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Read the strength and direction of a correlation coefficient, and know why no value of it establishes that one variable causes the other.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
In this lesson you learn to read the correlation coefficient $r$ as two pieces of information — its size for how tightly the points follow a line, its sign for which way that line goes — and to remember that it sees straight lines only. You then learn the limit that matters most in practice: a correlation, however strong, never shows that one thing causes another, because a lurking variable can produce the whole pattern.
You have just fitted a line to a scatter plot and read its slope, and you know that residuals measure what the line misses. This lesson puts a single number on how well a line describes the cloud of points — and then asks the harder question of what that number does and does not entitle anyone to say.
Correlation coefficient $r$: a number between $-1$ and $1$ measuring how close the points lie to a straight line.
Strength: the size of $r$, ignoring its sign. Near $1$ or $-1$ is strong; near $0$ is weak.
Direction: the sign of $r$. Positive means the two rise together; negative means one rises as the other falls.
Causation: one variable actually producing a change in the other.
Lurking variable: a third quantity affecting both, which can create a correlation with no causal link between them at all.
The correlation coefficient $r$ measures how close a scatter plot lies to a straight line, and it says two things at once. Its size gives the strength: $0.95$ and $-0.99$ are both very tight patterns, $0.4$ is a loose one, and anything near $0$ means the points are essentially a cloud as far as a straight line is concerned. Its sign gives the direction: positive if the two rise together, negative if one falls as the other rises. It only ever describes linear association, so a strong curved relationship can produce an $r$ near zero. And no value of $r$, however extreme, says that one variable causes the other. A lurking variable can drive both — hot weather drives ice cream sales and swimming, and so drownings — and the causation may run the other way, or be a coincidence. Establishing a cause needs a study designed to rule those out, which usually means assigning the treatment at random.
Another way: diagram
Four small scatter plots in a row labelled $r = 0.95$, $r = 0.4$, $r = 0.05$ and $r = -0.9$: a tight rising band, a loose rising cloud, a shapeless blob, and a tight falling band.
Another way: story
Read $r$ as two words. The size is 'how tight', the sign is 'which way'. Then stop: the number describes the picture and says nothing whatever about why the picture looks like that.
"$r = -0.9$ is weaker than $r = 0.5$, because it is negative." Strength is the size: $|-0.9| = 0.9$ is much stronger. The minus sign only says the relationship goes downhill.
"$r = 0$ means the two variables are unrelated." It means no linear relationship. Points lying neatly on a U-shaped curve can have $r$ close to zero.
"A very strong correlation must mean causation." Strength is not evidence of mechanism. Ice cream sales and drownings correlate strongly every summer, and neither causes the other.
Size first: $0.9$ is close to $1$, so the points lie close to a straight line — a strong association.
Strength is the size of $r$.
Sign second: it is negative, so the line falls — one variable decreases as the other increases.
Direction is the sign.
So: a strong negative linear association. Nothing about cause.
The correlation is real: both quantities do rise together.
The pattern is not in dispute.
But hot weather increases both ice cream sales and time spent in water.
Look for a third variable driving both.
Temperature is a lurking variable, so ice cream does not cause drowning.
$0.02$ is very close to zero, so there is almost no linear association.
It does not rule out a curved relationship, and it says nothing about cause either way.
Which of these correlation coefficients shows the strongest linear relationship?
Over $7$ summers, ice cream sales and drownings rose and fell together. Does buying ice cream cause drowning?
Two variables have $r = 0.03$. What can you say?
Match each correlation coefficient to its description.
| a strong positive linear association | a moderate negative linear association | essentially no linear association | every point exactly on a falling straight line | |
|---|---|---|---|---|
| $r = 0.9$ | ||||
| $r = -0.5$ | ||||
| $r = 0.04$ | ||||
| $r = -1$ |
A survey of $481$ people finds that those who take a vitamin have fewer colds, with $r = -0.6$. What would actually be needed to conclude that the vitamin **causes** fewer colds?
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
In a study of $197$ homes, the warmer the outside temperature was, the **less** was spent on heating. Which value of $r$ fits that?
You can interpret a correlation coefficient and explain why it is not evidence of cause. Say why $r = -0.9$ is stronger than $r = 0.5$, and why ice cream sales correlate with drownings without causing them.
8. Your turn: $r = 0.02$ for two variables. What can you say?, step 2