Interpreting Correlation

Read the strength and direction of a correlation from r and from a scatter plot, and describe it in a clear sentence.

Intermediate25 min18 problems
Practice Problems

Definition

Interpreting correlation means turning the number rr (or a scatter plot) into a clear statement about two variables. Every interpretation answers two questions:
  • Direction (the sign of rr): r>0r > 0 is positive (as xx goes up, yy tends to go up), r<0r < 0 is negative (as xx goes up, yy tends to go down).
  • Strength (the size of ∣r∣|r|): the closer ∣r∣|r| is to 11, the closer the points lie to a straight line.
Rule of thumb for strength:
∣r∣\lvert r \rvertStrength
0.70.7 to 11strong
0.40.4 to just under 0.70.7moderate
below 0.40.4weak (very close to 00: almost no linear relationship)
This is a guide, not a law: different books draw the lines in slightly different places.
Three facts to remember:
  • rr has no units, and it does not change when you change units (°C to °F, cm to m, euros to cents).
  • rr measures only linear (straight-line) relationships. A clear curve can have r=0r = 0.
  • One outlier can change rr a lot, so always look at the scatter plot too.

Try it now

On a mountain hike, the correlation between altitude and air temperature is r=−0.92r = -0.92. How do you describe it?

Worked Examples

An energy company finds r=−0.86r = -0.86 between the outdoor temperature and the amount of gas homes use for heating. Describe the correlation.
1

Look at the sign for the direction

r=−0.86r = -0.86 is negative → Negative: warmer days go with less gas
2

Take the absolute value for the strength

∣−0.86∣=0.86\lvert -0.86 \rvert = 0.86 → ∣r∣=0.86\lvert r \rvert = 0.86
3

Compare with the rule of thumb

0.86≥0.70.86 \ge 0.7 → Strong
4

Put direction and strength together

negative + strong → Strong negative linear correlation

Common Mistakes

Thinking r=−0.8r = -0.8 is weaker than r=0.5r = 0.5 because −0.8-0.8 is a smaller number

Why it's wrong: The sign only tells the direction. The strength is the distance from 00, which is ∣r∣\lvert r \rvert.

Correct: Compare ∣−0.8∣=0.8\lvert -0.8 \rvert = 0.8 with ∣0.5∣=0.5\lvert 0.5 \rvert = 0.5: r=−0.8r = -0.8 is the stronger correlation.

Concluding "no relationship" whenever rr is close to 00

Why it's wrong: rr only measures straight-line relationships. A hill or U-shaped curve can have r=0r = 0 (plot D).

Correct: Say "no linear relationship" and always look at the scatter plot for curves.

Believing rr changes when you switch units, for example from cm to inches

Why it's wrong: Changing units multiplies every value by the same positive number (and may add a constant). The shape of the scatter plot stays the same.

Correct: rr has no units: the same data in cm, m or inches give the same rr.

Reading rr without checking for outliers

Why it's wrong: A single unusual point can make a strong correlation look weak (plot E) or a weak one look strong.

Correct: Plot the data first. If there is an outlier, report rr with and without it.

Interactive Visual

Scatter Plot

A: Temperature and cold drinks sold

10152025305060708090100Temperature (°C)Drinks sold

r = 0.99 · strong positive correlation

Tap a point to see its coordinates.

Scatter Plot

B: Screen time and sleep

024686.577.588.59Screen time (hours per day)Sleep (hours)

r = -0.356 · weak negative correlation

Tap a point to see its coordinates.

Scatter Plot

C: Shoe size and math test score

3638404244465060708090Shoe size (EU)Test score

r = 0.025 · no linear correlation

Tap a point to see its coordinates.

Scatter Plot

D: Fertilizer and tomato harvest (curve, r = 0)

024610152025Fertilizer (kg)Tomatoes (kg)

r = 0 · no linear correlation

Tap a point to see its coordinates.

Scatter Plot

E: Practice and quiz score, with one outlier

02468405060708090Practice (hours)Quiz scoreOutlier

r = 0.139 · weak positive correlation

Tap a point to see its coordinates.

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Practice Problems

18 problems
Problem 1 of 18
Easy
On a mountain hike, the correlation between altitude and air temperature is r=−0.92r = -0.92. How do you describe it?

Why It Matters

News reports, science articles and sports analyses often quote a correlation: "r=0.5r = 0.5", "a strong link", "no relationship". Knowing how to read rr lets you check whether such claims make sense:
  • Health: Is the link between screen time and sleep strong or weak? The answer changes what advice makes sense.
  • Business: A shop that sees a strong positive link between temperature and drink sales can plan its stock.
  • Science: Before trusting a result, researchers look at the scatter plot for curves and outliers that rr alone hides.
  • School projects: When you collect your own data, a good interpretation sentence turns a number into a conclusion.
Reading rr well is also the first step toward the next lessons on regression and prediction.

Real World Applications

Planning Energy Use

Energy companies use the correlation between temperature and gas use to plan how much gas they need in winter.

Example:

With r=−0.86r = -0.86, colder days strongly tend to go with higher gas use, so a cold forecast is a good warning sign.

1Try It Yourself

An energy company finds r=−0.86r = -0.86 between the outdoor temperature and daily gas use in 200 homes.

How would you describe this correlation?

Step 1: Choose the right answer

Choose the description:

Fuel Use and Driving Speed

Car engineers know that fuel use is high at very low and very high speeds and lowest in between. This curved pattern fools $r$.

Example:

Speeds of 30, 50, 70, 90, 110, 130 and 150 km/h give fuel use of 8.0, 6.5, 5.5, 5.2, 5.5, 6.5 and 8.0 liters per 100 km: a U-shaped curve.

2Try It Yourself

The data are 30, 50, 70, 90, 110, 130, 150 km/h with fuel use 8.0, 6.5, 5.5, 5.2, 5.5, 6.5, 8.0 liters per 100 km. The pattern is perfectly symmetric around 90 km/h.

What is the correlation coefficient rr for these data?

Step 1: Write the mathematical expression

Enter rr:

Checking Data for Outliers

Data analysts always check scatter plots for outliers before they report a correlation, because one wrong or unusual value can change $r$ a lot.

Example:

In plot E, six students show r≈0.999r \approx 0.999, but one ill student with 7 hours of practice and a score of 40 brings rr down to about 0.140.14.

3Try It Yourself

A teacher reports r≈0.14r \approx 0.14 between practice hours and quiz scores for 7 students. Without the one ill student, r≈0.999r \approx 0.999.

Rounded to two decimals, by how much does rr go up when the outlier is removed?

Step 1: Write the mathematical expression

Compute 1.00−0.141.00 - 0.14:

Key Takeaways

  • 1The sign of rr gives the direction: positive means both tend to increase together, negative means one tends to decrease as the other increases
  • 2The size of ∣r∣\lvert r \rvert gives the strength. Rule of thumb: 0.70.7 or more strong, 0.40.4 to 0.70.7 moderate, below 0.40.4 weak
  • 3A good interpretation sentence names strength, direction, both variables and uses "tend to"
  • 4rr has no units and does not change when you change the units of xx or yy
  • 5rr only measures linear relationships: a clear curve can have r=0r = 0
  • 6One outlier can change rr a lot, so always look at the scatter plot and report rr with and without it

Frequently Asked Questions

No. The minus sign only means the direction is negative. ∣−0.9∣=0.9\lvert -0.9 \rvert = 0.9, so it is a strong negative correlation, stronger than r=0.5r = 0.5.
No. The minus sign only means the direction is negative. ∣−0.9∣=0.9\lvert -0.9 \rvert = 0.9, so it is a strong negative correlation, stronger than r=0.5r = 0.5.
Where "strong" begins is a convention, not a mathematical fact. This lesson uses 0.70.7 and 0.40.4 as a rule of thumb. Other tables, like the one in the Correlation Coefficient lesson, use more bands, for example "very strong" for 0.80.8 and above: in this lesson, "very strong" is simply part of strong. In every table, a larger ∣r∣\lvert r \rvert means a stronger linear relationship.
No. Dividing every height by 100100 does not change the shape of the scatter plot, only the numbers on the axis. rr has no units, so it stays exactly the same.
Yes. A cloud of points with no pattern (rr near 00) plus one point far away in the top right corner can give rr above 0.90.9. That is why you always check the plot.

Glossary

Direction of a correlation
Positive (r>0r > 0) or negative (r<0r < 0), given by the sign of rr
Strength of a correlation
How close the points are to a straight line, given by ∣r∣\lvert r \rvert (closer to 11 means stronger)
Linear relationship
A relationship whose scatter plot follows a straight line
Non-linear relationship
A relationship whose scatter plot follows a curve; rr can be close to 00 even when it is strong
Outlier
A data point far away from the pattern of the other points; it can change rr a lot
Interpretation sentence
A sentence that gives strength, direction and context, for example: "There is a strong positive linear correlation between ... and ..."

Formula Card

Strength (rule of thumb)

∣r∣≥0.7\lvert r \rvert \ge 0.7 strong,   0.4≤∣r∣<0.7\; 0.4 \le \lvert r \rvert < 0.7 moderate,   ∣r∣<0.4\; \lvert r \rvert < 0.4 weak

The sign of $r$ gives the direction, $\lvert r \rvert$ gives the strength.

Interpretation sentence

[strength] + [direction] + linear correlation between x and y\text{[strength] + [direction] + linear correlation between } x \text{ and } y

Add the context: "as $x$ increases, $y$ tends to increase (or decrease)".

Changing units

r(ax+b,  cy+d)=r(x,y)r(ax + b,\; cy + d) = r(x, y) for a>0, c>0a > 0,\ c > 0

Multiplying by a positive number and adding a constant (cm to m, °C to °F) never changes $r$.

Today's challengeOne problem, three tries

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