D. Statistical Analysis · Two-Variable Data Analysis and Validity

Interpreting r: the Coefficient of Determination and the Limits of r

D2.1 D2.5About 65 minutes

Learning goals

I can calculate and interpret the coefficient of determination r² in context.
I can list the properties of r: no units, symmetric in x and y, unchanged by changes of unit.
I can explain why r near 0 does not always mean "no relationship".
I can explain how outliers, small samples and a restricted range can change r and mislead.

Words to know

coefficient of determination (r²)
The square of r. It gives the fraction of the variation in y that is explained by the linear relationship with x.

nonlinear relationship
A relationship whose pattern is a curve rather than a straight line.

restricted range
Data that cover only a narrow part of the possible x-values. It usually makes |r| smaller than it would be over the full range.

influential point
A point whose removal would change r or the line of best fit a lot. Influential points are often far from the others in the x-direction.

Can r be fooled?

A weather report gives the average daily high temperature for each month of the year. If you compute r for month number and temperature, you get about 0.29: a weak correlation. Yet anyone in Ontario can tell you that temperature depends very strongly on the month! What went wrong?

Nothing is wrong with the arithmetic. The problem is that r only measures linear (straight-line) relationships. This lesson is about what r does and does not tell you.

Month	1	2	3	4	5	6	7	8	9	10	11	12
High (°C)	−3	−2	3	11	18	24	27	26	21	14	6	0

Average daily high temperature by month for a made-up southern Ontario town.

The points rise and then fall in a smooth arch. The relationship is very strong, but it is not linear, so r is small. Always graph before you trust r.

The coefficient of determination, r²

Squaring r gives the coefficient of determination, r². It answers a useful question:

What fraction of the variation in y is explained by the straight-line relationship with x?

For the study-time data in the last lesson, r ≈ 0.979, so r² ≈ 0.958. About 96% of the variation in marks is explained by the linear relationship with study time. The other 4% comes from everything else (sleep, prior knowledge, luck on the test day...).

Notice that r² is always between 0 and 1, and that it is smaller than |r| (unless r is 0, 1 or −1). An r of 0.5 sounds moderate, but r² = 0.25 says the line explains only a quarter of the variation.

Squaring makes the difference between correlations easier to feel. r = 0.7 sounds close to r = 0.9, but the line explains 49% of the variation in one case and 81% in the other.

Properties of r

No units. Changing centimetres to metres, or °C to °F, does not change r.
Symmetric. The correlation of x with y equals the correlation of y with x.
Only linear. r measures how well the points fit a straight line. A strong curve can have r near 0 (as with the monthly temperatures).
Sensitive to outliers. One unusual point can raise or lower r a lot.
Not reliable with tiny samples. With 3 or 4 points, a large |r| can happen by chance.
Affected by a restricted range. If you only look at a narrow slice of x-values (for example, heights of professional basketball players only), the correlation inside that slice is usually weaker than in the whole population.
Not a measure of cause. A large r never proves that x causes y (see the lesson on types of relationships).

Left: four points with no linear pattern at all (r = 0). Right: the same four points plus one point far away. That single point pushes r up to 0.9.

Worked example

Example 1: Interpreting r²

For 30 used cars of one model, r = −0.86 between age (years) and price ($). Find and interpret r².

r² = (−0.86)² = 0.7396 ≈ 0.74.
Interpret in context: about 74% of the variation in price among these cars is explained by the linear relationship with age.
The other 26% is due to other factors, such as kilometres driven, condition and options.

r² ≈ 0.74. About 74% of the variation in price is explained by the linear relationship with age. (r² does not tell you the direction; the negative sign of r does: older cars tend to cost less.)

Worked example

Example 2: One point, big effect

Show by calculation that adding the point (10, 10) to (1, 3), (2, 1), (3, 4), (4, 2) changes r from 0 to 0.9.

Four points: Σx = 10, Σy = 10, Σxy = 3 + 2 + 12 + 8 = 25. Numerator: 4(25) − 10(10) = 0, so r = 0.
Five points: Σx = 20, Σy = 20, Σxy = 25 + 100 = 125, Σx² = 30 + 100 = 130, Σy² = 30 + 100 = 130.
Numerator: 5(125) − 20(20) = 625 − 400 = 225.
Brackets: 5(130) − 20² = 250 and 5(130) − 20² = 250.
r = 225 ÷ √(250 × 250) = 225 ÷ 250 = 0.9.

r goes from 0 to 0.9 because of one point. A report that only said "r = 0.9" would badly misrepresent the four ordinary points. Check for outliers and report r with and without them.

Worked example

Example 3: A restricted range

Over all students at a school, marks on a Grade 11 math test and a Grade 12 math test have r = 0.8. A university program looks only at students who scored above 90% on the Grade 11 test and finds r = 0.3 for them. Did the relationship become weaker?

Among students above 90%, the Grade 11 marks are squeezed into a narrow band (90 to 100).
Inside a narrow band of x, small differences in x are swamped by everything else that affects the Grade 12 mark, so the points form a rounder cloud.
The relationship over the whole school has not changed. Only the slice being examined has.

No. This is a restricted range effect. Correlations computed for a narrow, selected group are usually weaker than for the full population, so conclusions about "everyone" should not be drawn from the slice.

A checklist before you report r

Graph it. Is the pattern linear?
Look for outliers and check whether they are errors.
Count the points. Is n large enough to trust?
Check the range. Does it cover the x-values you care about?
Report r² too if you want to say how much the line explains.
Never write "x causes y" from r alone.

Just for fun

A famous 1973 data set by the statistician Francis Anscombe has four groups of 11 points. All four have the same means, the same r (about 0.82) and the same line of best fit. Graphed, they look completely different: one is a curve, one is a line with an outlier, and one is a vertical stack with a single faraway point. It is still used to teach the rule: graph first.

Hands-on activity

Make r lie

You need: graph paper, a pencil, and a spreadsheet or Desmos.

Plot 6 points that clearly follow a U-shape (for example (1, 9), (2, 4), (3, 1), (4, 1), (5, 4), (6, 9)). Find r with technology. It should be close to 0, even though the pattern is perfect.
Plot 6 points in a round cloud with no pattern. Find r. Then add one point far up and to the right. Find r again.
Take your arm-span and height data from the first lesson. Find r for everyone, then only for the adults (or only for the people taller than 165 cm). Does r change?
Write two sentences warning a classmate about trusting r without a graph.

What to look for: step 1 gives exactly r = 0 for the sample points (the pattern is symmetric). Step 2 usually jumps from near 0 to above 0.7. Step 3 usually gives a smaller r for the restricted group.

Your turn

r = 0.6. Find r² and interpret it for hours of practice and points scored by basketball players.

r² = 0.81 and the scatter plot falls to the right. What is r?

Why can r² never be negative?

A scatter plot of practice time and skill forms a curve that levels off. r = 0.75. Is a linear model the best choice?

A researcher finds r = 0.95 using only 3 data points. Why should you be cautious?

Is the correlation between height and mass the same as the correlation between mass and height?

A data set has r = 0.12. A student concludes there is no relationship. What should they check?

More practice: warm-up

r = 0.8. Find r².

r = −0.5. Find r².

r² = 0.49 and the relationship is positive. Find r.

Does changing kilograms to pounds change r?

True or false: r = 0 always means the variables are unrelated.

More practice: on your own

Correlation A has r = 0.6 and correlation B has r = 0.9. How many times as much of the variation in y does model B explain compared with model A?

Find r for (1, 1), (2, 4), (3, 9), (4, 16), (5, 25). Is the relationship linear?

A researcher reports r = 0.07 between monthly ice cream sales and month number for a whole year. Explain why this does not mean ice cream sales have nothing to do with the time of year.

Explain, using the idea of a restricted range, why the correlation between SAT-style entrance test scores and first-year marks is weaker among students admitted to a selective program than among all applicants.

For 20 regions, r = 0.55 between median income and a health score. A newspaper says income "explains most of" the health score. Is that accurate?

Challenge

Points (1, 2), (2, 4), (3, 6) and an extra point (k, 0). Without calculating exactly, explain how r changes as k goes from 0 to 10.

Show that r² = 0.958 for the study-time data using the numbers from the last lesson: numerator 810, brackets 225 and 3044.

A data set has 50 points with r = 0.30, and another has 5 points with r = 0.80. Which is better evidence that a positive linear relationship exists? Explain.

Check your understanding

Quiz

r = −0.7. What is r²?

r² = 0.64 for hours of sleep and reaction time. What does it mean?

A scatter plot shows a perfect U-shape, symmetric about the middle. What is r closest to?

Which change can never affect r?

Why are correlations within a narrow, selected group often weak?

A report gives r = 0.88 from 4 data points. What is the best response?

For parents and teachers

Big idea (D2.1, D2.5): r only measures straight-line relationships, and it can be distorted by curves, outliers, tiny samples and narrow ranges. r² tells what fraction of the variation in y the line explains.

How to help: Ask your teen to show you the monthly temperature graph and explain why r is small. Then ask, "If someone tells you r = 0.9, what three questions would you ask?" (Good answers: Is it linear? Any outliers? How many points? Is it cause or just correlation?)

Common difficulties:

Treating r² as "percent of people" or "percent chance". It is the percent of variation explained.
Forgetting that r = √(r²) needs the sign from the graph.
Believing r near 0 means no relationship at all.
Reporting r without ever looking at a scatter plot.