Correlation Coefficient

Q: Can r be bigger than 1 or smaller than -1?

No. By definition $r$ always lands between $-1$ and $1$. If a calculation gives you $r = 1.3$, something went wrong — recheck the work.

Q: Is r = -0.9 weaker than r = 0.7?

No — it's stronger. Strength is measured by absolute value, and $|-0.9| = 0.9$ beats $|0.7| = 0.7$. The negative sign only tells you the direction of the trend, not how tight it is.

Q: Does r = 0 mean the variables are unrelated?

It means there is no linear relationship. The variables could still be related by a curve — data shaped like a U can have $r$ near $0$. Always look at the scatterplot, not just the number.

Q: Does a large r prove that one variable causes the other?

No. Even $r = 0.99$ only proves the variables move together. A lurking variable could drive both — that's the correlation vs. causation lesson. Strength and causation are separate questions.

Describing a scatterplot as strong positive or weak negative is a judgment call. The correlation coefficient, written $r$ , turns that judgment into a number. It is always between $-1$ and $1$ , and it packs two pieces of information into one value: direction and strength.

The sign of $r$ gives the direction — positive $r$ means a positive association, negative $r$ means a negative one. The size of $r$ (how far it is from zero) gives the strength: values near $1$ or $-1$ mean the points hug a line tightly, and values near $0$ mean barely any linear pattern at all.

Reading r in two steps

Step 1: read the sign. If $r > 0$ , the points trend upward; if $r < 0$ , they trend downward. The sign says nothing about strength — $r = -0.9$ is a very strong relationship even though the number is negative.

Step 2: read the distance from zero. As a rough guide, $|r|$ near $1$ (say $0.8$ or above) is strong, around $0.5$ is moderate, and near $0$ (say below $0.3$ ) is weak. So $r = 0.85$ is strong positive, $r = -0.15$ is weak negative, and $r = 0$ means no linear relationship at all.

The scale below runs from $-1$ to $1$ . The sign places you on the left (negative) or right (positive) half, and the distance from the center $0$ shows the strength — the color fades toward gray near $0$ where the relationship is weakest.

The endpoints: r = 1, r = -1, and r = 0

$r = 1$ means every point lies exactly on a line with positive slope — a perfect positive relationship. $r = -1$ is the same but sloping downward. Real data almost never hits these perfectly; they are the ends of the scale.

$r = 0$ means no linear pattern. Careful: it does not mean no pattern at all. Points on a perfect U-shaped curve can have $r$ near $0$ , because $r$ only measures how well a straight line fits.

Comparing strengths

To decide which of several $r$ -values is strongest, compare absolute values — the distances from zero. Between $r = 0.6$ and $r = -0.8$ , the second is stronger, because $|-0.8| = 0.8$ is closer to $1$ than $|0.6| = 0.6$ . Ignore the sign while comparing strength; bring it back when you state the direction.

Worked examples

Example 1: describe a positive r

A data set comparing hours of practice and quiz scores has $r = 0.92$ . Describe the correlation.

Read the sign:

r

is positive, so the direction is positive

r > 0

Read the size:

0.92

is close to

1

, so the relationship is strong

|r| = 0.92

Answer: Strong positive correlation

Example 2: describe a negative r near zero

A data set has $r = -0.25$ . Describe the correlation.

Read the sign:

r

is negative, so the direction is negative

r < 0

Read the size:

0.25

is close to

0

, so the relationship is weak

|r| = 0.25

Answer: Weak negative correlation

Example 3: pick the strongest

Which shows the strongest correlation: $r = 0.55$ , $r = -0.78$ , or $r = 0.32$ ?

Take absolute values to measure strength

0.55,\; 0.78,\; 0.32

Find the value closest to

1

0.78

Report the original

r

, keeping its sign

r = -0.78

Answer: $r = -0.78$ — strength comes from distance to $1$ or $-1$ , not the sign.

Try one yourself

Common questions

Can r be bigger than 1 or smaller than -1?

No. By definition $r$ always lands between $-1$ and $1$ . If a calculation gives you $r = 1.3$ , something went wrong — recheck the work.

Is r = -0.9 weaker than r = 0.7?

No — it's stronger. Strength is measured by absolute value, and $|-0.9| = 0.9$ beats $|0.7| = 0.7$ . The negative sign only tells you the direction of the trend, not how tight it is.

Does r = 0 mean the variables are unrelated?

It means there is no linear relationship. The variables could still be related by a curve — data shaped like a U can have $r$ near $0$ . Always look at the scatterplot, not just the number.

Does a large r prove that one variable causes the other?

No. Even $r = 0.99$ only proves the variables move together. A lurking variable could drive both — that's the correlation vs. causation lesson. Strength and causation are separate questions.

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