Question

Each year thousands of high school students take either the SAT or the ACT, standardized tests used in the college admissions process. Combined SAT Math and Verbal scores go as high as 1600, while the maximum ACT composite score is 36 . Since the two exams use very different scales, comparisons of performance are difficult. A convenient rule of thumb is \(S A T=40 \times A C T+150\); that is, multiply an ACT score by 40 and add 150 points to estimate the equivalent SAT score. An admissions officer reported the following statistics about the ACT scores of 2355 students who applied to her college one year. Find the summaries of equivalent SAT scores. Lowest score \(=19 \quad\) Mean \(=27\) Standard deviation \(=3\) \(\mathrm{Q} 3=30\) Median \(=28 \quad \mathrm{IQR}=6\)

Short answer

Lowest SAT score: 910, Mean SAT: 1230, SAT SD: 120, Q3 SAT: 1350, Median SAT: 1270, SAT IQR: 240.

Step-by-step solution

Understand the Relationship Between SAT and ACT

The problem gives a formula to convert ACT scores to SAT scores: \( \text{SAT} = 40 \times \text{ACT} + 150 \). This formula will be used to convert all the given ACT statistics to the equivalent SAT statistics.

Convert the Lowest ACT Score to SAT

The lowest ACT score given is 19. To find the equivalent SAT score, substitute the ACT score into the formula: \( \text{SAT} = 40 \times 19 + 150 \). Calculating, we get \( \text{SAT} = 760 + 150 = 910 \). Thus, the lowest equivalent SAT score is 910.

Convert the Mean ACT Score to SAT

The mean ACT score is 27. Using the conversion formula: \( \text{SAT} = 40 \times 27 + 150 \). Calculating, we find \( \text{SAT} = 1080 + 150 = 1230 \). Therefore, the mean equivalent SAT score is 1230.

Convert the ACT Standard Deviation to SAT

The standard deviation of the ACT scores is 3. Since standard deviation is a measure of variability and follows linear transformation rules, it is converted by multiplying by 40: \( \text{SAT standard deviation} = 40 \times 3 = 120 \).

Convert ACT Q3 to SAT

The third quartile (\(Q3\)) for ACT scores is 30. Convert to SAT: \( \text{SAT} = 40 \times 30 + 150 \). This gives \( \text{SAT} = 1200 + 150 = 1350 \). So, \(Q3\) for SAT is 1350.

Convert Median ACT Score to SAT

The median ACT score is 28. Substitute into the conversion formula: \( \text{SAT} = 40 \times 28 + 150 \). Calculating gives \( \text{SAT} = 1120 + 150 = 1270 \). Therefore, the median equivalent SAT score is 1270.

Convert ACT IQR to SAT

The IQR for the ACT scores is 6. Since IQR is a measure of spread, convert it by multiplying by 40: \( \text{SAT IQR} = 40 \times 6 = 240 \).

Key concepts

SAT

The SAT is a standardized test that many students take as part of their college admissions process in the United States. It evaluates skills in math, evidence-based reading, and writing. Scores range between 400 and 1600. The SAT is designed to assess a student's readiness for college and provides colleges with a common data point that can be used to compare applicants. A higher SAT score can strengthen a student's application by showcasing their academic ability.
However, standardized tests like the SAT are just one part of the college admissions puzzle. It's important to understand SAT scores in the context of a student's entire application, including academic record, extracurricular activities, and personal essays.

ACT

The ACT is another standardized test used for college admissions, similar to the SAT. It covers four academic skill areas: English, mathematics, reading, and science reasoning, with an optional writing test. The composite score ranges from 1 to 36. Schools sometimes have no preference between the SAT or ACT, allowing students to choose which test showcases their strengths more effectively.
Unlike the SAT, which combines different parts to give a single score out of 1600, the ACT's structure allows colleges to see how a student performs in specific subject areas. Understanding how ACT scores are used can be beneficial when creating a test-taking strategy, as students can focus on improving certain sections based on their target colleges' requirements.

Score Conversion

Comparing SAT and ACT scores can seem complicated due to their distinct scales. Luckily, there are standardized score conversion tools and formulas, like the one provided above: \( \text{SAT} = 40 \times \text{ACT} + 150 \). This helps schools and students transform their ACT scores into equivalent SAT scores easily. Such conversions are particularly useful in college admissions, as they offer a common ground for evaluating applicants' standardized test performance.
Adequate score conversion aids students in identifying where they stand in comparison to typical applicant pools at their desired colleges, allowing them to adjust their preparation and application strategies as needed. Being familiar with how to convert scores can give applicants a clearer picture of their position in the admissions race.

Statistics

Statistics play a crucial role in evaluating standardized tests. Core statistical measures, such as the mean, median, standard deviation, and interquartile range (IQR) provide insight into score distributions.

• The mean gives an average score and reflects overall performance.
• The median provides the middle score in a dataset, offering another measure of central tendency.
• The standard deviation indicates the variability of scores—how spread out the scores are from the mean.
• The IQR measures the spread of the middle 50% of scores, showing data variability like the standard deviation, but is less affected by outliers.

These statistics help colleges understand applicant performance beyond raw scores, assisting them in making more informed admission decisions.

College Admissions

College admissions is a competitive process where standardized test scores like the SAT and ACT are important components. However, universities consider a variety of factors when evaluating applications.

• Academic performance: GPA, course rigor, and academic awards can be very significant.
• Extracurricular activities: Leadership roles, volunteer work, and specific interests can show a well-rounded student.
• Essays and personal statements: These offer insight into a student's character, aspirations, and motivations.
• Recommendations: Letters from teachers or mentors can provide a personal perspective on the student's abilities and character.

Understanding that not all weight is put solely on test scores can relieve some of the pressure students feel. Balancing strong academic performance with personal strengths and achievements is key to making a compelling college application.