Properties of Pearson's r

Range

A basic property of Pearson's r is that its possible range is from -1 to 1.

• A correlation of -1 means a perfect negative linear relationship,
• a correlation of 0 means no linear relationship, and
• a correlation of 1 means a perfect positive linear relationship.

Symmetry

Pearson's correlation is symmetric in the sense that the correlation of X with Y is the same as the correlation of Y with X.

• For example, the correlation of Weight with Height is the same as the correlation of Height with Weight.

Linear transformations

A critical property of Pearson's r is that it is unaffected by linear transformations.

• This means that multiplying a variable by a constant and/or adding a constant does not change the correlation of that variable with other variables.
• For instance, the correlation of Weight and Height does not depend on whether Height is measured in inches, feet, or even miles.
• Similarly, adding five points to every student's test score would not change the correlation of the test score with other variables such as GPA.

Quiz

Quiz

1 The correlation between temperature and number of ice cream cones bought is the same whether the temperature is measured in Celsius or Fahrenheit.

	True
	False

Answer >>

True

It will be the same because that is a linear transformation.

2 The correlation between two sets of numbers is the same as the correlation between the log of those two sets of numbers.

	True
	False

Answer >>

False

It won't be the same because a log transformation is not a linear transformation.

3 Which of the following is not a possible value for Pearson's correlation?

	-1.5
	-1
	0.0
	0.99

Answer >>

-1.5

Pearson's correlation can be any value between -1 and 1 inclusive.

4 Which is higher, the correlation between height and weight or the correlation between weight and height?

	weight and height
	They are about the same
	They are exactly the same
	height and weight

Answer >>

They are exactly the same. Correlations are symmetric so they are exactly the same.