Purpose of Statistical Analysis
Statistical analysis is used to examine data and draw conclusions about trends and distribution patterns, as well as the nature of relationships between variables. For this paper, we propose referring to Z-statistics, a standardized tool for estimating distributions that are close to normal (Andrade, 2021). In analysis, the Z-score is used in multiple contexts, including standardizing data, calculating probabilities, testing hypotheses, and identifying outliers in distributions (Nevil, 2023). In this assignment, the Z-score is used to calculate percentiles, determining the percentage of the entire distribution above or below the current parameter. IBM SPSS is used for the analysis to automate and optimize all calculations and avoid errors.
Data Description
The original dataset consisted of four columns (variables), as shown in Figure 1. The variable Month indicated the current month, and the remaining columns contained data on students’ average performance by class. For example, the value in the second row, second column indicates that, as of September, the average performance of students in Class A was 67 out of 100. Comparing within a single class seems straightforward.
One can see that each class’s performance rose or fell throughout the code. However, comparing performance across classes can be more difficult because the factors that affect this indicator are not taken into account: complexity, class composition, and other external variables. For this reason, it makes sense to standardize the values by converting them into Z-scores using IBM SPSS.
Data Manipulation and Analysis
Figure 2 shows the expansion of the data set by adding three new columns. Each value in these columns was automatically calculated using a formula and represents the corresponding Z-score (Nevil, 2023). So, for example, the previously mentioned achievement value of 67 units was transformed into a Z-score of -1.38642.
The next step in the analysis was to determine the percentile for each calculated Z-score (Kirk, 2023). On the one hand, this could be accomplished using a rule of thumb, namely a chart showing the corresponding Z-score percentiles (Figure 3). Based on this rule of thumb, the 67.00 performance for Class A in September (Z = -1.39) was above the 10% threshold (0.13% + 2.14% + ≈8%). Using the same rule, the September performance of 81.00 (Z = -1.64) for Class B was above about 5% of all grades for that class for the year (0.13% + 2.14% + ≈3%).
However, it is clear that such calculations are not precise and rely on rough percentile estimates. Therefore, on the other hand, we can use the properties of a normal distribution and find that the September performance of 67.00 for Class A (Z = -1.39) was above 8.28% of the distribution, and the September performance of 81.00 (Z = -1.64) for Class B was above 5.10%, as shown in Figure 4.
Percentiles are used to estimate the cumulative probability that a particular value lies above a given percentile of the distribution, using the properties of the normal distribution. Percentiles seem more straightforward to use than Z-scores because percentiles make more practical sense. First, percentiles provide information about how much a particular value differs from the rest of the distribution — in other words, the higher the percentile, the more prominent the value is (Kirk, 2023). Second, percentiles have only positive values, so comparing them is easier than with Z-scores.
References
Andrade, C. (2021). Z scores, standard scores, and composite test scores explained. Indian Journal of Psychological Medicine, 43(6), 555-557.
Kirk, D. (2023). Contemporary mathematics. OpenStax.
Nevil, A. (2023). How to calculate Z-score and its meaning. Investopedia.