RSCH FPX 7864 Assessment 1 Descriptive Statistics

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• RSCH FPX 7864 Assessment 1 Descriptive Statistics.

Descriptive Statistics

Capella University

RSCH-FPX7864

Professor Name

Date

Figure 1

Descriptive Statistics

Descriptive statistics reveal a difference between the upper and lower divisions, as indicated by large values such as mean, standard deviation, and data range. The upper division has a mean of M = 62.161, slightly higher than that of the lower division, M = 61.469. In addition, the lower division has a larger standard deviation (SD = 8.595), indicating a greater spread of scores. On the contrary, the upper division has a smaller standard deviation (SD = 6.747), indicating greater value homogeneity (Cooksey, 2020). The entire dataset ranges from 40 to 75, encompassing all the observed values. The implication is that the greater the mean of the upper division, the more dispersed the lower one is in the score, i.e., it has a more spread distribution among its points. Explore RSCH FPX 7864 Assessment 2 Correlation Application and Interpretation for more information.

Figure 2

Lower Division

The histogram shows the distribution of scores on a final exam compared with the achievement of a second sample of 49 lower-division students. In this case, the scores (40 to 75) are binned in 5-unit intervals, which provides a more accurate estimation of the score distribution. 60-65 were the most frequent students, with the frequency of two students 40-45, three students 45-50, eight students 50-55, seven students 55-60, twelve students 60-65, seven students 65-70, and ten students 70-75. The orientation of the t is left, as indicated; there is a longer tail on the left, which suggests that fewer students scored lower marks (Pajankar, 2021). Skewness in the distribution is also evident from the mean (M) of 61.469 and the median of 62. The average is greater than the mean. This indic. This indicates that most students scored above average, but there were abysmally low scores.

Figure 3 

Upper Division

The histogram displays the distribution of scores in the final examination for 56 upper-division students, with the independent variable being examination scores and the dependent variable being whether the student is an upper-division student. Scores are from 50 to 75 at 5-unit levels for better interpretation of the distribution. Above everything, student scores are divided as 11 in 50-55, 12 in 55-60, 14 in 60-65, 13 in 65-70, and 6 in 70-75. The bar graph shows a hump in the 60-65 area, or that there are primarily students in the general area. The histogram is bell-shaped, indicating a normal distribution, as the scores cluster in the middle and slope off symmetrically in both directions (Pajankar, 2021). The mean of 62.5 and the median score, M = 62.161, also go almost hand in hand, as confirmed with a standard deviation (SD) of 6.747. The concordance indicates that the data is highly symmetrical with zero high skewness. The histogram shows relatively homogeneous, uniform performance in the upper division group with normal variation in the examination scores around the mean score.

Data Set Interpretation

Part 2

Figure 4

Descriptive Statistics

Interpretation of Results

The dataset represents the typical performance of college students, based primarily on intensive statistical data, including the mean (M), standard deviation (SD), GPA, and Quiz 3 scores. Descriptive records report the central tendency and variation in college students’ overall performance. The most common rating is the advocate score, and the range of ratings from the average is the standard deviation, which indicates how frequently college students rate about average (Darling, 2022). Descriptive records variability and distributions for GPA and Quiz Three are illustrated. The advocate M = 2.862 and significant deviation SD = 713 for GPA. Variability is characterised by the heterogeneity of scholars’ performances, ranging from extreme to extreme. The skewness (0.220) indicates that students are leaning closer to the better part of the distribution. GPAicantly less kurtotic at -0.688 and, therefore, flatter than the ordinary distribution curve. Flatness indicates lower excessive GPA ratings in the distribution, with scores more concentrated around the suggested range. The purpose of Maths Quiz Three is 7.133, and the equal variance deviation is SD 1.600.

Maths -0.078 leaves Quiz 3 left-skewed, and, as a result, Maths has a symmetrical distribution with a fairly minute left skew. Maths. A low skew indicates that the distribution is nearly normal, albeit with a slight downward slope towards the more extreme scores. Quiz three has a kurtosis of 0.149, indicating an extraordinary yet more peaked distribution than GPA. Both rating distributions for Quiz 3 and GPA exhibit some non-normality, although the latter is slightly flatter with more left skew than the former in the case of Quiz 3. The statistical description provides insight into examining variability in ordinary, normal overall performance measurements between Quiz 3, GPA and presentations. Each variable is slightly left-skewed, but the distribution unfolds with lower Kurtosis than the latter in the case of Quiz 3. Proof has supported measures of central tendency and variability of the average overall student performance based on these measures.

Skewness and Kurtosis Distribution

Skewness and Kurtosis are beneficial in assessing the structure and distribution of statistics and normality. Skewness and Kurtosis are essential to consider when screening for the normality of a data set. Skewness is used to measure the degree of asymmetry, and readings ranging from -1 to +1 generally span a nearly symmetrical distribution. Kurtosis is measured in terms of the tail shape within the distribution, and measurements ranging from -2 to +2 are typically considered to indicate no extreme tail shapes in the data (Hatem et al., 2022). For instance, the skewness of GPA measurements is -0.220, i.e., drastically insignificant left skew. However, due to the reality that the diploma is in the ideal range, the distribution of GPA can be considered nearly symmetrical. Skewness in Quiz Three is towards zero at zero.078, i.e., roughly exceptional properly-allocated quantity with almost zero skewness, i.e., factors are lightly spaced across the endorse (Church et al., 2019).

Regarding Kurtosis, the GPA dataset’s Kurtosis is -0, indicating that the GPA distribution is much less skewed and has a milder tail than the regular distribution. Importing correctly yields proper endorsements. Inside an ordinary Quiz 3 course, the Kurtosis of 0.149 is closer to zero. It also indicates an everyday curve-type shape with recommended peaks and weights of tails. As a result of the truth, each skewness of -2 to +2 and Kurtosis of -2 to +2 is high-quality; GPA and Quiz three facts test a normal distribution to a significant degree. This would be helpful in future statistical assessments, i.e., parametric checks, where normality is a vital requirement for making accurate inferences.

Interpretation of Results

The dataset offers a detailed look at student performance by examining necessary statistical measures: Mean (M), Standard Deviation (SD), GPA, and Quiz 3 scores. The descriptive statistics help clarify the central tendencies and variability within student outcomes. The mean represents the average score, while the standard deviation indicates the spread of scores around the mean, showing how consistently students perform relative to the average (Darling, 2022). The descriptive analysis of the variables GPA and Quiz 3 provides insights into the distributions and variability,  with a 2.862 for GPA and a standard deviation of 0.713. The variability points to some diversity in student academic performance levels, but not an extreme range. The skewness of GPA is -0.220, indicating a slight leftward skew, suggesting that more students score on the higher GPA scale. Additionally, the kurtosis for GPA is -0.688, which implies a flatter distribution than a standard curve. The flattening suggests fewer extreme values in the GPA distribution, with scores clustering more closely around the mean. 

In comparison, Quiz 3 shows a mean M = 7.133 with a standard deviation SSD = The skewness for Quiz 3 is -0.078, indicating a near-symmetric distribution with only a slight leftward tendency. The mild negative skew shows that the distribution is close to normal but has a slight lean towards higher scores. The kurtosis of Quiz 3 is 0.149, which suggests a distribution closer to normal than GPA, though it is slightly more peaked. Both distributions for GPA and Quiz 3 scores show a mild deviation from normality, with GPA exhibiting more flattening and a somewhat stronger left skew than Quiz 3. The statistical summary provides a foundation for understanding the differences in performance measures between GPA and Quiz 3. While both variables exhibit minor leftward skewness, GPA has a slightly wider spread and lower kurtosis than Quiz 3. The information helps assess the central tendencies and variability in student performance based on the metrics.

Skewness and Kurtosis Distribution

Skewness and kurtosis provide valuable insights into the shape and distribution of data, particularly concerning normality. Skewness and kurtosis are crucial metrics for evaluating the normality of a data distribution. Skewness measures the asymmetry, with values between -1 and +1 often indicating a distribution close to symmetric. Kurtosis assesses the shape of the distribution tails, where values between -2 and +2 are commonly interpreted as usual, indicating that the data does not exhibit extreme tail behaviour (Hatem et al., 2022). For example, the GPA data shows a skewness of -0.220, suggesting a slight left skew. However, because the value is within the standard range, the GPA distribution can still be considered almost symmetric. Quiz 3’s skewness is even closer to zero at -0.078, which reflects an almost perfectly balanced distribution with minimal skewness, meaning that the data points are symmetrically distributed around the mean (Church et al., 2019).

Regarding kurtosis, the GPA data has a kurtosis of -0.688, suggesting that GPA has lighter tails and a flatter peak than a normal distribution. The data implies a lower concentration of data points around the mean, but the distribution is still close to normal. Quiz 3’s kurtosis value of 9 is nearer zero, indicating a shape similar to a standard curve, with average peaks and tail weights. Given skewness and kurtosis values within the acceptable ranges of -2 to +2, it is reasonably assumed that the GPA and Quiz 3 data follow a near-normal distribution. This is valuable for further statistical tests, particularly parametric tests, which require normality for accuracy in inference.

References

Church, B. V., Williams, H. T., & Mar, J. C. (2019). Investigating skewness to understand gene expression heterogeneity in large patient cohorts. BioMed Central Bioinformatics, 20(4), 4–18. https://doi.org/10.1186/s12859-019-3252-0 

Cooksey, R. W. (2020). Descriptive statistics for summarising data. Illustrating Statistical Procedures: Finding Meaning in Quantitative Data, 6(3), 61–139. https://doi.org/10.1007/978-981-15-2537-7_5

Darling, H. S. (2022). Do you have a standard way of interpreting the standard deviation? A narrative review. Cancer Research, Statistics, and Treatment, 5(4), 728. https://doi.org/10.4103/crst.crst_284_22 

Hatem, G., Zeidan, J., Goossens, M., & Moreira, C. (2022). Normality testing methods and the importance of skewness and kurtosis in statistical analysis. Science and Technology, 3(2), 2-8.. https://doi.org/10.54729/ktpe9512 

Pajankar, A. (2021). Histograms, contours, and stream plots. Apress EBooks, 99–110. https://doi.org/10.1007/978-1-4842-7410-1