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Admit Rate vs. Average SAT: Correlation and the Least-Squares Line

Students plot the admit rate of 723 U.S. colleges against their average SAT, describe the association, compute the correlation, fit a least-squares line and interpret residuals for real colleges.

AP StatisticsUnit 2: Exploring Two-Variable Data; Extension: Unit 9: Inference for Quantitative Data: Slopes
TimeOne 50-minute class period
Common CoreHSS.ID.B.6, HSS.ID.C.7, HSS.ID.C.8, HSS.ID.C.9
DataCSV, 723 rows

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Learning objectives

  • Describe the direction, form, strength and unusual features of a scatterplot in context.
  • Compute and interpret the correlation coefficient r and the coefficient of determination r squared.
  • Find the least-squares regression line and interpret its slope and y-intercept in context.
  • Compute and interpret a residual, and explain why a prediction outside the data is extrapolation.

Materials

  • The lesson CSV (one row per college)
  • A graphing calculator, spreadsheet or statistics software
  • The student worksheet below

CSV: https://collegecalcai.com/api/data-csv/teach/admit-rate-vs-sat

About the data: Admit rate: IPEDS fall 2023 (admitted divided by applied, as a percent, one decimal). Average SAT: College Scorecard SAT_AVG, release 2026-06-10. Colleges with at least 1,000 applicants that report an average SAT.

Columns: college, state, type, admit_rate_pct, sat_avg.

Student worksheet

Name: ______________________ Period: ______ Open the CSV in a spreadsheet, a graphing calculator or statistics software.

  1. Make a scatterplot with admit_rate_pct on the x-axis and sat_avg on the y-axis. Describe the direction, form and strength of the association in context, and name anything unusual.
  2. Compute the correlation r. Interpret it in context.
  3. Find the least-squares regression line predicting average SAT from admit rate. Interpret the slope in context.
  4. Interpret the y-intercept. Does it make sense in this context?
  5. Use your line to predict the average SAT of a college that admits 50% of applicants.
  6. Find the college with the largest positive residual and the one with the largest negative residual. What does each residual mean?
  7. What fraction of the variation in average SAT is accounted for by the linear relationship with admit rate?

Teacher key

Every value below is computed by code from the same CSV students download.

  1. Negative, roughly linear and moderately strong: colleges that admit a smaller share of applicants tend to have higher average SATs. Admit rates run from 3.5% to 100.0%. The spread is not even: among the 69 colleges admitting under 25%, average SATs have a standard deviation of 127.7 points; among the 358 admitting 75% or more it is 98.3. Look for colleges far from the pattern (question 6 finds the two farthest).
  2. r = -0.673 (n = 723). There is a moderately strong negative linear association between a college's admit rate and its average SAT.
  3. predicted SAT = 1489.8 minus 4.217 x (admit rate). For each additional percentage point of admit rate, the predicted average SAT decreases by about 4.22 points.
  4. The intercept, 1489.8, is the predicted average SAT for a college with a 0% admit rate. No college admits nobody, and the lowest admit rate in the data is 3.5%, so this is extrapolation with no practical meaning.
  5. 1489.8 minus 4.217 x 50 = 1278.9, about 1279.
  6. Largest positive residual: Wheaton College (IL) (admit rate 90.0%, average SAT 1380, residual 269.7): its average SAT is 269.7 points higher than the line predicts for its admit rate. Largest negative residual: Alcorn State University (admit rate 24.7%, average SAT 904, residual -481.6): 481.6 points lower than predicted.
  7. r squared = 0.452: about 45.2% of the variation in average SAT among these colleges is accounted for by the linear relationship with admit rate.

Extension

  • Treat these colleges as if they were a random sample and test whether the population slope is negative (Unit 9). Discuss why that assumption is questionable: these are nearly all U.S. four-year colleges that report SAT averages, not a sample.
  • Make a residual plot. Is a line the right model, or is there curvature?
  • Split the data by type (public or private) and compare the two correlations.

All free lessons

Free to use and share (CC BY 4.0). Source: CollegeCalcAI, collegecalcai.com/teach/admit-rate-vs-sat, from U.S. Department of Education data (IPEDS and the College Scorecard). Unit names follow the College Board AP Statistics Course and Exam Description; not endorsed by the College Board.