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Textbook by: Vicky Moyle. E-mail the author

# Correlation Coefficient and Coefficient of Determination

Module by: Roberta Bloom. E-mail the author

Summary: This module provides an overview of Linear Regression and Correlation: The Correlation Coefficient and Coefficient of Determination. It is part of the Roberta Bloom's Custom Collection of Collaborative Statistics (col10617). It is based on module m17092 as a part of Collaborative Statistics collection (col10522) by Barbara Illowsky and Susan Dean. This revised has been expanded from the original to include coverage of the coefficient of determination and to include a discussion of properties of the correlation coefficient previously included in Illowsky and Dean's module. Some material previously in module m17077 Facts about the Correlation Coefficient have been moved to this module in Bloom's custom edition of Collaborative Statistics.

## The Correlation Coefficient r

Besides looking at the scatter plot and seeing that a line seems reasonable, how can you tell if the line is a good predictor? Use the correlation coefficient as another indicator (besides the scatterplot) of the strength of the relationship between xx and yy.

The correlation coefficient, r, developed by Karl Pearson in the early 1900s, is a numerical measure of the strength of association between the independent variable x and the dependent variable y.

The correlation coefficient is calculated as r = n Σ x y - ( Σ x ) ( Σ y ) [ n Σ x 2 - ( Σ x ) 2 ] [ n Σ y 2 - ( Σ y ) 2 ] r= n Σ x y - ( Σ x ) ( Σ y ) [ n Σ x 2 - ( Σ x ) 2 ] [ n Σ y 2 - ( Σ y ) 2 ]

where nn = the number of data points.

If you suspect a linear relationship between xx and yy, then rr can measure how strong the linear relationship is.

### What the VALUE of r tells us:

• The value of rr is always between -1 and +1: -1r1-1r1.
• The closer the correlation coefficient rr is to -1 or 1 (and the further from 0), the stronger the evidence of a significant linear relationship between xx and yy; this would indicate that the observed data points fit more closely to the best fit line. Values of rr further from 0 indicate a stronger linear relationship between xx and yy. Values of rr closer to 0 indicate a weaker linear relationship between xx and yy.
• If r=0r=0 there is absolutely no linear relationship between xx and yy (no linear correlation).
• If r=1r=1, there is perfect positive correlation. If r=-1r=-1, there is perfect negative correlation. In both these cases, all of the original data points lie on a straight line. Of course, in the real world, this will not generally happen.

### What the SIGN of r tells us

• A positive value of rr means that when xx increases, yy increases and when xx decreases, yy decreases (positive correlation).
• A negative value of rr means that when xx increases, yy decreases and when xx decreases, yy increases (negative correlation).
• The sign of rr is the same as the sign of the slope, bb, of the best fit line.

### Note:

Strong correlation does not suggest that xx causes yy or yy causes xx. We say "correlation does not imply causation." For example, every person who learned math in the 17th century is dead. However, learning math does not necessarily cause death!

The formula for rr looks formidable. However, computer spreadsheets, statistical software, and many calculators can quickly calculate rr. The correlation coefficient rr is the bottom item in the output screens for the LinRegTTest on the TI-83, TI-83+, or TI-84+ calculator (see previous section for instructions).

## The Coefficient of Determination

r 2 r 2 is called the coefficient of determination. r 2 r 2 is the square of the correlation coefficient , but is usually stated as a percent, rather than in decimal form. r 2 r 2 has an interpretation in the context of the data

• r 2 r 2 , when expressed as a percent, represents the percent of variation in the dependent variable y that can be explained by variation in the independent variable x using the regression (best fit) line.
• 1- r 2 r 2 , when expressed as a percent, represents the percent of variation in y that is NOT explained by variation in x using the regression line. This can be seen as the scattering of the observed data points about the regression line.

### Consider the third exam/final exam example introduced in the previous section

• The line of best fit is: y ^ = -173.51 + 4.83x y ^ =-173.51+4.83x
• The correlation coefficient is r = 0.6631 r=0.6631
• The coefficient of determination is r 2 r 2 = 0.6631 2 0.6631 2 = 0.4397
• Interpretation of r 2 r 2 in the context of this example:
• Approximately 44% of the variation in the final exam grades can be explained by the variation in the grades on the third exam, using the best fit regression line.
• Therefore approximately 56% of the variation in the final exam grades can NOT be explained by the variation in the grades on the third exam, using the best fit regression line. (This is seen as the scattering of the points about the line.)

In the next section, we will learn more about the correlation coefficient and will examine rr in the context of the example about grades on the third exam and final exam.

## Glossary

Coefficient of Correlation:
A measure developed by Karl Pearson (early 1900s) that gives the strength of association between the independent variable and the dependent variable. The formula is:
r = n xy ( x ) ( y ) [ n x 2 ( x ) 2 ] [ n y 2 ( y ) 2 ] , r = n xy ( x ) ( y ) [ n x 2 ( x ) 2 ] [ n y 2 ( y ) 2 ] , size 12{r= { {n Sum { ital "xy"} - $$Sum {x$$ $$Sum {y$$ } } } over { sqrt { $n Sum {x rSup { size 8{2} } - $$Sum {x$$ rSup { size 8{2} }$ $n Sum {y rSup { size 8{2} } - $$Sum {y$$ rSup { size 8{2} }$ } } } } } } } ,} {}
(1)
where n is the number of data points. The coefficient cannot be more then 1 and less then -1. The closer the coefficient is to ±1±1 size 12{ +- 1} {}, the stronger the evidence of a significant linear relationship between xx size 12{x} {} and yy size 12{y} {}.

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