RSCH FPX 7864 Assessment 1 Descriptive Statistics

Name

Capella university

RSCH-FPX 7864 Quantitative Design and Analysis

Prof. Name

Date

Descriptive Statistics

Part 1

Lower Division

The histogram presents a detailed distribution of final exam results for a cohort of 49 lower-division students, illustrating the relationship between their scores and the corresponding score ranges. Exam results are the independent variable in this analysis, and the lower-division category is the dependent variable. Two students obtained scores between 40 and 45, and three students scored between 45 and 50, according to the data, which shows the score distribution. Seven pupils received a score between 55 and 60, and eight students fell between 50 and 55. The most populated range is between 60 and 65, with twelve students achieving scores in that interval.

Seven students scored between 65 and 70, and ten students secured scores ranging from 70 to 75. This distribution indicates a notable clustering of scores around the higher end, suggesting that many students performed well in their final assessments. (Yağcı, 2022). The largest number of students, 12 in total, scored between 60 and 65, suggesting this is the most common score range. The data reveals a left-skewed distribution, with a longer tail extending to the lower score range, indicating that more students scored closer to the higher end of the distribution (Liu et al., 2024).

This is further reinforced by the fact that the score is left-skewed, with the median (62.5) being higher than the mean (61.469). This distribution implies that although the majority of students did well, a smaller fraction scored much poorly, which depressed the average. This insight is valuable for identifying performance patterns in lower-division students and could inform future exam preparation strategies.

Upper Division

The distribution of final test scores among 56 upper-division students is effectively depicted by an examination results histogram, which also highlights the relationship between the dependent variable (exam results) and the independent variable (student performance categories). The histogram indicates that the score ranges provide important information about student performance: Eleven students had scores in the range of 50 to 55, while twelve students received scores in the range of 55 to 60. Moreover, fourteen pupils were in the 60–65 range. Indicating a stronger grasp of the material.

Thirteen students scored between 65 and 70, demonstrating solid performance, and six students excelled with scores between 70 and 75. This detailed breakdown provides a clear view of how students performed across different scoring brackets, facilitating targeted interventions and strategies for academic improvement. Therefore, the range between 60 and 65 has the highest concentration of pupils, meaning that most upper-level students received their scores in this range (Dhal et al., 2020).

The histogram’s bell-shaped curve suggests a normal distribution, with the frequency peaking in the center and tapering off at both extremes. The average score is calculated to be 62.161, which is close to the median score of 62.5, further supporting the conclusion of a normally distributed data set. The alignment between the mean and median, combined with the symmetrical distribution of scores, reinforces the observation that student performance in this group follows a typical bell curve pattern, with a central tendency toward the middle range of scores and fewer students scoring at the extremes.

Data Set Interpretation

Part 2

The GPA distribution’s skewness values, which range from -0.220 to 0.220, indicate a minor negative skew, indicating that the distribution’s left tail is marginally longer than its right. This subtle skewness suggests that lower GPA values are more frequent, though not excessively so. Additionally, the distribution’s kurtosis, ranging from -0.688 to 0.688, indicates that it is flatter than the normal curve. Negative kurtosis values suggest a distribution that is less peaked and more dispersed compared to a normal distribution, reflecting a wider range of GPA values without a sharp concentration around the mean (Jammalamadaka et al., 2020).

Despite minor deviations from a perfectly normal distribution, the skewness and kurtosis values fall within acceptable limits for normality, typically considered -1 to +1 for skewness and -2 to +2 for kurtosis. This implies that the GPA distribution is often near to a normal distribution despite exhibiting a minor asymmetry and a flatter shape. These findings provide useful insights for interpreting GPA data, indicating that the distribution does not deviate significantly from expectations and can still be considered reasonably normal for most statistical purposes.

Quiz 3 Distribution

The skewness of the Quiz 3 distribution is negative, with values ranging from -0.078 to 0.078. This suggests a minor asymmetry in the distribution, where the left and right tails exhibit a slight difference in length. In addition to this, the kurtosis for Quiz 3 ranges between -0.149 and 0.149, implying that the distribution is marginally more peaked than a standard normal curve. Despite these small deviations, the Quiz 3 distribution can still be considered approximately normal. This conclusion is supported by the skewness and kurtosis values, both of which fall within the accepted thresholds for normality, typically defined as -1 to +1 for skewness and -2 to +2 for kurtosis.

Although the distribution exhibits modest deviations from full normalcy, these numbers imply that the distribution’s form remains mostly consistent with a normal curve. The combined analysis of skewness and kurtosis, as noted by Mohammed et al. (2020), provides valuable insights into the overall behavior of the distribution. These statistical measures help better to understand the form and characteristics of the data, ensuring that it aligns with expectations of normality within permissible limits for analytical purposes.

References

Dhal, K. G., Das, A., Ray, S., Gálvez, J., & Das, S. (2020). Histogram equalization variants as optimization problems: A review. Archives of Computational Methods in Engineering, 28(3), 1471–1496. https://doi.org/10.1007/s11831-020-09425-1

Jammalamadaka, S. R., Taufer, E., & Terdik, G. H. (2020). On multivariate skewness and kurtosis. Sankhya A, 83. https://doi.org/10.1007/s13171-020-00211-6

Liu, A., Cheng, W., & Guan, R. (2024). A novel skewed generalized normal distribution: Properties, statistical inference, and its applications. Communications in Statistics – Simulation and Computation, 1–38. https://doi.org/10.1080/03610918.2024.2378952