Unit 5 · Topic 5.2 Beta

Correlation

The correlation r = 1/(n − 1) ∑ z_xz_y gives the direction (sign) and strength (closeness to ±1 ) of a linear association.

Practice 3: Analyze DataPractice 4: Interpret Results

Part 1 · Hook

Why this matters

Two scatterplots both show "a strong positive association". Which is stronger? Words like strong and weak are hard to compare. The correlation coefficient rr turns the direction and strength of a linear pattern into one number between −1-1 and 1, and it works the same way for any two quantitative variables, in any units.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. A value is 1 standard deviation below the mean. What is its z-score?

  1. −1-1
  2. 11
  3. 00
  4. −0.5-0.5
Show the answer

A z-score counts standard deviations from the mean; below is negative.

  • Correct: −1-1:
  • 11:
  • 00:
  • −0.5-0.5:

2. Which word describes a scatterplot whose points fall from upper left to lower right?

  1. negative
  2. positive
  3. curved
  4. strong
Show the answer

One variable rises as the other falls: a negative direction.

  • Correct: negative:
  • positive:
  • curved:
  • strong:

Part 4 · See it

See it first

Four small scatterplots with their correlations computed from the data: strong positive, r = 0.99; moderate negative, r = -0.79; weak positive, r = 0.29; and a strong curved arch with r = 0.04.
Four data sets and their correlations. The arch has rr near 0 even though xx and yy are closely related. LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

  1. Standardize each value: zx=x−x¯sxz_x = \frac{x - \bar{x}}{s_x} and zy=y−y¯syz_y = \frac{y - \bar{y}}{s_y}units disappear, so rr is the same in any units
  2. Points above average on both or below average on bothgive positive products zxzyz_x z_y; mixed points give negative products
  3. Average the products: r=1n−1∑zxzyr = \frac{1}{n - 1}\sum z_x z_yrr is positive for a rising pattern, negative for a falling one, and near ±1\pm 1 when points hug a line
  4. rr measures only how well a straight line fitsalways plot first: a curve can have rr near 0, and one outlier can change rr a lot

Part 6 · Key ideas

Key ideas

  • The correlation rr measures the direction and strength of a linear association; −1≤r≤1-1 \le r \le 1.
  • Interpret: "r=0.95r = 0.95: a strong, positive, linear association between hours studied and quiz score."
  • rr has no units, does not change when units change, and is the same if xx and yy swap.
  • rr is not resistant, measures only linear patterns, and never shows cause.

Part 7 · Misconception

A common mistake

The wrong idea: A correlation near 0 means the two variables are unrelated.

What actually happens: A correlation near 0 means there is no linear association. The variables can still be strongly related along a curve, so look at the scatterplot before trusting rr.

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

Graph

Study time and quiz scores

A teacher records how many hours each of 8 students studied for a quiz and their score out of 100. (Invented data.)
0123456760708090hours studiedquiz score

1. Use technology to find the correlation rr between hours studied and quiz score. Give three decimal places.

Type a number. A table answer and a calculator answer are both accepted.

Show the answer

Technology (a linear regression or correlation command on the two lists) gives r=0.951r = 0.951.

  • Answer: 0.951

2. Which interpretation of r=0.95r = 0.95 is correct?

  1. The quiz scores are about 95% accurate measures
  2. About 95% of the students scored higher because they studied more
  3. A strong, positive, linear link between study time and score
  4. Each extra hour of study raises the score by 0.95 points
Show the answer

rr gives direction (sign) and strength (closeness to 1 or −1-1) of a linear association.

  • The quiz scores are about 95% accurate measures: Correlation measures a linear association, not accuracy.
  • About 95% of the students scored higher because they studied more: rr is not a percent of students, and it says nothing about cause.
  • Correct: A strong, positive, linear link between study time and score: Right: rr near 1 means a strong positive linear association.
  • Each extra hour of study raises the score by 0.95 points: That would be a slope. rr has no units and does not measure change per hour.

3. The teacher recomputes rr with study time in minutes instead of hours. What happens to rr?

  1. It becomes 60 times smaller, 0.016
  2. It stays 0.95
  3. It can no longer be computed
  4. It becomes 60 times larger, 57.0
Show the answer

rr is unitless: it does not change when either variable changes units.

  • It becomes 60 times smaller, 0.016: Rescaling a variable does not change how tightly the points follow a line.
  • Correct: It stays 0.95: Right: changing units changes every z-score not at all, so rr is unchanged.
  • It can no longer be computed: Any units work; rr is based on standardized values.
  • It becomes 60 times larger, 57.0: rr is always between −1-1 and 1; it has no units to convert.

4. If quiz score were put on the horizontal axis and hours on the vertical axis, what would the correlation be?

  1. The negative, −0.95-0.95
  2. The same, 0.95
  3. The reciprocal, 1.05
  4. Zero, since a score is unable to explain hours
Show the answer

Correlation is symmetric in xx and yy.

  • The negative, −0.95-0.95: Switching axes does not reverse the direction: high scores still go with more hours.
  • Correct: The same, 0.95: Right: rr treats the two variables the same way, so switching them leaves it unchanged.
  • The reciprocal, 1.05: A correlation can never exceed 1, and switching axes leaves it as it was.
  • Zero, since a score is unable to explain hours: Correlation does not depend on which variable is called explanatory.

5. A ninth student studied 6 hours but scored 40. What would adding this student do to rr?

  1. It would rise, since the new student studied a lot
  2. It would drop sharply, to about 0.14: rr is not resistant
  3. It would become negative, since 40 is the lowest score
  4. It would barely change, since one point out of nine is small
Show the answer

Like the mean and SD, rr is pulled by outliers.

  • It would rise, since the new student studied a lot: The new point sits far below the trend, so the association gets weaker, not stronger.
  • Correct: It would drop sharply, to about 0.14: rr is not resistant: Right: with the ninth point, technology gives r=0.14r = 0.14. Correlation is not resistant to outliers.
  • It would become negative, since 40 is the lowest score: It drops a lot, to about 0.14, but the other eight points still trend upward.
  • It would barely change, since one point out of nine is small: One point far from the line can change rr a lot. Correlation is not resistant.

6. Which statement about the correlation rr is true?

  1. It is large for any strong curved pattern
  2. It is 1 whenever yy increases with xx
  3. It is between −1-1 and 1 and has no units
  4. It is measured in the units of yy per unit of xx
Show the answer

−1≤r≤1-1 \le r \le 1, unitless, linear only.

  • It is large for any strong curved pattern: A strong curve can give rr near 0.
  • It is 1 whenever yy increases with xx: r=1r = 1 only when the points lie exactly on a rising line.
  • Correct: It is between −1-1 and 1 and has no units: Right: it is computed from z-scores, which have no units.
  • It is measured in the units of yy per unit of xx: That describes a slope.

Part 9 · Summary

Summary

The correlation r=1n−1∑zxzyr = \frac{1}{n - 1}\sum z_x z_y gives the direction (sign) and strength (closeness to ±1\pm 1) of a linear association. It has no units, ignores which variable is explanatory, is pulled by outliers, misses curved patterns, and says nothing about cause.

Part 10 · Up next

What comes next

Part 11 · Connections

Connections

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