Correlation: Study Guide (Statistics 5.2)

One number for a linear association

In Topic 5.1 you described a scatterplot in words: form, direction, strength and unusual features. Words like “strong” and “moderate” are useful but vague. When the form is linear, one number can summarise the direction and strength: the correlation coefficient, r.

• Direction: a positive r means a positive association; a negative r means a negative association.
• Strength: the closer r is to −1 or 1, the more closely the points follow a straight line.

In this course you find r with technology. Your job is to interpret it and to know when it can be trusted.

Properties of r

Property	What it means
−1 ≤ r ≤ 1	r can never be bigger than 1 or smaller than −1. A value like 1.3 is always a mistake.
Sign	r > 0: positive association. r < 0: negative association.
Size	The closer r is to −1 or 1, the stronger the linear association. r = −0.85 is stronger than r = 0.60.
r = 1 or r = −1	A perfect linear association: every point lies on one straight line.
r = 0	No linear association. There may still be a strong curved pattern.
No units	r is a pure number. Changing units (cm to inches, kg to pounds) does not change r.
Order of variables	Swapping x and y gives the same r.

A rough working guide (not an official rule): |r| above about 0.8 is often called strong, about 0.5 to 0.8 moderate, and below about 0.5 weak. Always check the scatterplot too.

Background: where r comes from

You will not be asked to calculate r by hand, but the idea behind it helps you interpret it. Technology converts every x-value and y-value to a z-score (how many standard deviations it is from its mean). Then:

r = (1 / (n − 1)) Σ zₓ z_y

If a point is above average on both variables, or below average on both, the product zₓz_y is positive. If it is above on one and below on the other, the product is negative. When most points are “both high or both low”, r is positive. Because z-scores have no units, r has no units either.

Worked example 1: interpreting r in context

Question. A fictional orchard measured the trunk diameter (cm) and the apple yield (kg) of 10 trees of the same variety.

Trunk diameter (cm)	12	15	17	18	20	22	24	25	28	30
Yield (kg)	38	49	45	52	60	53	72	62	79	74

(a) The scatterplot shows a linear pattern with no unusual points. Use technology to find r and interpret it. (b) The orchard’s export manager converts diameters to inches and yields to pounds. What is r now?

(a) Enter diameter as x and yield as y. Technology gives r = 0.9271… ≈ 0.93.

Interpretation. r = 0.93 shows a strong, positive, linear association between trunk diameter and apple yield for these 10 trees: trees with wider trunks tend to produce more apples.

(b) r has no units, so changing units does not change it: r = 0.93 still. (Technology confirms 0.9271 with diameters ÷ 2.54 and yields × 2.20462.) Swapping the axes would also give 0.93.

Check. A full interpretation names the strength, the direction, the word linear, and both variables in context.

When r misleads (1): a curved pattern

r close to −1 or 1 does not mean a straight line is the right model. Always look at the scatterplot.

Worked example 2: a cooling drink

Question. A student recorded the temperature of a cup of tea every 2 minutes in a fictional science lesson. Technology gives r = −0.98. A classmate says: “r is almost −1, so the relationship is linear.” Is the classmate right?

Time (min)	0	2	4	6	8	10	12	14	16	18	20
Temperature (°C)	90.0	79.9	71.4	64.1	57.9	52.6	48.0	44.2	40.9	38.1	35.7

1. Look at the rate of change. In the first 2 minutes the tea cools by 10.1 °C. Between 18 and 20 minutes it cools by only 2.4 °C. A linear pattern would have a roughly constant drop every 2 minutes.
2. Look at the plot. The dots curve; all the middle dots sit below the straight line joining the first and last readings.
3. Conclusion. The classmate is wrong. r = −0.98 tells us there is a strong negative association, and that a straight line follows it fairly well, but the form is clearly non-linear. A value of r near −1 or 1 never proves that the form is linear; you must check the scatterplot. (Later in Unit 5 you will use residual plots to check this more carefully.)

When r misleads (2): unusual points

r is not resistant. One unusual point can change it a lot. Eight fictional lizards had lengths 10, 11, 12, …, 17 cm and masses 21, 24, 26, 30, 31, 35, 37 and 40 g, lying almost on a straight line: r = 0.997. A ninth record, length 11 cm and mass 62 g (probably a typing error for 26 g), drops r to 0.18. A ninth point that fits the pattern, such as 25 cm and 62 g, leaves r at about 0.999. So report any unusual points alongside r, and check whether they are errors.

When r misleads (3): correlation is not causation

A strong correlation shows that two variables are associated. It does not show that changes in one variable cause changes in the other. Often a lurking variable (a third variable not in the study) affects both. Only a well-designed experiment, with random assignment of treatments (Topic 1.13), can give good evidence of cause and effect.

Worked example 3: cafés and bike thefts

Question. In a fictional region, a researcher counted the number of cafés and the number of reported bicycle thefts last year in 14 towns. The scatterplot is linear with r = 0.91. A local newspaper writes: “Cafés cause bike crime. Close cafés to make streets safer.” Respond.

1. Interpret r. r = 0.91 shows a strong, positive, linear association: towns with more cafés tend to have more reported bike thefts.
2. Look for a lurking variable. Town population is a likely one. Bigger towns have more cafés and more people, more bikes and more thefts. Population could explain the association without cafés causing anything.
3. Type of study. This is an observational study. Nobody assigned cafés to towns at random, so it cannot show cause and effect.
4. Conclusion. The newspaper’s claim is not justified. The correlation is real, but correlation does not imply causation. A fairer comparison would at least look at thefts per 1,000 residents; even then, only an experiment could establish a cause.

Common misconceptions

• “r = −0.85 is weaker than r = 0.60 because it is negative.” The sign is direction only. Strength depends on distance from 0: |−0.85| = 0.85 is stronger.
• “r = 0 means the variables are unrelated.” r = 0 means no linear association. Figure 1, panel D, has a strong curved pattern with r = 0.
• “r near 1 means a line is the right model.” A curve can give r = 0.98 (Worked example 2). Check the scatterplot.
• “r has units.” It does not. “r = 0.93 kg per cm” is wrong.
• “r can be bigger than 1 if the association is very strong.” Never: −1 ≤ r ≤ 1.
• “High correlation proves cause.” Correlation does not imply causation; look for lurking variables.
• “r tells me the slope.” r measures how closely points follow a line, not how steep the line is. A perfect line with a gentle slope still has r = 1.
• “One odd point won’t matter.” r is not resistant; one point can change it greatly.
• Interpreting r without context. Say “strong, positive, linear association between trunk diameter and yield”, not just “strong”.

Where this leads

Once you know a linear association is strong, you can fit a line and use it to predict. That is Topic 5.3: Linear Regression Models. Later, in least-squares regression, you will also meet r², which builds on r. Try the practice questions now, then use the revision notes and the checklist to consolidate.