Statistics for Decision Makers - 04.02 - Pearson Correlation

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04.02 - Pearson Correlation

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Bernard Szlachta (NobleProg Ltd) bs@nobleprog.co.uk

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Pearson's correlation。

Pearson product-moment correlation coefficient

A measure of the strength of the linear relationship between two variables.

Other names

Pearson's correlation

Correlation coefficient

• If the relationship between the variables is not linear, then the correlation coefficient does not adequately represent the strength of the relationship between the variables

Symbols for Pearson's correlation

"ρ" in the population

"r" when it is measured in a sample

Pearson's r 。

Pearson's r can range from -1 to 1.

• r = 1 indicates a perfect positive linear relationship between variables
• r = -1 indicates a perfect negative linear relationship between variables
• r = 0 indicates no linear relationship between variables

r = 1	r = -1	r = 0
perfect positive linear relationship	A perfect negative linear relationship	No relationship

-1<r<1 。

With real data, you would not expect to get values of r of exactly -1, 0, or 1.

r = 0.97	r = 0.63

Symmetry。

Pearson's correlation is symmetric.

• 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。

Pearson's r is 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。

Please find the Quiz here

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
	0.99

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.

5 The scatter plot below represents

	a positive association
	a negative association
	no association

Answer >>

a positive association

As X increases, Y tends to increase, so it is a positive association.

6 The scatter plot below represents

	a positive association
	a negative association
	no association

Answer >>

a negative association

As X increases, Y tends to decrease, so it is a negative association.