Teacher Guide: Interpreting Correlation

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

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10 questions on Correlation. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives

Determine the direction and strength of a correlation from the value of $r$
Describe the correlation shown in a scatter plot and estimate whether $\lvert r \rvert$ is large or small
Write an interpretation sentence that names strength, direction and both variables in context
Explain why $r$ has no units and does not change when units change
Recognize that $r$ measures only linear relationships and that outliers can change it strongly

Prerequisites

• Introduction to Correlation: positive, negative and no correlation in scatter plots
• Correlation Coefficient: what $r$ is and that $-1 \le r \le 1$
• Absolute value of a number
• Reading points on a scatter plot

Discussion Starters

1. Which is stronger, $r = -0.75$ or $r = 0.55$ ? How do you know?
2. Why do you think the cutoffs for "strong" and "weak" differ between books?
3. If you measured temperatures in °F instead of °C, would the scatter plot look different? Would $r$ change?
4. Can you think of two variables that are clearly related but would have $r$ close to $0$ ?
5. A classmate entered one value wrongly, 400 instead of 40. What could that do to $r$ ?

Common Misconceptions

A negative $r$ means a weak or bad correlation

Remediation: Place values on a line from $-1$ to $1$ and mark the distance from $0$ . Show that $-0.9$ is as far from $0$ as $0.9$ : just as strong, only in the other direction.

$r = 0$ means the variables have nothing to do with each other

Remediation: Show plot D (fertilizer and tomatoes): $r = 0$ but the pattern is a clear curve. Practice saying "no linear relationship".

$r$ is measured in the units of the data

Remediation: Convert one data set from °C to °F or cm to m with the class and compute $r$ with technology both times: the value is identical.

Differentiation Ideas

For Struggling Students:

• Start with only the sign: sort cards with $r$ values into positive and negative
• Give a printed strength table to use during all exercises
• Use a sentence frame: "There is a ___ ___ linear correlation between ___ and ___"

For On-Level Students:

• Rank sets of $r$ values from strongest to weakest
• Match scatter plots with $r$ values
• Write interpretation sentences for real data from the news

For Advanced Students:

• Use a spreadsheet to see how adding or moving one point changes $r$
• Show why $r$ does not change under $x \to ax + b$ with $a > 0$ using the formula for $r$
• Find a real data set with a curved pattern and explain why $r$ is misleading

Standards Alignment

HSS.ID.C.8 (CCSS.MATH.CONTENT.HSS.ID.C.8)
Compute (using technology) and interpret the correlation coefficient of a linear fit
8.SP.A.1 (CCSS.MATH.CONTENT.8.SP.A.1)
Construct and interpret scatter plots for bivariate measurement data; describe patterns such as clustering, outliers, positive or negative association, linear association and nonlinear association
HSS.ID.B.6 (CCSS.MATH.CONTENT.HSS.ID.B.6)
Represent data on two quantitative variables on a scatter plot, and describe how the variables are related

Lesson Resources

visualFive Scatter Plots
Plots A to E: strong positive, weak negative, none, a curve with $r = 0$ and the effect of an outlier
activityGuess the r
Students estimate $r$ for printed scatter plots, then compare with the computed values and rank them by strength
worksheetInterpretation Sentences
Students write full interpretation sentences for given $r$ values in different contexts

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

Interpreting correlation means turning the number

r

(or a scatter plot) into a clear statement about two variables. Every interpretation answers two questions:

Direction (the sign of $r$ ): $r > 0$ is positive (as $x$ goes up, $y$ tends to go up), $r < 0$ is negative (as $x$ goes up, $y$ tends to go down).
Strength (the size of $|r|$ ): the closer $|r|$ is to $1$ , the closer the points lie to a straight line.

Rule of thumb for strength:

$\lvert r \rvert$	Strength
$0.7$ to $1$	strong
$0.4$ to just under $0.7$	moderate
below $0.4$	weak (very close to $0$ : almost no linear relationship)

This is a guide, not a law: different books draw the lines in slightly different places.

Three facts to remember:

$r$ has no units, and it does not change when you change units (°C to °F, cm to m, euros to cents).
$r$ measures only linear (straight-line) relationships. A clear curve can have $r = 0$ .
One outlier can change $r$ a lot, so always look at the scatter plot too.

Worked Examples

An energy company finds

r = -0.86

between the outdoor temperature and the amount of gas homes use for heating. Describe the correlation.

Look at the sign for the direction

r = -0.86

is negative → Negative: warmer days go with less gas

Take the absolute value for the strength

\lvert -0.86 \rvert = 0.86

→

\lvert r \rvert = 0.86

Compare with the rule of thumb

0.86 \ge 0.7

→ Strong

Put direction and strength together

negative + strong → Strong negative linear correlation

Answer: There is a strong negative linear correlation ( $r = -0.86$ ): on warmer days, homes tend to use less gas for heating.

Common Mistakes

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

Why it's wrong: The sign only tells the direction. The strength is the distance from $0$ , which is $\lvert r \rvert$ .

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

Concluding "no relationship" whenever $r$ is close to $0$

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

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

Believing $r$ 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: $r$ has no units: the same data in cm, m or inches give the same $r$ .

Reading $r$ 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 $r$ with and without it.

Why It Matters

News reports, science articles and sports analyses often quote a correlation: "

r = 0.5

", "a strong link", "no relationship". Knowing how to read

r

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 $r$ alone hides.
School projects: When you collect your own data, a good interpretation sentence turns a number into a conclusion.

Reading

r

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 $r = -0.86$ , 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 $r = -0.86$ 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 $r$ for these data?

Step 1: Write the mathematical expression

Enter $r$ :

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 $r \approx 0.999$ , but one ill student with 7 hours of practice and a score of 40 brings $r$ down to about $0.14$ .

3Try It Yourself

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

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

Step 1: Write the mathematical expression

Compute $1.00 - 0.14$ :

Key Takeaways

1The sign of $r$ gives the direction: positive means both tend to increase together, negative means one tends to decrease as the other increases
2The size of $\lvert r \rvert$ gives the strength. Rule of thumb: $0.7$ or more strong, $0.4$ to $0.7$ moderate, below $0.4$ weak
3A good interpretation sentence names strength, direction, both variables and uses "tend to"
4 $r$ has no units and does not change when you change the units of $x$ or $y$
5 $r$ only measures linear relationships: a clear curve can have $r = 0$
6One outlier can change $r$ a lot, so always look at the scatter plot and report $r$ with and without it

Frequently Asked Questions

Is $r = -0.9$ a weak correlation because it is negative?

No. The minus sign only means the direction is negative.

\lvert -0.9 \rvert = 0.9

, so it is a strong negative correlation, stronger than

r = 0.5

Why are there different strength tables in different books?

Where "strong" begins is a convention, not a mathematical fact. This lesson uses

0.7

and

0.4

as a rule of thumb. Other tables, like the one in the Correlation Coefficient lesson, use more bands, for example "very strong" for

0.8

and above: in this lesson, "very strong" is simply part of strong. In every table, a larger

\lvert r \rvert

means a stronger linear relationship.

If I measure height in meters instead of centimeters, does $r$ change?

No. Dividing every height by

100

does not change the shape of the scatter plot, only the numbers on the axis.

r

has no units, so it stays exactly the same.

Can an outlier make a correlation look stronger?

Yes. A cloud of points with no pattern (

r

near

0

) plus one point far away in the top right corner can give

r

above

0.9

. That is why you always check the plot.

Glossary

Direction of a correlation: Positive ( $r > 0$ ) or negative ( $r < 0$ ), given by the sign of $r$
Strength of a correlation: How close the points are to a straight line, given by $\lvert r \rvert$ (closer to $1$ means stronger)
Linear relationship: A relationship whose scatter plot follows a straight line
Non-linear relationship: A relationship whose scatter plot follows a curve; $r$ can be close to $0$ even when it is strong
Outlier: A data point far away from the pattern of the other points; it can change $r$ 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)

\lvert r \rvert \ge 0.7

strong,

\; 0.4 \le \lvert r \rvert < 0.7

moderate,

\; \lvert r \rvert < 0.4

weak

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

Interpretation sentence

\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)

for

a > 0,\ c > 0

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

Teacher Guide: Interpreting Correlation

Use this lesson with your class

For Teachers

For Struggling Students:

For On-Level Students:

For Advanced Students:

Lesson Content

Definition

Worked Examples

Common Mistakes

Why It Matters

Real World Applications

Planning Energy Use

Fuel Use and Driving Speed

Checking Data for Outliers

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Strength (rule of thumb)

Interpretation sentence

Changing units

More in This Topic

Introduction to Correlation

Correlation Coefficient

Correlation vs Causation

Correlation Word Problems