# Coefficient of Determination The coefficient of determination is r-squared.    It is important to note that a correlation coefficient is not a proportion and cannot be interpreted as such.  A correlation of 0.50 cannot be interpreted as half or 50% of anything—correlations simply do not work like that.

The coefficient of determination can be interpreted as a proportion or a fraction.  If a correlation is computed to be 0.50, then r-squared will be equal to 0.5 x 0.5 = 0.25.  This result of 0.25 is interpreted as the proportion of variance explained or the amount of variance accounted for.  We can easily convert this to a percentage by multiplying r2 by 100.  Thus, with a correlation of r = .50, 25% of the variance in one variable is accounted for by knowing the value of the other variable.

`The coefficient of determination tells us what proportion of the variance in one variable that can be accounted for by the variance in another variable.`

Note that we can compute the percentage of variance not accounted for by subtracting the r2 percentage from 100%.  In the above example where r = .05, we can determine that 25% of the variance is accounted for and that 75% of the variance is not accounted for.

It is also important to note that correlation coefficients can be misleading when they are small.  This is because the percentage of variance explained shrinks faster than the correlation coefficient seems to indicate.

### Pearson’s r and Coefficients of Determination

Pearson’s r Pearson’s r-square % of variance

Accounted For

% of variance Not

Accounted For

1.00

.90

.80

.70

.60

.50

.40

.30

.20

.10

1.00

.81

.64

.49

.36

.25

.16

.09

.04

.01

100%

81%

64%

49%

36%

25%

16%

9%

4%

1%

0%

19%

36%

51%

64%

75%

84%

91%

96%

99%

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`Last Modified:  06/03/2021`

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