Sample Assessments

The term descriptive statistics refers to a group of methods that are utilized to provide summaries about the sample and the measures. These methods include measures of central tendency, such as mean, median, and mode, as well as measures of variability or dispersion, such as range, variance, and standard deviation (Cooksey, 2020). Additionally, descriptive statistics include the use of graphical representations, such as histograms, bar charts, and box plots, to visually present data distributions (Mondal et al., 2022). This section employs histograms to show the distribution of final exam scores among students in the lower and upper divisions. These graphic depictions offer a comprehensive analysis, highlighting patterns and inclinations found in the data. Simultaneously, the following section explores central tendency and dispersion metrics, emphasizing skewness and kurtosis, to provide further context for understanding the distribution characteristics of Quiz 3 scores and Grade Point Average (GPA).

Part 1: Histograms for Visual Interpretation

Figure 01

Descriptive Statistics

Lower Division

Figure 02

Histogram

Figure 02 presents the distribution of final exam scores for the lower division for the 49 students as a histogram. The lower division category is the dependent variable, and the test result is the independent variable (Lee, 2022). The scores for the students are shown on the histogram as follows: 2 falls between 40 and 45, 3 between 45 and 50, 8 between 50 and 55, 7 between 55 and 60, 12 between 60 and 65, 7 between 65 and 70, and 10 between 70 and 75.

Most lower-division students received final test scores in the 65–67 range, based on the histogram. The distribution of the peak is left-skewed in the histogram because its left tail is longer than its right. The median score for the lower division is 62.5, higher than the mean score of 61.469, according to the results. The histogram’s left-skewed distribution is becoming more and more evident.

Upper Division

Figure 03

Histogram

The distribution of the 56 upper-division students’ final test scores is shown by the histogram in Figure 3, which uses the students’ category as the independent variable and the exam results as the dependent variable. Eleven students had scores between fifty-five and sixty, twelve between sixty and seventy, thirteen between sixty and seventy-five, and six between seventy-five and seventy-five, according to the data displayed by the histogram. The majority of upper-level students had final test scores in the 65–67 range, according to the histogram analysis. According to Yu et al. (2021), the bell-shaped histogram displays a normal distribution with the highest frequency counts in the middle and a rapid drop-off towards the tails (Yu et al., 2021). The median (62.5) and average (62.161) scores for the top division are similar. In cases where the mean and median are represented by a consistently distributed histogram.

Part 2: Measures of Central Tendency and Dispersion

Figure 04

Descriptive Statistics

According to Figure 04, the sample of students who completed Quiz 3 had an average score of 7.133 and a standard deviation of 1.792. On the other hand, the average score for the group of students evaluated using their GPA was 2.862, with a standard deviation of 0.343. With a skewness score of -0.220, the GPA distribution exhibits a small negative skewness, pointing to a longer tail on the left side of the distribution. Quiz 3 also shows a slight negative skewness of -0.078. Furthermore, the distributions of Quiz 3 and GPA also show flatter features than the norm, with kurtosis values of -0.149 and -0.688, respectively.

For the GPA distribution, a skewness rating in the range of -0.220 to 0.220 denotes a little negative shift. This indicates that the GPA distribution has a longer left tail than a shorter right tail. Furthermore, the kurtosis value, which varies from -0.688 to 0.688, shows that the GPA distribution is flatter than a normal distribution. Compared to a distribution with a positive kurtosis value, one with a negative one is more dispersed and has fewer peaks (J. Einasto et al., 2021). Despite some deviations from a fully normal distribution, the skewness and kurtosis values are both within the allowed ranges, suggesting that the GPA distribution can be considered fairly typical.

Quiz 3 Distribution

Quiz 3’s distribution is somewhat skewed negatively, with a skewness value between -0.078 and 0.078. This suggests that the left tail of the distribution is slightly longer than the right tail. The kurtosis score of Quiz 3 displays a somewhat more peaked distribution than usual, ranging from -0.149 to 0.149. Apart from these anomalies, the Quiz 3 distribution’s skewness and kurtosis values fall between the normality ranges of -1 and +1 for skewness and -2 to +2 for kurtosis. We can therefore presume that it is roughly average.

The GPA and Quiz 3 distributions show considerable deviation from a totally normal distribution, but their skewness and kurtosis values are within acceptable bounds for requirements related to normalcy (Demir, 2022). These metrics provide insight into the structure and properties of the distributions, which aids in our understanding of their statistical aspects.

References

Cooksey, R. W. (2020). Descriptive statistics for summarising data. Illustrating Statistical Procedures: Finding Meaning in Quantitative Data, 1, 61–139. https://doi.org/10.1007/978-981-15-2537-7_5

Demir, S. (2022). Comparison of normality tests in terms of sample sizes under different skewness and kurtosis coefficients. International Journal of Assessment Tools in Education, 9(2), 397–409. https://doi.org/10.21449/ijate.1101295

Einasto, J.  Klypin, A. Hütsi, G. Liivamägi, L. J., & M. Einasto. (2021). Evolution of skewness and kurtosis of cosmic density fields. Astronomy and Astrophysics, 652, A94–A94. https://doi.org/10.1051/0004-6361/202039999

Lee, S. W. (2022). Regression analysis for continuous independent variables in medical research: statistical standard and guideline of Life Cycle Committee. Life Cycle, 2. https://doi.org/10.54724/lc.2022.e3

Mondal, H., Swain, S., & Mondal, S. (2022). How to conduct descriptive statistics online: A brief hands-on guide for biomedical researchers. Indian Journal of Vascular and Endovascular Surgery, 9(1), 70. https://doi.org/10.4103/ijves.ijves_103_21

Yu, J. C., Shukla, D., Doroodchi, A., Doroodchi, R., Khodadadi, H., & Linder, D. (2021). Descriptive biostatistics and epidemiology for craniofacial surgeons: the what, who, when, where, how, and why in data description, analysis, presentation, and interpretation. FACE, 2(4), 414–425. https://doi.org/10.1177/27325016211052939