mean, Standard Deviation and Standard Error Mean) of the Mathematics pre-tests and post-tests are shown in the Table 4.2 below.

Table 4.2

Group Statistics of Mathematics Tests

Test N Mean Std. Deviation Std. Error Mean Experimental Group, Mathematics Pre-test 22 39.41 8.86 1.89 Experimental Group, Mathematics Post-test 22 43.64 9.45 2.01 Control Group, Mathematics Pre-test 22 29.5 8.73 1.86 Control Group, Mathematics Post-test 22 31.64 11.89 2.53

I interpreted the Table 4.2 above of Group Statistics as follows: There were 22 valid observations in both experimental and control group for both mathematics tests. The means, the central tendencies of achievements of the experimental group are 39.41 marks for the mathematics pre-test and 43.64 marks for the mathematics post-test. The means, the central tendencies of achievements of the control group are 29.50 marks for the mathematics pre-test and 31.64 marks for the mathematics post-test. The standard deviations, expressing the spread of the data are 8.86 and 9.45 marks for the experimental group, and 8.73 and 11.89 marks for the control group. Howell (2010) defines the standard error as “the standard deviation of any sampling distribution is called the standard error of that distribution” (p. 205). The standard

errors of the means are calculated relative to N, in our case as the corresponding ratios of the standard deviations and the square root of N = 22.

It can be observed that in the previous observation, based on percentage differences between the pre-test and post-test achievements, the increase of the understanding of the concept of function of the participant students in the experimental group was nearly the double of the increase of the understanding of the participant students in the control group which is reflected in the increase of the mean values as well. The increase of the mean of the experimental group is 43.64 - 39.41 = 4.23 marks, and the increase of the mean of the control group is 31.64 – 29.50 = 2.14 marks, which gives the ratio of mean increases Experimental Group Mean Increase / Control Group Mean Increase = 4.23 / 2.14 ≈ 1.98, this being in accordance with the percentage result presented above and is expressing a nearly double improvement of the understanding of the function concept of the students in the experimental group compared to that of the students in the control group. I believe that this result can be attributed to the treatment intervention, showing that this type of active integration of mathematics and music has improved my senior secondary mathematics students' understanding of function.

The standard deviations, expressing the spread of the measure of understanding of the function concept, have increased in different modes. The increase of the standard deviation of the experimental group is 9.45 - 8.86 = 0.59 marks, or (9.45 – 8.86) / 8.86 × 100% = 6.62% increase. The increase of the standard deviation of the control group is 11.89 – 8.73 = 3.16 marks or (11.89 – 8.73) / 8.73 × 100% = 36.20% increase. If we compare the two increases of standard deviations, then we obtain the control group / experimental group ratio of 36.20% / 6.62% = 5.47, which means that the increase of the standard deviation of the control group is nearly 5.5 times bigger than the increase of the standard deviation of the experimental group. These figures indicate that the distribution of the measure of understanding of the two groups became very different. While the pre-tests standard deviations were fairly close to each other, 8.86 marks for the experimental group and 8.73 marks for the control group, showing the equivalence of the two groups also in this way, the post-test standard deviations are different. The experimental group has kept the original distribution of the measures of understanding of the concept of function in a much

more accentuated way than the control group. This means that the students in the experimental group have improved their understanding of function proportionally, but the students in the control group gained improvement of their understanding of function differently, they have spread their improvement of understanding differently than their colleagues in the experimental group, which means that some students improved very well, other students perhaps didn't improve at all, or some students perhaps have decreased their understanding, showing a negative improvement.

The following Table 4.3 presents the statistics regarding the two groups' mathematics pre-tests and post-tests.

Table 4.3

Mathematics Independent Samples Test

Levene’s Test for Equality of Variances t-test for Equality of Means

95% CI of the diff. F Sig. t df Sig. (2- tailed) Mean Diff. Std. Err.

Diff. Lower Upper

Pre 0.32 0.58 3.74 42 .001 9.91 2.65 4.56 15.26 Post 1.14 0.29 3.71 42 .001 12 3.24 5.47 18.53

We interpret the above Table 4.3 of Mathematics Independent Samples Test as follows.

Levene's Test for Equality of Variances

The first column, “F”, provides the F-statistic values (for the mathematics pre-tests and for the mathematics post-tests) of Levene's Test for Equality of Variances, the ratios which underline the analysis of the equality of variances tests.

The second column, “Sig.”, provides the two-tailed p-values (significance value probabilities), regarding the null hypotheses that there are no differences in the variances of the two groups. Because both p-values are greater than the a priori set α = 0.05 alpha level, we don't reject these hypotheses, and we state that Levene's statistics for equality of variances show that there is no statistically significant difference in the variance for both mathematics pre-test (p = .58) and mathematics post-test (p = .29). These results indicated that the variances of both mathematical tests were equal.

t-Test for Equality of Means

In the “t” and “df” columns provide the t-statistic, respective the df-statistic values. In Table 4.3 the df-statistic values, the degrees of freedom, are calculated as df = N – 2. In my study N = 44, so df = 44 - 2 = 42 in both cases. We can see that the t and df statistics are: t = 3.736, df = 42 for the mathematics pre-tests, and t = 3.707, df = 42 for the mathematics post-tests. The equality of the degrees of freedom is a possible sign of normality.

The “Sig. (2-tailed)” column provides the p-values (significance values, two-tailed probabilities) – calculated by using the corresponding t-distribution, and expressing the probability of observing a t-value of equal or greater value under the null hypothesis. If this p-value is less than the pre-set alpha significance level of α = 0.05, then we conclude that the difference in means is significantly different from zero. In our case p = .001 < .05 = α for both mathematics tests, and so we can state that the difference in means is statistically significantly different to zero.

The “Mean Diff.” column provides the mathematics achievement mean differences between the experimental and control groups for both pre-test and post-test, which in this case are 9.91 marks for the mathematics pre-tests, and 12.00 marks for the mathematics post-tests.

The “Std. Error Diff.” column provides the statistics expressing the estimated standard deviations of the differences between the sample means. These are 2.65 marks for the mathematics pre-test and 3.24 for the mathematics post-test. These statistics provide a measure of the variability of the sample means.

The “Lower” and “Upper” columns provide the 95% confidence intervals of the differences of means, which in our case are [4.56, 15.26] for the mathematics pre-test and [5.47, 18.54] for the mathematics post-test.