Properties and assumptions of Pearson's r

Properties of Pearson's r

Pearson's r summarises the direction and strength of a linear association between two numerical variables. Its value is bounded, its sign gives direction, and its magnitude gives linear strength

1. Defined values For paired numerical observations with nonzero variation in each variable, −1≤r≤1. Pearson’s r is unitless and measures linear association. A constant variable makes its ordinary denominator zero, so r is undefined
2. Use the sign for direction A positive r indicates a rising direction and a negative r indicates a falling direction \(\displaystyle r>0\Rightarrow\text{positive direction},\quad r<0\Rightarrow\text{negative direction}\)
3. Interpret the display Inspect the scatterplot before summarising. A coefficient can be calculated for a nonconstant curved dataset, but it can miss that relationship. Near-zero r does not rule out a strong non-linear pattern. Investigate unusual cases and separate groups without automatically deleting them
4. State the exercise convention For these exercises, |r|<0.25 is reported as little or no linear association; 0.25≤|r|<0.50 is weak, 0.50≤|r|<0.75 is moderate, and 0.75≤|r|≤1 is strong. These are reporting conventions, not universal boundaries. At |r|=1, also state perfect linear association. A small r does not establish independence or rule out a curved relationship

Worked example

A numerical data set has the Pearson correlation coefficient shown below. State whether the value is possible and classify the direction and strength of the linear association. Use the declared exercise bands

1. \(\displaystyle -1\le-0.64\le1\)The value lies in the inclusive range, so it is possible for Pearson's r
2. \(\displaystyle r<0\Rightarrow\text{negative direction}\)The negative sign means the response tends to decrease as the explanatory variable increases
3. \(\displaystyle \lvert-0.64\rvert=0.64,\quad0.5\le0.64<0.75\Rightarrow\text{moderate strength}\)The magnitude 0.64 is in the moderate band, so the association is moderate negative and linear

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