RSCH FPX 7864 Assessment 1 Descriptive Statistics

Name

Capella university

RSCH-FPX 7864 Quantitative Design and Analysis

Prof. Name

Date

Descriptive Statistics

Lower Division

The histogram illustrates the final exam results distribution for a cohort of 49 lower-division students, showing the relationship between their scores and the corresponding score ranges. The exam results serve as the independent variable, while the lower-division category acts as the dependent variable. The data reveals that two students scored between 40 and 45, while three students obtained scores between 45 and 50. Additionally, seven students fell within the 55 to 60 range, and eight students scored between 50 and 55. The most frequently occurring score range is between 60 and 65, where twelve students achieved scores.

Further analysis shows that seven students scored between 65 and 70, while ten students secured scores ranging from 70 to 75. The concentration of scores in the higher range suggests that many students performed well in their final assessments (Yağcı, 2022). With the highest number of students (12) scoring between 60 and 65, this range represents the most common performance level. The left-skewed distribution, where the longer tail extends toward the lower score range, indicates that a majority of students scored closer to the higher end of the distribution (Liu et al., 2024).

This observation is further confirmed by the median score (62.5) being slightly higher than the mean (61.469), which signifies that while most students performed well, a smaller subset obtained significantly lower scores, thereby reducing the average. Understanding this distribution pattern is essential for assessing performance trends among lower-division students and can help shape future exam preparation strategies.

Upper Division

The histogram depicting the final test scores for 56 upper-division students effectively captures the relationship between exam results and student performance categories. The data reveals that eleven students scored between 50 and 55, while twelve students fell within the 55 to 60 range. Furthermore, fourteen students scored between 60 and 65, indicating a solid understanding of the course material.

A closer examination shows that thirteen students scored between 65 and 70, representing strong performance, whereas six students excelled by achieving scores between 70 and 75. The highest concentration of students is observed in the 60 to 65 range, signifying that most upper-division students attained their scores in this interval (Dhal et al., 2020).

The histogram exhibits a bell-shaped curve, suggesting a normal distribution, with a peak frequency at the center and a gradual decrease at both extremes. The calculated average score of 62.161 aligns closely with the median score of 62.5, reinforcing the notion of a normally distributed dataset. The symmetry of the distribution, along with the alignment of the mean and median, indicates that student performance follows a typical bell curve pattern, with most students scoring within the middle range and fewer students positioned at the extremes.

Data Set Interpretation

The GPA distribution exhibits skewness values ranging from -0.220 to 0.220, suggesting a slight negative skew. This minor leftward skew implies that lower GPA values are marginally more prevalent but not to a significant extent. The kurtosis values, ranging from -0.688 to 0.688, indicate that the distribution is flatter than a normal curve, meaning that GPA values are more dispersed rather than tightly concentrated around the mean (Jammalamadaka et al., 2020).

Despite these small deviations from perfect normality, the skewness and kurtosis values remain within the acceptable range for normality, typically considered -1 to +1 for skewness and -2 to +2 for kurtosis. This suggests that the GPA distribution is approximately normal, with only slight asymmetry and a relatively flat shape. These insights are valuable for analyzing GPA trends, indicating that while the distribution is not perfectly normal, it remains within acceptable statistical limits.

For Quiz 3, the distribution’s skewness is negative, ranging from -0.078 to 0.078, signifying a minor asymmetry in the dataset. Additionally, the kurtosis values, which range from -0.149 to 0.149, indicate that the distribution is slightly more peaked than a standard normal curve. While these deviations are minimal, the distribution still largely conforms to normality. The skewness and kurtosis values remain within the standard thresholds for normality (-1 to +1 for skewness and -2 to +2 for kurtosis), confirming that the distribution maintains an overall normal shape (Mohammed et al., 2020).

Although the distribution exhibits slight deviations from a perfectly normal shape, the combined skewness and kurtosis analysis offers a deeper understanding of the dataset’s overall characteristics. These statistical indicators are crucial for determining whether the data aligns with the expectations of normality, which is essential for accurate data interpretation and analysis.

Table Representation

Category	Findings	Interpretation
Lower Division	– Most common score range: 60-65 (12 students) – Skewness: Left-skewed – Mean: 61.469 – Median: 62.5	The left-skewed distribution suggests that while most students performed well, a small number scored significantly lower, affecting the overall average. The highest concentration of scores was in the 60-65 range.
Upper Division	– Most common score range: 60-65 (14 students) – Distribution: Bell-shaped, normal – Mean: 62.161 – Median: 62.5	The normal distribution suggests that students’ scores were evenly spread, with most students scoring near the average. The symmetry between the mean and median supports this observation.
GPA Distribution	– Skewness: -0.220 to 0.220 (minor negative skew) – Kurtosis: -0.688 to 0.688 (flatter than normal)	The data distribution is close to normal but slightly flatter and negatively skewed. This suggests that lower GPAs are slightly more common, but the deviation is within acceptable limits.
Quiz 3	– Skewness: -0.078 to 0.078 (minor negative skew) – Kurtosis: -0.149 to 0.149 (slightly more peaked)	The distribution is nearly normal, with minor asymmetry and slightly increased peak concentration. These variations do not significantly affect overall data interpretation.

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