SAT Scores The SAT is an exam used by colleges and universities to evaluate undergraduate applicants. The test scores are normally distributed. In a recent year, the mean test score was 1509 and the standard deviation was 312 . The test scores of four students selected at random are 1924, 1241,2202, and 1392. (Source: The College Board)

Certainly! Let's analyze the SAT scores you've provided using the principles of normal distribution. Here's a step-by-step breakdown: ### **Given Information:** - **Mean (μ):** 1509 - **Standard Deviation (σ):** 312 - **Sample Scores:** 1924, 1241, 2202, 1392 ### **1. Understanding the Normal Distribution:** The SAT scores are normally distributed, which implies that most students score around the mean, and the probabilities of scoring significantly higher or lower than the mean decrease symmetrically. ### **2. Calculating Z-Scores:** A **z-score** indicates how many standard deviations an element is from the mean. The formula for calculating a z-score is: \[ z = \frac{{X - \mu}}{{\sigma}} \] Where: - \( X \) = individual's score - \( \mu \) = mean score - \( \sigma \) = standard deviation Let's calculate the z-scores for each of the four students: | **Student** | **Score (X)** | **Z-Score Calculation** | **Z-Score (z)** | |-------------|---------------|--------------------------|------------------| | 1 | 1924 | (1924 - 1509) / 312 | 415 / 312 ≈ **1.33** | | 2 | 1241 | (1241 - 1509) / 312 | -268 / 312 ≈ **-0.86** | | 3 | 2202 | (2202 - 1509) / 312 | 693 / 312 ≈ **2.22** | | 4 | 1392 | (1392 - 1509) / 312 | -117 / 312 ≈ **-0.38** | ### **3. Interpreting the Z-Scores:** - **Z-Score = 1.33 (1924):** - **Interpretation:** This score is 1.33 standard deviations **above** the mean. - **Percentile:** Approximately the 91st percentile. This means the student scored better than about 91% of test-takers. - **Z-Score = -0.86 (1241):** - **Interpretation:** This score is 0.86 standard deviations **below** the mean. - **Percentile:** Approximately the 19th percentile. This means the student scored better than about 19% of test-takers. - **Z-Score = 2.22 (2202):** - **Interpretation:** This score is 2.22 standard deviations **above** the mean. - **Percentile:** Approximately the 98.7th percentile. This indicates an exceptionally high score, better than about 98.7% of test-takers. - **Z-Score = -0.38 (1392):** - **Interpretation:** This score is 0.38 standard deviations **below** the mean. - **Percentile:** Approximately the 35th percentile. This means the student scored better than about 35% of test-takers. ### **4. Visual Representation:** To better understand where these scores lie on the normal distribution curve, here's a simple representation: ``` Left Tail Mean (1509) Right Tail <--|---------|---------|---------|---------|---------|--> -2σ -1σ 0 +1σ +2σ ≈ 900 1200 1500 1800 2100 Student Scores: 1241 (-0.86σ), 1392 (-0.38σ), 1924 (+1.33σ), 2202 (+2.22σ) ``` ### **5. Implications for College Admissions:** - **Higher Scores (Above Mean):** - Students with scores like 1924 and 2202 are likely to be competitive candidates, especially in selective colleges. - A score of 2202 places a student in a very high percentile, which can be advantageous for scholarships and honors programs. - **Scores Around or Below Mean:** - A score of 1241 is below average, which might require supplementary materials or a strong overall application to enhance competitiveness. - A score of 1392 is slightly below the mean but still within a reasonable range for many institutions. ### **6. Additional Considerations:** - **Composite Evaluation:** While SAT scores are important, colleges also consider other factors like GPA, extracurricular activities, essays, and recommendation letters. - **Score Trends:** Improvement over time can also be a positive indicator for admissions committees. - **Test-Optional Policies:** Some universities have adopted test-optional policies, placing less emphasis on SAT scores. ### **Conclusion:** Understanding where individual SAT scores fall within the normal distribution helps in assessing one's academic standing compared to peers. It's essential to aim for continual improvement and present a well-rounded application to maximize college admission opportunities. If you have specific questions or need further analysis on these scores, feel free to ask!