Does adding the same number to every value change $r$?

No. Adding a constant moves all points together, so the deviations stay the same. Multiplying by a positive number (for example hours to minutes) does not change $r$ either. Multiplying one variable by a negative number flips the sign of $r$.

When is it fine to remove an outlier?

When you have a clear reason that is not about the result: a measuring error, a typing mistake, a plant eaten by a snail. Always say that you removed it and, if possible, report $r$ both with and without the point.

How can we ever show causation?

With a randomized controlled experiment: assign the treatment (for example fertilizer) at random to some subjects and not to others, keep everything else the same, and compare. Random assignment spreads confounding variables evenly over both groups.

Teacher Resources

Correlation Word Problems

Solve real correlation problems: compute r step by step from a small data set, interpret it in context, judge causal claims and see how an outlier changes r.

Advanced35 minLesson

Printable worksheet

Definition

A correlation word problem gives you paired data from a real situation and asks you to measure, interpret and judge the relationship. You already know the formula for the correlation coefficient

r

from the lesson *Correlation Coefficient*. Here you use it as a full method.

The five-step method for a small data set:

1.Find the means $\bar{x}$ and $\bar{y}$ .
2.Find each deviation $x_i - \bar{x}$ and $y_i - \bar{y}$ .
3.Add the products: $S_{xy} = \sum(x_i - \bar{x})(y_i - \bar{y})$ .
4.Add the squares: $S_{xx} = \sum(x_i - \bar{x})^2$ and $S_{yy} = \sum(y_i - \bar{y})^2$ .
5.Divide:
$r = \frac{S_{xy}}{\sqrt{S_{xx} \cdot S_{yy}}}$

Then answer the question in context:

Direction: the sign of $r$ (positive: both rise together; negative: one rises while the other falls).
Strength: the size of $|r|$ (scale below).
Causation: a strong $r$ alone never proves that one variable causes the other. Look for a confounding variable (also called a lurking variable) and ask whether the data come from a controlled experiment.
Outliers: one unusual point can make $r$ much stronger, much weaker or even change its sign.

	r		Strength
$0.00$ to $0.19$	very weak
$0.20$ to $0.39$	weak
$0.40$ to $0.59$	moderate
$0.60$ to $0.79$	strong
$0.80$ to $1.00$	very strong

Try it now

A survey of teenagers finds

r = -0.85

between daily screen time and hours of sleep. Which description is correct?

Worked Examples

Five students recorded their study hours and quiz scores:

(1, 64)

(2, 63)

(3, 77)

(4, 71)

(5, 75)

. Compute

r

and interpret it.

Find the means

\bar{x} = \frac{1+2+3+4+5}{5} = 3

\bar{y} = \frac{64+63+77+71+75}{5} = \frac{350}{5} = 70

→

\bar{x} = 3

\bar{y} = 70

Find the deviations

x_i - \bar{x}

-2, -1, 0, 1, 2

y_i - \bar{y}

-6, -7, 7, 1, 5

→ Each deviation sum is

0

(a quick check)

Multiply the deviations and add

(-2)(-6) + (-1)(-7) + (0)(7) + (1)(1) + (2)(5) = 12 + 7 + 0 + 1 + 10

→

S_{xy} = 30

Square the deviations and add

S_{xx} = 4 + 1 + 0 + 1 + 4 = 10

S_{yy} = 36 + 49 + 49 + 1 + 25 = 160

→

S_{xx} = 10

S_{yy} = 160

Apply the formula

r = \frac{30}{\sqrt{10 \cdot 160}} = \frac{30}{\sqrt{1600}} = \frac{30}{40}

→

r = 0.75

Interpret in context

Positive sign,

|r| = 0.75

is in the band

0.60

0.79

→ Strong positive correlation

Answer: $r = 0.75$ : a strong positive correlation. Students who studied longer tended to score higher, although the student who studied 1 hour scored better than the one who studied 2 hours, so the pattern is not perfect.

Common Mistakes

Multiplying the raw values $x_i y_i$ instead of the deviations

Why it's wrong: The formula $r = \frac{S_{xy}}{\sqrt{S_{xx} S_{yy}}}$ uses the distances from the means. Raw products are always large and positive for positive data and give a wrong $r$ .

Correct: Subtract the means first. A quick check: the deviations of each variable add up to $0$ .

Forgetting the square root or rounding too early

Why it's wrong: $\frac{30}{10 \cdot 160} \approx 0.019$ is not $r$ . Rounding $\sqrt{520}$ to $23$ gives $0.957$ instead of $0.965$ .

Correct: Take the square root of $S_{xx} \cdot S_{yy}$ and round only the final answer to 2 decimal places.

Treating a strong $r$ as proof that one variable causes the other

Why it's wrong: Observational data can show a strong correlation because a confounding variable drives both, as rainy weather drives umbrella sales and traffic accidents.

Correct: Say "is associated with". Only a randomized controlled experiment can justify a causal claim.

Ignoring an outlier or deleting it without a reason

Why it's wrong: One point can change $r$ from $1$ to $0$ . Ignoring it hides the problem; deleting it just because it is inconvenient is dishonest.

Correct: Plot the data, compute $r$ with and without the outlier and remove it only with a clear reason (a measuring error, a snail), stating that you did.

Interactive Visual

Scatter Plot

Study hours and quiz scores (r = 0.75)

ŷ = 3x + 61

r = 0.75 · strong positive correlation

Tap a point to see its coordinates.

Scatter Plot

One outlier: r drops from 1 to 0

ŷ = 4

r = 0 · no linear correlation

Tap a point to see its coordinates.

Interactive Sandbox

Expression Calculator

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History

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Practice Problems

18 problems

Problem 1 of 18

Easy

A survey of teenagers finds

r = -0.85

between daily screen time and hours of sleep. Which description is correct?

Practice Problems: Correlation Word Problems

1. A survey of teenagers finds r = -0.85 between daily screen time and hours of sleep. Which description is correct?
2. Four studies report these correlations. Which one shows the strongest linear relationship?
3. For a data set, S_{xy} = 27, S_{xx} = 20 and S_{yy} = 45. Compute r = \frac{S_{xy}}{\sqrt{S_{xx} \cdot S_{yy}}}.
4. Five students practised a song for 2, 3, 4, 5 and 6 hours and got these scores: 50, 55, 65, 70, 80. The first step to find r is the means. What is \bar{y}, the mean score?
5. A bakery records the number of hours it is open on five days: 2, 4, 6, 8, 10. Find the mean and the deviations needed for r.
6. A café finds r = -0.72 between the outdoor temperature and the number of hot chocolates sold per day. Interpret this value.
7. Data from many towns show r = 0.90 between the number of pharmacies and the number of hairdressers in a town. Which conclusion is justified?
8. A company wants to show that its new fertilizer causes faster plant growth. Which study could justify that causal claim?
9. For five data points the deviations are x_i - \bar{x}: -2, -1, 0, 1, 2 and y_i - \bar{y}: -4, -2, 1, 3, 2. Compute S_{xy} = \sum(x_i - \bar{x})(y_i - \bar{y}).
10. A study of car age and fuel efficiency gives S_{xy} = -36, S_{xx} = 40 and S_{yy} = 45. Compute r and round to 2 decimal places.
11. A football coach records training sessions and goals for five players: (1, 2), (2, 6), (3, 4), (4, 5), (5, 8). Compute r to 2 decimal places.
12. A teacher compares days absent with test scores (out of 10) for four students: (1, 9), (3, 8), (5, 4), (7, 3). Compute r to 2 decimal places.
13. In a primary school with children aged 6 to 12, shoe size and reading score have r = 0.88. A parent says: "Bigger feet make children read better." Judge this claim.
14. Five points lie close to a rising line and have r = 0.95. A sixth point is added far to the right and far below the line. What happens to r?
15. A shop sells four umbrella models and compares each price (in tens of dollars) with the number sold per week: (2, 9), (4, 7), (6, 3), (8, 1). Compute r and round to 2 decimal places.
16. A basketball coach records extra practice hours and points scored for five players: (1, 2), (2, 4), (3, 6), (4, 8), (10, 0). The last player was injured early in the game. Compute r with and without that player.
17. A city planner compares the distance from the city center (km) with the number of cafés within walking distance: (2, 8), (4, 10), (6, 5), (8, 5). Compute r and round to 2 decimal places.
18. For study hours and test scores of five students, S_{xy} = 45, S_{xx} = 10 and S_{yy} = 250. Compute r, then see what happens when the data are changed.

Why It Matters

Correlation numbers appear in news articles, school reports and science projects every day, and many of them are misread:

Health news: "People who drink coffee live longer" is usually a correlation from a survey, not proof.
Sports: coaches check whether training hours really go with better race times before changing a plan.
Business and energy: a shop compares temperature with sales, a family compares outdoor temperature with heating use.
Science projects: one broken measurement (an outlier) can ruin a result if nobody checks the scatter plot.

If you can compute

r

yourself, describe it in plain words and spot a false causal claim or a misleading outlier, you can judge statistics instead of just believing them.

Real World Applications

Sports Science

Coaches compare weekly training hours with race times to check whether more training goes with faster runs.

Example:

For a running club the sums are $S_{xy} = -45$ , $S_{xx} = 25$ and $S_{yy} = 100$ . A negative $r$ means: more training, lower (faster) times.

1Try It Yourself

A running club records training hours per week and 5 km times (minutes) for its members. The sums are $S_{xy} = -45$ , $S_{xx} = 25$ and $S_{yy} = 100$ .

What is the correlation coefficient $r$ ?

Step 1: Write the mathematical expression

Compute $r = \frac{S_{xy}}{\sqrt{S_{xx} \cdot S_{yy}}}$ :

Reading the News Critically

Journalists and readers must decide whether a reported correlation supports a claim about cause and effect.

Example:

A report finds $r = 0.83$ between the number of TVs per 100 people in a country and life expectancy, and a blog says TVs make people live longer.

2Try It Yourself

Across many countries, the number of TVs per 100 people and life expectancy have $r = 0.83$ . A blog concludes that buying more TVs makes people live longer.

Is the causal claim justified?

Step 1: Choose the right answer

Choose your verdict:

Quality Control of Sensors

Engineers check measurement data for faulty readings, because one bad value can hide a perfect relationship.

Example:

A temperature sensor is tested at five settings: $(2, 10)$ , $(4, 14)$ , $(6, 18)$ , $(8, 22)$ and a faulty reading $(10, 6)$ .

3Try It Yourself

A sensor gives the readings $(2, 10)$ , $(4, 14)$ , $(6, 18)$ , $(8, 22)$ , $(10, 6)$ . The last reading is a known error. The means are $\bar{x} = 6$ and $\bar{y} = 14$ .

What is $r$ once the faulty reading is removed?

Step 1: Write the mathematical expression

Compare $r$ with and without the error:

Key Takeaways

1To compute $r$ by hand: means, deviations, $S_{xy}$ , $S_{xx}$ , $S_{yy}$ , then $r = \frac{S_{xy}}{\sqrt{S_{xx} S_{yy}}}$
2Check your work: the deviations of each variable add up to $0$ , and $r$ must lie between $-1$ and $1$
3Round only the final value of $r$ , usually to 2 decimal places
4Interpret $r$ in context: the sign gives the direction, $|r|$ gives the strength
5A strong correlation from observational data does not justify a causal claim; look for confounding variables
6One outlier can strengthen, weaken or even reverse $r$ : plot the data and compare $r$ with and without it

Frequently Asked Questions

The mean is the balance point of the data: the values above it and below it cancel out exactly. If your deviations do not add up to

0

, you have a mistake in the mean or in a subtraction.

Glossary

Deviation: The distance of a value from the mean, $x_i - \bar{x}$ ; it can be positive or negative
$S_{xy}$: The sum of the products of deviations, $\sum(x_i - \bar{x})(y_i - \bar{y})$ ; its sign is the sign of $r$
$S_{xx}$ and $S_{yy}$: The sums of squared deviations of $x$ and of $y$ ; they are never negative
Outlier: A data point far from the pattern of the others; it can change $r$ a lot
Confounding variable: A variable not included in the study that affects both variables and can create a misleading correlation
Controlled experiment: A study in which the researcher assigns the treatment at random; the only kind of study that can justify a causal claim

Formula Card

Correlation coefficient

r = \frac{S_{xy}}{\sqrt{S_{xx} \cdot S_{yy}}}

$S_{xy} = \sum(x_i - \bar{x})(y_i - \bar{y})$, $S_{xx} = \sum(x_i - \bar{x})^2$, $S_{yy} = \sum(y_i - \bar{y})^2$

Checks

\sum(x_i - \bar{x}) = 0, \quad -1 \le r \le 1

If a deviation sum is not $0$ or $|r| > 1$, there is a calculation error

Strength scale

|r|: \ 0.2 \ \text{weak}, \ 0.4 \ \text{moderate}, \ 0.6 \ \text{strong}, \ 0.8 \ \text{very strong}

Each word starts at its value; below $0.20$ the correlation is very weak

Today's challengeOne problem, three tries

Correlation Word Problems

Definition

Worked Examples

Common Mistakes

Interactive Visual

Scatter Plot

Study hours and quiz scores (r = 0.75)

Scatter Plot

One outlier: r drops from 1 to 0

Interactive Sandbox

Expression Calculator

History

Practice Problems

Why It Matters

Real World Applications

Sports Science

Reading the News Critically

Quality Control of Sensors

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Correlation coefficient

Checks

Strength scale

More in This Topic

Introduction to Correlation

Correlation Coefficient

Interpreting Correlation

Correlation vs Causation