Scatter Plots and Correlation

Do students who study longer get higher marks? Do taller people have longer arms? Questions like these are about how two attributes of the same person or thing are related. A scatter plot lets you see the relationship, and the correlation coefficient $r$ puts a number on how strong it is.

Key ideas

Two-variable data

In two-variable (bivariate) data, you measure two attributes for each individual, like hours studied and test mark for each student. The goal is to study the relationship between the two.

• The independent variable ($x$) is the one you think might explain or influence the other. It goes on the horizontal axis.
• The dependent variable ($y$) is the one that might respond. It goes on the vertical axis.

For hours studied and test mark, hours studied is independent and test mark is dependent.

Scatter plots

A scatter plot shows each individual as one point $(x, y)$. Don’t join the dots: each point is a separate individual.

Describing a scatter plot

Describe four things:

Feature	What to look for
Direction	Positive ($y$ tends to rise as $x$ rises) or negative ($y$ tends to fall)
Form	Linear (points follow a line) or non-linear (a curve)
Strength	Strong (points close to a line or curve), moderate, or weak (widely scattered)
Outliers	Points that sit far away from the overall pattern

The correlation coefficient

The correlation coefficient $r$ measures how closely the points fit a straight line:

$$-1 \le r \le 1$$

• $r = 1$: all points lie exactly on a rising line. $r = -1$: exactly on a falling line.
• $r$ near $0$: no linear relationship.
• The sign gives the direction; the size $|r|$ gives the strength.

Four sets of points and their correlation coefficients.

Many Ontario textbooks use these cut-offs (others draw the lines a little differently):

Value of r	Description
$0.67 \le r \le 1$	strong positive
$0.33 \le r \lt 0.67$	moderate positive
$0 \lt r \lt 0.33$	weak positive
$r = 0$	no linear correlation
$-0.33 \lt r \lt 0$	weak negative
$-0.67 \lt r \le -0.33$	moderate negative
$-1 \le r \le -0.67$	strong negative

Finding r with technology

You’ll almost always find $r$ with technology:

• Spreadsheet: =CORREL(A2:A11, B2:B11), with the $x$-values in column A and the $y$-values in column B.
• Graphing calculator (TI-83/84): enter the data in lists L1 and L2, turn on DiagnosticOn, then choose STAT, CALC, LinReg(ax+b).
• Desmos: enter a table, then type y1 ~ m x1 + b. Desmos shows $r$.

For the curious, the formula is:

$$r = \frac{n\sum xy - \left(\sum x\right)\left(\sum y\right)}{\sqrt{\left[n\sum x^2 - \left(\sum x\right)^2\right]\left[n\sum y^2 - \left(\sum y\right)^2\right]}}$$

You aren’t expected to use it by hand for real data sets. Answers on this page are rounded to three decimals; your technology may differ in the last digit if you round along the way.

Comparing groups with side-by-side boxplots

When one variable is categorical (like grade) and the other is numerical (like hours of sleep), a scatter plot doesn’t work. Instead, draw a boxplot for each category on the same scale and compare their medians, spreads (IQR and range), and outliers. You met boxplots in quartiles and percentiles.

Worked examples

Example 1: Which variable is which?

For each pair, identify the independent and dependent variables.

• (a) The outdoor temperature and the number of hot chocolates a café sells.
• (b) A car’s age and its resale value.

Solution.

(a) Temperature might affect sales, not the other way around. Independent: temperature. Dependent: hot chocolates sold.

(b) The age of the car influences its value. Independent: age. Dependent: resale value.

Example 2: Describing a scatter plot

Ten students recorded how long they studied for a test and their mark.

Hours studied	$1$	$2$	$2$	$3$	$4$	$4$	$5$	$6$	$7$	$8$
Mark (%)	$55$	$52$	$64$	$60$	$71$	$63$	$74$	$70$	$85$	$80$

Draw a scatter plot, describe it, and find $r$.

Solution.

Test mark against hours studied for ten students.

• Direction: positive; students who studied longer tended to score higher.
• Form: roughly linear.
• Strength: fairly strong; the points stay close to a line, with some scatter.
• Outliers: none.

With =CORREL or a calculator:

$$r \approx 0.900$$

That’s a strong positive linear correlation, which matches the picture.

Example 3: Reading values of r

Describe each correlation using the table above: (a) $r = -0.92$ (b) $r = 0.45$ (c) $r = -0.18$.

Solution.

(a) $|r| = 0.92 \ge 0.67$ and $r$ is negative: strong negative.

(b) $0.33 \le 0.45 \lt 0.67$: moderate positive.

(c) $|r| = 0.18 \lt 0.33$ and $r$ is negative: weak negative.

Example 4: Side-by-side boxplots

A survey asked Grade 9 and Grade 12 students how many hours they sleep on a school night. Compare the groups.

Hours of sleep on a school night for two grades.

Solution.

• Centre: the Grade 9 median is $8$ hours; the Grade 12 median is $7$ hours. Grade 9 students typically sleep about an hour more.
• Spread: Grade 9 IQR $= 8.5 - 7.5 = 1$ hour; Grade 12 IQR $= 7.5 - 6 = 1.5$ hours. Ranges are $3$ and $4$ hours. Grade 12 sleep times are more spread out.
• Overall: in fact, $75\%$ of Grade 12 students sleep $7.5$ hours or less, while $75\%$ of Grade 9 students sleep $7.5$ hours or more.

So there does seem to be a relationship between grade and sleep in this survey.

Common mistakes

Swapping the axes. The independent variable goes on the $x$-axis. Ask: “Which one might influence the other?”

Thinking a negative r means a weak relationship. The sign is only the direction. $r = -0.9$ is just as strong as $r = 0.9$.

Using r for a curved pattern. $r$ measures linear fit only. Points that follow a clear curve can have $r$ near $0$ (see Practice Question 9). Always look at the scatter plot first.

Ignoring outliers. A single outlier can change $r$ a lot (see Practice Question 6). Check whether it’s a data-entry error or a genuinely unusual individual before deciding what to do.

Reading r as a percentage. $r = 0.5$ does not mean “50% related”. It’s just a number on the scale from $-1$ to $1$.

Concluding that x causes y. A strong correlation alone doesn’t prove cause and effect. See correlation and causation.

Practice

1. (Warm-up) Identify the independent and dependent variables.

• (a) The number of hours a phone is used and its remaining battery percentage.
• (b) A person’s height and their shoe size.

Solution

(a) Independent: hours of use. Dependent: battery percentage.

(b) Height is usually treated as independent and shoe size as dependent, since shoe size is thought of as following from overall body size. (Either choice can be defended here, as long as you explain it.)

2. (Warm-up) Describe each correlation: (a) $r = 0.82$ (b) $r = -0.45$ (c) $r = 0.12$ (d) $r = -0.93$

Solution

(a) Strong positive. (b) Moderate negative. (c) Weak positive. (d) Strong negative.

3. (Warm-up) Which shows a stronger linear relationship, $r = -0.85$ or $r = 0.70$? Explain.

Solution

$r = -0.85$, because $|-0.85| = 0.85$ is larger than $0.70$. The sign only tells you the direction.

4. (Core) Eight people measured their arm span and height in centimetres.

Arm span (cm)	$152$	$160$	$165$	$170$	$174$	$178$	$183$	$190$
Height (cm)	$155$	$158$	$167$	$168$	$177$	$175$	$186$	$188$

Use technology to find $r$, and describe the correlation.

Solution

$r \approx 0.976$. This is a strong positive linear correlation: people with longer arm spans tend to be taller, and the points lie very close to a line.

5. (Core) A used-car website lists the asking prices of eight cars of the same model.

Age (years)	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$
Price (thousands of $)	$27.5$	$24.0$	$22.5$	$18.0$	$17.5$	$13.0$	$12.5$	$9.0$

Find $r$ and describe the scatter plot (direction, form, strength, outliers).

Solution

$r \approx -0.991$. The scatter plot shows a strong, negative, linear relationship with no outliers: older cars have lower prices, and the price falls steadily with age.

6. (Core) An eleventh student is added to the data in Example 2: they studied $8$ hours but scored $45\%$. Find the new value of $r$. What does this show?

Solution

With the extra point, $r \approx 0.402$ (down from $0.900$). One outlier changed a strong correlation into a moderate one. Outliers can have a big effect on $r$, especially in small data sets.

7. (Core) A class recorded their travel times to school. Five-number summaries, in minutes:

	Min	Q1	Median	Q3	Max
Bus	$10$	$18$	$25$	$30$	$45$
Walk	$5$	$8$	$12$	$15$	$20$

Describe how side-by-side boxplots of these would compare the two groups.

Solution

The bus box would sit well to the right of the walking box. The median bus trip ($25$ min) is more than twice the median walk ($12$ min). Bus times are also more spread out: IQR $30 - 18 = 12$ min versus $15 - 8 = 7$ min, and range $35$ min versus $15$ min. Even the slowest walker ($20$ min) is faster than the median bus rider, so method of travel and travel time seem to be related.

8. (Challenge) Use the formula to calculate $r$ by hand for the points $(1, 2)$, $(2, 4)$, $(3, 5)$, $(4, 4)$, $(5, 6)$. Check with technology.

Solution

$n = 5$, $\sum x = 15$, $\sum y = 21$, $\sum xy = 2 + 8 + 15 + 16 + 30 = 71$, $\sum x^2 = 55$, $\sum y^2 = 4 + 16 + 25 + 16 + 36 = 97$.

$$\begin{aligned} n\sum xy - \left(\sum x\right)\left(\sum y\right) &= 5(71) - 15(21) = 40 \\ n\sum x^2 - \left(\sum x\right)^2 &= 5(55) - 15^2 = 50 \\ n\sum y^2 - \left(\sum y\right)^2 &= 5(97) - 21^2 = 44 \end{aligned}$$$$r = \frac{40}{\sqrt{50 \times 44}} = \frac{40}{\sqrt{2200}} \approx 0.853$$

Technology gives the same value: a strong positive correlation.

9. (Challenge) Find $r$ for the points $(-2, 4)$, $(-1, 1)$, $(0, 0)$, $(1, 1)$, $(2, 4)$. Does this mean $x$ and $y$ are unrelated?

Solution

$r = 0$. In the formula, $\sum x = 0$ and $\sum xy = -8 - 1 + 0 + 1 + 8 = 0$, so the numerator is $5(0) - 0(10) = 0$.

But $x$ and $y$ are perfectly related: every point is on $y = x^2$. The relationship is just not linear, so $r$ can’t detect it. This is why you should always look at the scatter plot, not just $r$.