Scatter Plots and Correlation

A scatter plot is a graph of paired data, with one point for each pair. Correlation describes how the two variables move together. If the points rise from left to right, the correlation is positive. If they fall, it is negative. If there is no pattern, there is no correlation.

Before this lesson: You should know slope, slope-intercept form and two-way tables.

What Is a Scatter Plot?

Paired data means two numbers for each person or thing. For example, a teacher records how many hours each student studied and their quiz score. Each student gives one point: (hours, score).

Hours studied	1	2	2	3	4	4	5	6	6	7
Quiz score	62	65	70	72	75	80	84	85	90	92

Put hours on the x-axis and scores on the y-axis, and plot each point. Don’t connect the dots. The shape of the cloud of points tells the story.

Types of Correlation

Type	What the points do	Example
Positive	Rise from left to right	More study time, higher scores
Negative	Fall from left to right	Warmer days, less hot cocoa sold
None	Show no clear pattern	Shoe size and test score

Correlation can also be strong or weak. In a strong correlation, the points lie close to a line. In a weak one, they are spread out but still lean one way.

The correlation coefficient, r

The correlation coefficient, r, is a number from −1 to 1 that measures how close the points are to a straight line. A calculator or app finds it for you.

r is close to…	Meaning
1	Strong positive linear correlation
0	Little or no linear correlation
-1	Strong negative linear correlation

A common rule of thumb: |r| \ge 0.7 is strong, 0.3 \le |r| \lt 0.7 is moderate, and |r| \lt 0.3 is weak. The sign shows the direction. The size shows the strength.

Steps for describing a scatter plot

1. Look at the direction: do the points rise, fall or neither?
2. Look at the strength: are the points close to a line or spread out?
3. Look at the form: is the pattern straight or curved?
4. Look for outliers, points far from the rest.

Worked Examples

Example 1: Describe a scatter plot

Describe the study-time scatter plot above.

The points rise from left to right, so the correlation is positive. They lie close to a line, so it is strong and linear. A calculator gives r \approx 0.98, which agrees.

Example 2: Positive, negative or none?

Decide the type of correlation for each pair.

• Temperature and hot cocoa sales: as it gets warmer, fewer people buy cocoa. Negative.
• Height and arm span: taller people tend to have longer arms. Positive.
• Shoe size and test score: one doesn’t tell you anything about the other. None.

Example 3: Read r values

Describe each correlation: r = -0.92, r = 0.15 and r = 0.55.

• r = -0.92: strong negative. It is close to −1.
• r = 0.15: weak positive, close to none.
• r = 0.55: moderate positive.

Example 4: Describe data from a table

A store tracks the price of a snack (in dollars) and how many it sells each day.

Price	1	2	3	4	5
Number sold	20	18	15	13	10

As the price goes up, sales go down, and the drops are steady. This is a strong negative linear correlation. A calculator gives r \approx -0.998.

Example 5: Correlation is not causation

In one town, ice cream sales and swimming accidents rise and fall together. Does ice cream cause accidents?

No. Both go up in hot weather, when more people swim and buy ice cream. A hidden cause like this is called a lurking variable. A correlation shows that two things are linked, but not that one causes the other.

Common Mistakes

Mistake	Example of the error	Fix
Thinking correlation means cause	“More ice cream causes more accidents.”	Look for a lurking variable. Correlation only shows a link.
Thinking negative means weak	r = -0.9 is weaker than r = 0.5	Compare sizes, not signs. |-0.9| \gt |0.5|, so r = -0.9 is stronger.
Mixing up r and slope	r = 0.98, so the line rises 0.98 per unit	r measures how close points are to a line, not how steep the line is.
Connecting the dots	Drawing a zigzag through every point	Leave the points alone. Look at the overall pattern.
Calling a curve “no correlation”	Points form a U shape, so r is near 0	r only measures straight-line patterns. A curve can still be a strong pattern.

Practice Problems

1. Is the correlation positive, negative or none: a car’s age and its value?
2. Is the correlation positive, negative or none: a person’s height and arm span?
3. Is the correlation positive, negative or none: a person’s birth month and height?
4. Which shows the strongest correlation: r = 0.45, r = -0.88 or r = 0.70?
5. Describe a correlation with r = 0.05.
6. Describe the correlation in this data: x = 1, 2, 3, 4, 5 and y = 3, 5, 6, 9, 10.
7. Students who own more books tend to get higher grades. Does owning books cause higher grades? Explain.
8. Match each description to an r value from -0.95, 0.10 and 0.80: (a) strong positive, (b) strong negative, (c) almost none.
9. Points on a scatter plot fall from left to right and are very spread out. Describe the correlation.
10. In a scatter plot, every point is close to a rising line except one point far below the rest. What is that point called?

Show answers

1. Negative. Older cars tend to be worth less.
2. Positive. Taller people tend to have longer arm spans.
3. None. Birth month doesn’t predict height.
4. r = -0.88, because |-0.88| = 0.88 is the largest size.
5. Little or no linear correlation. 0.05 is very close to 0.
6. The y-values go up as x goes up, close to a line. This is a strong positive linear correlation (r \approx 0.99).
7. Not necessarily. It is a correlation, not proof of cause. A lurking variable, like how much a family reads together, could affect both.
8. (a) 0.80, (b) −0.95, (c) 0.10.
9. A weak negative correlation.
10. An outlier.

FAQ

What does a scatter plot show?

It shows paired data as points, so you can see if two variables are related. The pattern tells you the direction and strength of the relationship.

What is the difference between correlation and causation?

Correlation means two variables move together. Causation means one actually makes the other change. A correlation alone does not prove causation.

What does the correlation coefficient r tell you?

It tells you how close the points are to a straight line, from −1 to 1. Values near 1 or −1 are strong, and values near 0 are weak. You can also describe the center and spread of each variable with mean, median, mode and range.

Summary

• A scatter plot shows paired data as points.
• Rising points mean positive correlation. Falling points mean negative correlation.
• The correlation coefficient r goes from −1 to 1. Its size shows strength.
• Strong correlations have points close to a line.
• Correlation does not prove causation. Watch for lurking variables.

Previous lesson: Two-Way Tables

Next lesson: Line of Best Fit

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