AP Statistics · Lesson 1

Scatterplots and correlation

Know Know it

A scatterplot shows two quantitative variables measured on the same individuals. Put the explanatory variable on the x-axis and the response variable on the y-axis.

Describe a scatterplot with four features, always in context. Direction: positive (y tends to rise as x rises) or negative. Form: linear or curved. Strength: how tightly the points follow the form. Unusual features: outliers, clusters or gaps.

The correlation r measures the direction and strength of a linear relationship only. It runs from -1 to 1; values near -1 or 1 mean the points lie close to a line, and values near 0 mean no linear pattern.

r has no units. It does not change if you swap x and y, or if you change units (for example, inches to centimeters). It is not resistant: one unusual point can change it a lot.

Rough guide to reading r
rLinear relationship
close to 1strong and positive
about 0.5moderate and positive
close to 0no linear pattern (a curve is still possible)
close to -1strong and negative

Trap r near 0 does not mean no relationship

r only measures straight-line patterns. Fuel use versus speed can fall and then rise in a clear U shape and still have r close to 0. Always look at the scatterplot before you trust r. And a strong r in observational data never shows that x causes y.

Worked example Easy

Which correlation shows the strongest linear relationship?

  1. r = 0.62
  2. r = -0.91Answer
  3. r = 0.08
  4. r = 0.85

How to solve it

  1. Strength depends on the size of r, not its sign.
  2. Compare the absolute values: 0.62, 0.91, 0.08 and 0.85.
  3. The largest is 0.91, so r = -0.91 shows the strongest linear relationship (a negative one).

Why each choice is right or wrong

  • A. Incorrect. r = 0.62 is a moderate positive relationship; its distance from 0 is smaller than 0.91.
  • B. Correct. Strength is how far r is from 0, ignoring the sign, and 0.91 is the largest of the four.
  • C. Incorrect. r = 0.08 is very close to 0, which means almost no linear pattern at all.
  • D. Incorrect. r = 0.85 is strong, but 0.85 is closer to 0 than 0.91, so -0.91 is stronger.

Explain Explain it

Task verbs: describe, interpret

Describe in context: name both variables. Not 'there is a strong positive relationship', but 'there is a strong, positive, linear relationship between weekly practice hours and free-throw percentage: players who practice more tend to have higher percentages'.

Interpret r with its sign, its size and the word linear: r = -0.80 means a strong, negative, linear relationship between the two named variables.

Say tend to: a relationship describes a pattern, not a rule for every individual.

Worked example Medium

A fictional cafe records the daily high temperature and the number of hot drinks sold on 10 autumn days. The scatterplot falls in a tight straight band from upper left to lower right, and r = -0.997.

Which description of the relationship is right?

  1. Strong, positive and linear, because r is very close to 1 in size
  2. Strong, negative and linear: on warmer days the cafe tends to sell fewer hot drinksAnswer
  3. Weak and negative, because a negative r means the variables are only loosely linked
  4. Strong and negative, which shows that warm weather causes the cafe to sell fewer hot drinks

How to solve it

  1. Sign of r: negative, so higher temperatures go with fewer hot drinks.
  2. Size of r: 0.997 is very close to 1, so the relationship is strong and the points lie close to a line.
  3. The data are observational, so describe a tendency, not a cause.

Why each choice is right or wrong

  • A. Incorrect. The size is close to 1, but the sign of r is negative, so the direction is negative, not positive.
  • B. Correct. r = -0.997 is close to -1, so the pattern is a strong, negative, straight-line one, described here in context.
  • C. Incorrect. The sign of r gives the direction, not the strength; -0.997 is very close to -1, so the relationship is strong.
  • D. Incorrect. These are observational data, so a strong correlation cannot show cause; other factors may change with the season too.

Describe Your turn

A coach records weekly practice hours (x) and free-throw percentage (y) for the 10 players on a fictional team. The scatterplot rises from lower left to upper right in a roughly straight band, r = 0.97, and no point sits far from the pattern. Describe the relationship in context.

Write it, then compare. Nothing you type here is saved or graded.

Compare with a model answer

A full answer does this

  • States a positive direction in context: players who practice more tend to have higher free-throw percentages
  • States that the form is linear and the relationship is strong, using r = 0.97
  • Says there are no unusual points such as outliers

Model answer

There is a strong, positive, linear relationship between weekly practice hours and free-throw percentage for these 10 players: players who practice more hours tend to have higher free-throw percentages. The points follow a fairly tight straight band, and r = 0.97 confirms the strength. There are no outliers; no player sits far from the pattern.

Apply Apply it

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