In 2022, a total of 1,737,678 students who graduated high school in the year took the SAT critical reading test. The mean score for all the students who took the test was 1050, while the standard deviation was 216. For a normally distributed dataset, the first and third quartiles can be calculated based on the provided mean and standard deviation using the formula:
Q1= µ- (.675)σ (Holmes, Illowsky & Dean, 2018).
Q3= µ+ (.675)σ
Where µ is the population mean, and σ is the standard deviation of the entire population of the students who took the SAT test in the period (Johnson & Bhattacharyya, 2017). The first quartile (Q1) represents the 25th percentile, and the third quartile (Q3) is the 75th percentile of the dataset (Johnson & Bhattacharyya, 2019).
Therefore:
Q1 = µ- (.675)σ = 1050 – (.675)*216 = 904.2
Q2 = µ- (.675)σ = 1050 + (.675)*216 = 1,195.8
Thus, the first quartile is 904.2, and the third quartile of the dataset is 1,195.8 based on the standard deviation and the mean score of the population. However, the data provided also shows the percentile table for the total group, which gives the first and third real quartiles of the data. These values are compared greatly with the scores obtained by using the formula above. In the table, the real first quartile is 880, while the third quartile has a value of 1200. For the first quartile, the calculated value for the normally distributed dataset is higher (904.2) than the real quartile (880). However, the real third quartile is larger (1200) than the calculated (1,195.8). Consequently, this provides evidence that the distribution of critical reading scores is not normal (Fan & Luo, 2022).
The data that the College Board has used to provide the statistics captures various aspects of the students who took the test and some of which are personal and confidential. This data captures the personal characteristics of ethnicity, race and color, family financial background, gender, and grade level (College Board, 2022). In addition, it captures data on individual scores and the number of times the test is taken. Therefore, the College Board must have considered certain ethical and legal considerations, such as obtaining permission from the SAT and the education department. Also, it must have agreed to conceal the names and identities of the students.
References
College Board. (2022). SAT suite of assessments annual report. Web.
Fan, M., & Luo, C. (2022). Generalized skew-normal distribution model of partition curves based on quartile. International Journal of Coal Preparation and Utilization, 42(7), 2219-2231. Web.
Holmes, A., Illowsky, B., & Dean, S. (2018). Introductory to business statistics. Samurai Media.
Johnson, R. A., & Bhattacharyya, G. K. (2017). Statistics: Principles and methods. Samurai Media.
Johnson, R. A., & Bhattacharyya, G. K. (2019). Statistics: Principles and methods. John Wiley & Sons.