# Parametric and Non-Parametric Tests in Statistics

## Non-Parametric Procedures

Non-parametric tests are choice tests if the dataset does not assume normality in its distribution (for skewed data) (Field, 2009). In addition, data that does not heed to the assumption of homogeneity of variances calls for the use of non-parametric rather than parametric tests. The preference for non-parametric tests in place of parametric tests would also be based on the fact that non-parametric tests have the ability to give exact probabilities in spite of how the population from which the sample was taken is distributed. In addition, a non-parametric test is valuable for analyzing data with small samples sizes. When working with data that is classificatory, non-parametric tests become vary indispensable just as they are when working with data which is collected from a number of different populations. Data that is essentially arranged in ranks or data which gets more strength by arranging its numerical scores in ranks is suitably handled by non-parametric tests (Hebel, 2002).

Non-parametric tests are less powerful compared to their parametric counterparts, unless when testing significance when power is almost equal. The low power of non-parametric tests is attributed to the fact that the data is usually not normally distributed, but instead the data is skewed towards one direction. Parametric tests calculate their statistical power by use of formulas as well as dependence on tables as well as graphs and hence they have more power. On the other hand, non-parametric tests register a lower statistical power since power calculation is done in less straightforward approaches (Hebel, 2002).

 Parametric Non-parametric Dependent t-test Wilcoxon signed Ranks Independent samples t-test Mann Whitney U Repeated measures ANOVA (one-variable) Friedman’s Test One-way ANOVA (independent) Kruskal Wallis H Pearson Correlation Spearman Rho

## Non-Parametric Tests for Parametric Versions

### Wilcoxon’s Test

The non-parametric test that was selected in place of the dependent t test (parametric test) is the Wilcoxon’s test. This was used to determine whether taking part in a creative writing course causes an increase scores as assessed in a creativity assessment. Activity 6a.sav data was used to conduct this analysis. Table 1 indicates that there were 9 cases where creativity pre-test scores were greater than creativity post-test scores. In 28 cases, the creativity test scores were greater than creativity pre-test scores and only in 3 cases were creativity pre-test scores equal to the creativity post-test scores. The Wilcoxon’s test was conducted to establish whether 40 students registered higher post-test scores than pre-test scores. It was identified that the difference between pre-test and post-test creativity score were significantly different, Wilcoxon’s Z = -3.179, p =.001 (2-tailed significance) (Table 2). It was clear than post-test scores were significantly higher than creativity pre-test scores.

### Mann Whitney U test

A new dataset (New Activity 6.sav/or Activity 6b.sav) was created from activity 6a.sav for use in between-subjects design. The new data set contained two variables only (creativity test scores while pre-test and post-test scores became the grouping variable). The most suitable non-parametric test that was selected to conduct the independent t test (parametric version) was the Mann Whitney test. This new dataset was used to conduct a Mann Whitney U test. A Mann Whitney U test was conducted on creativity test scores and it was identified that the average rank (pretest) scores was 36.22 whereas the mean rank scores for post-test scores was 44.78 (Table 3). This implies that creativity test scores conducted after the test were higher than creativity test scores registered before the creativity test was conducted. Table 4 displays the Mann Whitney U which was conducted at 2-tailed level of significance. The Mann Whitney U when applied to creativity test scores found that there existed no significant differences between the pre-test creativity scores and the post-test creativity scores, Mann Whitney U (N = 80) = 929.00, p =.10, which is greater than the significance level of.05.

### Friedman’s test

The Friedman’s test was a choice non-parametric version of the single factor ANOVA (parametric test) for identifying differences in blood pressures depending on the setting. Activity 6c.sav was used on this analysis. On conducting a Friedman test to find out differences in systolic and diastolic blood pressures based on the setting, the mean rank for systolic blood pressure was 3.00 while the mean rank for diastolic blood pressure was 2.00 (Table 5). The Friedman test was significant, Chi-Square (N = 30) = 60.00 p =.001 (Table 6). As a result, a Wilcoxon test was conducted as a follow up test to enable pairwise comparisons and the Z score was significant, Wilcoxon Z = -4.784, p =.001 (Table 8). It was established that the median for systolic blood pressure was significantly greater than diastolic pressure, p <.05. In fact in all the 30 cases (Table 7), the systolic blood pressure was higher than the diastolic blood pressure.

## References

Field, A. (2009). Discovering statistics using SPSS, Third edition. San Diego, CA: Sage Publications Ltd.

Hebel, A. (2002). Statistics-when to use them and which is more powerful? Lecture Notes. Department of Natural Sciences, University of Maryland Eastern Shore

## Appendix

Table 1: Ranks for Creativity Pre- and Post-test Scores

 Ranks N Mean Rank Sum of Ranks Creativity post-test – Creativity pre-test Negative Ranks 9a 15.67 141.00 Positive Ranks 28b 20.07 562.00 Ties 3c Total 40 a. Creativity post-test < Creativity pre-test b. Creativity post-test > Creativity pre-test c. Creativity post-test = Creativity pre-test

Table 2: Wilcoxon’s Test for Creativity Pre- and Post-test Scores

 Test Statisticsb Creativity post-test – Creativity pre-test Z -3.179a Asymp. Sig. (2-tailed) .001 a. Based on negative ranks. b. Wilcoxon Signed Ranks Test

Table 3: Ranks for Pre- and Post-Test Score

 Ranks pre-test and post-test group N Mean Rank Sum of Ranks creativity test score Pre-test scores 40 36.22 1449.00 Post-test scores 40 44.78 1791.00 Total 80

Table 4: Mann Whitney U Test for Creativity Test Scores

 Test Statisticsa creativity test score Mann-Whitney U 629.000 Wilcoxon W 1449.000 Z -1.647 Asymp. Sig. (2-tailed) .100 a. Grouping Variable: pre-test and post-test group

Table 5: Ranks for Systolic and Diastolic Blood Pressures

 Ranks Mean Rank Setting 1.00 Systolic Blood Pressure 3.00 Diastolic Blood Pressure 2.00

Table 6: Friedman Test for Systolic and Diastolic Blood Pressures

 Test Statisticsa N 30 Chi-Square 60.000 df 2 Asymp. Sig. .000 a. Friedman Test

Table 7: Follow up Wilcoxon Signed Ranks Test for Blood Pressures

 Ranks N Mean Rank Sum of Ranks Diastolic Blood Pressure – Systolic Blood Pressure Negative Ranks 30a 15.50 465.00 Positive Ranks 0b .00 .00 Ties 0c Total 30 a. Diastolic Blood Pressure < Systolic Blood Pressure b. Diastolic Blood Pressure > Systolic Blood Pressure c. Diastolic Blood Pressure = Systolic Blood Pressure

Table 8: Follow up Wilcoxon’s test for Blood Pressures

 Test Statisticsb Diastolic Blood Pressure – Systolic Blood Pressure Z -4.784a Asymp. Sig. (2-tailed) .000 a. Based on positive ranks. b. Wilcoxon Signed Ranks Test