Success TutoringWorksheets
Study Guide12th GradeLevel 3 · Challenge

Free 12th Grade Bivariate Data: Correlation and Lines of Best Fit

A senior study guide to analyzing two numerical variables: describing scatterplots, interpreting Pearson's r and the coefficient of determination, finding the least-squares line from summary statistics or a small data set, interpreting its slope and intercept, calculating residuals and knowing the limits of extrapolation. Includes a six-question self-check with answers.

Download Free PDF

Instant download · No sign-up · Free for personal, classroom and homeschool use

Grade
12th Grade
Subject
Math
Topic
Statistics
Difficulty
Level 3 · Challenge
Estimated time
25 minutes
Curriculum
Texas grade levels (TEKS)
Answers
Not applicable
Format
PDF (A4) + print

Students will practice

  • describing the direction, form and strength of association on a scatterplot and interpreting Pearson's r
  • calculating the least-squares regression line from summary statistics or raw data
  • interpreting the slope, intercept and coefficient of determination in context
  • using the line to predict, calculating residuals and recognizing the risk of extrapolation

Curriculum: Texas grade levels (TEKS). We show specific outcome codes only where they have been verified against the official curriculum document.

What's next?

Completed: Bivariate Data: Correlation and Lines of Best Fit

  1. 1The Binomial Distribution
  2. 2Continuous Random Variables and Probability Density Functions
Study guide

How to use this study guide

  1. Read it together first, pausing at each worked example to try the step before reading the answer.
  2. Attempt the "Check yourself" questions at the end without looking back.
  3. Then practice with a worksheet from the pathway above and finish with the topic test.

Common questions

Who is this study guide for?

12th Grade students (typically ages 17–18) working on statistics. It is pitched at level 3 · challenge.

Are the answers included?

This is a study guide, so there is no separate answer sheet; the 'Check yourself' questions include answers.

How long does it take?

About 25 minutes. Short, regular sessions work best: two or three a week beats one long one.

Do I need to sign up to download?

No. Click Download Free PDF and it opens immediately. It is free for personal, classroom and homeschool use.

What should we do next?

Try The Binomial Distribution.

Preview

Success TutoringWorksheets12th Grade · Mathematics · Statistics

Bivariate Data: Correlation and Lines of Best Fit

What you need to know

Bivariate data records two variables for each individual, such as hours studied and exam mark. The explanatory (independent) variable x is the one you think influences the other; the response (dependent) variable y goes on the vertical axis. A scatterplot is described by its direction (positive or negative), form (linear or non-linear) and strength (strong, moderate, weak).

Pearson's correlation coefficient r measures the strength and direction of a linear association and always lies between −1 and 1. The coefficient of determination r[sup 2] is the fraction of the variation in y that is explained by the variation in x. The least-squares line y = a + bx is the line that minimizes the sum of the squared vertical distances (residuals) from the points: b = r × [frac SD(y)/SD(x)] and a = ȳ − b x̄. A residual is actual y − predicted y.

|r|Strength (common descriptors)Example
0.75 to 1strongr = −0.9: strong negative
0.5 to 0.75moderater = 0.6: moderate positive
0.25 to 0.5weakr = 0.3: weak positive
0 to 0.25no (or negligible) linear associationr = 0.1

Find the least-squares line from summary statistics: x̄ = 5, SD(x) = 2, ȳ = 20, SD(y) = 6, r = 0.8. Then predict y when x = 7.

  1. Slope: b = r × = 0.8 × = 2.4
  2. Intercept: a = ȳ − b x̄ = 20 − 2.4 × 5 = 8
  3. Line: y = 8 + 2.4x
  4. Prediction at x = 7: y = 8 + 2.4 × 7 = 24.8. This is interpolation if 7 is inside the range of the data.

Interpret the slope, intercept and r2 for mark = 40 + 5 × hours, with r = 0.8

  1. Slope 5: on average, each extra hour of study is associated with an increase of 5 marks.
  2. Intercept 40: the predicted mark for a student who studies 0 hours is 40 (only meaningful if 0 hours is within the data).
  3. r2 = 0.82 = 0.64: 64% of the variation in marks is explained by the variation in hours studied; the other 36% is due to other factors.

A student studied 7 hours and scored 30, while the line predicts 24.8. Find and interpret the residual.

  1. Residual = actual − predicted = 30 − 24.8 = 5.2
  2. The residual is positive, so the point lies above the line: this student scored 5.2 marks more than the model predicts.

From raw data: x = 1, 2, 3, 4, 5 and y = 2, 4, 5, 4, 5. Find r and the least-squares line.

  1. Means: x̄ = 3, ȳ = 4. Work out the deviations (x − x̄) = −2, −1, 0, 1, 2 and (y − ȳ) = −2, 0, 1, 0, 1.
  2. Σ(x − x̄)(y − ȳ) = (−2)(−2) + (−1)(0) + 0 + (1)(0) + (2)(1) = 6
  3. Σ(x − x̄)2 = 4 + 1 + 0 + 1 + 4 = 10 and Σ(y − ȳ)2 = 4 + 0 + 1 + 0 + 1 = 6
  4. r = = = 0.77 (moderate to strong positive)
  5. b = = = 0.6 and a = 4 − 0.6 × 3 = 2.2, so y = 2.2 + 0.6x
  6. In an exam you would normally get these from a calculator's statistics mode — but check that your x and y lists are entered the right way around.
Common mistake: predicting far outside the range of the data (extrapolation). A line fitted to 1–10 hours of study says nothing reliable about 40 hours, and a negative predicted mark is a sign that you have extrapolated too far.
Common mistake: saying that a strong correlation proves that x causes y. Correlation is not causation: ice-cream sales and drowning rates are correlated because both rise in summer. Also note that r only measures linear association; a clear curve can have r near 0.

Check yourself

  1. Describe the association when r = −0.9.
  2. Find the coefficient of determination when r = 0.6, and interpret it.
  3. A least-squares line has slope b = 1.5, with x̄ = 10 and ȳ = 25. Find the intercept a.
  4. Using y = 10 + 1.5x, predict y when x = 12.
  5. The actual value at x = 12 was 26. Find the residual.
  6. Given x̄ = 4, SD(x) = 1.5, ȳ = 30, SD(y) = 6 and r = 0.5, find the slope of the least-squares line.

Answers: 1. strong negative linear association 2. r2 = 0.36, so 36% of the variation in y is explained by x 3. a = 25 − 1.5 × 10 = 10 4. 10 + 18 = 28 5. 26 − 28 = −2 (the point is below the line) 6. b = 0.5 × = 2

Free worksheets at worksheets.successtutoring.com · © Success Tutoring · Free for personal and classroom use

Preview — the PDF contains the full resource.

More 12th Grade statistics resources

  • Study Guide

    The Normal Distribution and z-scores

    A senior study guide to the bell curve: calculating z-scores, comparing results from different tests, using the 68–95–99.7 rule, reading…

    Difficulty
    Level 3 · Challenge
    Time
    25 min
  • Worksheet

    The Binomial Distribution

    A senior math worksheet on the binomial distribution: recognizing Bernoulli trials and binomial situations, calculating P(X = k) with the…

    Difficulty
    Level 3 · Challenge
    Time
    50 min
    Questions
    16 questions
    Answers
    Answers included
  • Worksheet

    Continuous Random Variables and Probability Density Functions

    A senior math worksheet on continuous random variables: checking and completing probability density functions, finding probabilities by…

    Difficulty
    Level 3 · Challenge
    Time
    50 min
    Questions
    17 questions
    Answers
    Answers included
Was this resource helpful?

About this resource

Created by
Success Tutoring
Last reviewed
1 October 2026
How it was made
Written by our team

Free to print and share for personal, classroom and homeschool use. Please don't resell. Spotted an error? Let us know.