**Descriptive Statistics**

**Part 1**

**RSCH FPX 7864 Assignment 1 Descriptive Statistics**

The histogram above shows the distribution for the final exams for 49 students in the lower division. The final test scores are the independent variable, and the lower division category is the dependent variable. From the histogram, we can see that:

- There are two students with a final score between 40 and 45.
- There are three students with final scores ranging from 45 to 50.
- There are 8 students with a final score between 50 and 55.
- There are seven students with final scores ranging from 55 to 60.
- There are 12 students with a final score between 60 and 65.
- There are seven students with final scores ranging from 65 to 70.
- There are 10 students with a final score between 70 and 75.

From the histogram, it can be concluded that a majority of students in the lower division scored between 60.1 and 65 in the final exam. The above histogram is skewed to the left. According to Sulewski (2020), a left-skewed histogram is one in which the left-side tail of the peak is longer than the right-side tail. The mean of the lower division is 61.469, while the median is (60 + 65)/2 = 62.5. The median is greater than the mean, and thus the histogram is a skewed-left histogram.

**RSCH FPX 7864 Assignment 1 Descriptive Statistics**

The histogram above shows the distribution for the final exams for 56 students in the upper division. The independent variable is the final exam results, while the dependent variable is the upper division category. From the histogram, we can see that:

- There are 11 students with a final score between 50 and 55.
- There are 12 students with final scores ranging from 55 to 60.
- There are 14 students with final scores ranging from 60 to 65.
- There are 13 students with final scores ranging from 65 to 70.
- There are six students with a final score between 70 and 75.

From the histogram, it can be concluded that a majority of students in the upper division scored between 60.1 and 65 in the final exam. The above histogram is a normal histogram. There is uniformity around the median, and the histogram follows the conventional bell shape, with the majority of the frequency counts concentrated in the center and tails that gradually decrease in size. The mean of the upper is 62.161, while the median is (60 + 65)/2 = 62.5. The mean of the histogram is almost equal to the median, and this is a characteristic of a normally distributed histogram (Sulewski, 2020).

**Data Set Interpretation**

**Part 2**

**RSCH FPX 7864 Assignment 1 Descriptive Statistics**

The means of the GPA and Quiz 3 are 2.862 and 7.133, respectively.

Skewness is a metric for a distribution’s asymmetry (Eberl & Klar, 2021). This number might be either positive or negative. A graph with a normal distribution has skewness equal to zero. This indicates that the graph is symmetric around the mean and that its left side is a mirror image of its right side. Eberl and Klar (2021) reported that skewness values less than -0.5 are considered to be left-skewed, greater than 0.5 are right-skewed, and between -0.5 and 0.5 are considered to be approximately symmetric. The skewness of the GPA was found to be -.22, which indicates that the distribution is approximately symmetric. The skewness of quiz 3 was found to be -.08, and this indicates that the distribution is approximately symmetric.

Kurtosis quantifies how heavily or lightly a dataset deviates from a normal distribution (Budny, 2022). Sets of data that have a high kurtosis often have heavy tails and a greater number of outliers, whereas data sets with a low kurtosis typically have light tails and fewer outliers. A normal distribution has a kurtosis value of 3. A distribution is considered to be platykurtic if the kurtosis value is less than 3. Leptokurtic distributions are those that have kurtosis values larger than 3. Using Fisher’s definition, a normal distribution has a kurtosis equal to zero (Budny, 2022). The kurtosis of the gpa was found to be -.69, which indicates that the distribution is light-tailed in comparison with the normal distribution. The kurtosis quiz3 value was .15, indicating that the distribution is heavy-tailed when compared to the normal distribution.

**RSCH FPX 7864 Assignment 1 Descriptive Statistics **

**References**

Budny, K. (2022). Improved probability inequalities for Mardia’s coefficient of kurtosis. *Statistics & Probability Letters*, *191*, 109664.

https://doi.org/10.1016/j.spl.2022.109664

Eberl, A., & Klar, B. (2021). Expectile‐based measures of skewness. *Scandinavian Journal of Statistics*, *49*(1), 373-399.

https://doi.org/10.1111/sjos.12518

Sulewski, P. (2020). Equal-bin-width histogram versus equal-bin-count histogram. *Journal of Applied Statistics*, *48*(12), 2092-2111.