Scatterplots and correlation

About this lesson

Integrated Math I › Descriptive statistics: two variables

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Scatterplots and correlation

Warm-up

Three quick questions from earlier lessons.

Discover

Points lie close to a line going down from left to right. Which correlation fits?

  1. r=−0.95r = -0.95
  2. r=0.2r = 0.2

Key idea

Describe a scatterplot by direction, form, strength and unusual points. Correlation measures linear strength.

The correlation rr is between −1-1 and 11: its sign gives the direction, ∣r∣|r| the strength of a linear relationship. It has no units, does not change when units change, and is the same if xx and yy are swapped. It is sensitive to outliers and misleading for curved patterns.

Worked example

Hours of study vs test score: r=0.78r = 0.78.

Direction: positive (more study, higher scores)
Positive r.
Strength: fairly strong, linear
|r| = 0.78.
Scores out of 50 or 100: same rr
Unitless.

Your turn, guided

Your turn: r=0.6r = 0.6 for (x,y)(x, y). What is rr for (y,x)(y, x)?

0.60.6
Symmetric in x and y.

Practice ★

  1. A scatterplot of car age against resale value has r=−0.94r = -0.94. How would you describe the relationship?
    1. Strong, negative, linear: older cars tend to be worth less
    2. Weak, because r is negative
    3. Each extra year causes a fixed loss of value
  2. A scatterplot of car age against resale value has r=−0.92r = -0.92. How would you describe the relationship?
    1. Strong, negative, linear: older cars tend to be worth less
    2. Weak, because r is negative
    3. Each extra year causes a fixed loss of value
  1. Heights (cm) and weights (kg) of students have r=0.58r = 0.58. Heights are converted to inches. What is the new correlation?
    1. Still 0.580.58: changing units does not change rr
    2. Smaller, since inches are bigger units
    3. It cannot be known without the data

Watch out!

Practice ★★

  1. Which statements about the correlation rr of 50 pairs (x,y)(x, y) are true? Tick all that are.
    1. −1≤r≤1-1 \le r \le 1
    2. rr has no units
    3. Swapping xx and yy does not change rr
    4. rr measures the strength of any relationship, curved or straight
    5. r=0r = 0 means there is no relationship at all

Exit ticket

Three questions to check you have got it.

  1. A scatterplot of car age against resale value has r=−0.94r = -0.94. How would you describe the relationship?
    1. Strong, negative, linear: older cars tend to be worth less
    2. Weak, because r is negative
    3. Each extra year causes a fixed loss of value
  2. A scatterplot of car age against resale value has r=−0.92r = -0.92. How would you describe the relationship?
    1. Strong, negative, linear: older cars tend to be worth less
    2. Weak, because r is negative
    3. Each extra year causes a fixed loss of value
  3. A scatterplot of car age against resale value has r=−0.90r = -0.90. How would you describe the relationship?
    1. Strong, negative, linear: older cars tend to be worth less
    2. Weak, because r is negative
    3. Each extra year causes a fixed loss of value

Summary card

Correlation

  • −1≤r≤1-1 \le r \le 1; sign = direction, size = linear strength.
  • Unitless; symmetric in xx and yy.

Watch out: rr only sees straight lines.

End of the lesson

Next lesson: The least-squares regression line Back to the chapter

Answers: Scatterplots and correlation

A

  1. Strong, negative, linear: older cars tend to be worth less
  2. Strong, negative, linear: older cars tend to be worth less

B

  1. Still 0.580.58: changing units does not change rr

C

  1. −1≤r≤1-1 \le r \le 1 · rr has no units · Swapping xx and yy does not change rr

D

  1. Strong, negative, linear: older cars tend to be worth less
  2. Strong, negative, linear: older cars tend to be worth less
  3. Strong, negative, linear: older cars tend to be worth less

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