### May

#### 8

# R Squared - What is it? From Phillip McDonnell

May 8, 2007 |

A seeker of knowledge inquires:

Q: Can you give me a good layman's term "working" definition of R squared? Questions that pop up are why is it important? What does it reflect? Is it predictive?

A: R Squared is simply the square of the correlation coefficient. As most will recall the correlation coefficient can range from +1 for a strong positive correlation to -1 for a strongly negative relationship between two variables. Two variables which are unrelated will usually have a correlation around zero.

So when we square the correlation coefficient we get a number between 0 and +1. Remember that the negative correlations become positive so there are no more negative numbers. We also almost always get a smaller absolute number because multiplying two numbers less than 1 always gives a small number (except for zero and 1).

There is another interpretation. The R^2 also generally associated with a regression model (but need not be). The R^2 can be thought of as representing the percent of variance which is explained by the model. Mathematically things are linear in the variance but not in the square roots such as the standard deviation and correlation. So we can decompose the total variance of a regression into the part that is explained by the model and the part that is unexplained (the error or residual variance). The three relationships are:

Total variance = Explained Variance + Unexplained variance

100% = R^2 + (1 - R^2)

Total SSq = Explained SSq + Unexplained SSq

In the last line the term SSq means the Sum of Squares. The sum of squares relationship simply comes about because the variance is simply the average sum of squares. The bottom line is that if you have an R^2 of 25% you know that it explains 25% of the variance in the variable you wish to predict. You also know that the correlation coefficient is +/-50% because .5 * .5 = .25. Conversely if you know that a correlation coefficient is 90% then you know the R Squared will be 81%.

From just the R Squared you do not know if the correlation is positive or negative however. For that you have to look at the beta coefficient of the regression which tells you which sign to choose for the correlation coefficient.

## Yishen Kuik adds:

I've found an intuitive interpretation of correlation coefficient (R) to be a measure of how in phase two datastreams are.

For two dataseries X and Y:

R = (sum of Zx * Zy)/(N-1), where Zx is the z-normalized series X and Zy is same for Y

Hence, the more the below-mean datapoints in X and Y coincide, the greater the value in the numerator, since the product of two negative Z scores is positive. The corollary is that above-mean datapoints will also coincide, and since the sumproduct of two positive numbers is also positive, also contributes to a larger numerator.

Hence I think of this coincidence of above/below mean datapoints as in phase.

# Comments

## Archives

- June 2016
- May 2016
- April 2016
- March 2016
- February 2016
- January 2016
- December 2015
- November 2015
- October 2015
- September 2015
- August 2015
- July 2015
- June 2015
- May 2015
- April 2015
- March 2015
- February 2015
- January 2015
- December 2014
- November 2014
- October 2014
- September 2014
- August 2014 2064
- July 2014
- June 2014
- May 2014
- April 2014
- March 2014
- February 2014
- January 2014
- December 2013
- November 2013
- October 2013
- September 2013
- August 2013
- July 2013
- June 2013
- May 2013
- April 2013
- March 2013
- February 2013
- January 2013
- December 2012
- November 2012
- October 2012
- September 2012
- August 2012
- July 2012
- June 2012
- May 2012
- April 2012
- March 2012
- February 2012
- January 2012
- December 2011
- November 2011
- October 2011
- September 2011
- August 2011
- July 2011
- June 2011
- May 2011
- April 2011
- March 2011
- February 2011
- January 2011
- December 2010
- November 2010
- October 2010
- September 2010
- August 2010
- July 2010
- June 2010
- May 2010
- April 2010
- March 2010
- February 2010
- January 2010
- December 2009
- November 2009
- October 2009
- September 2009
- August 2009
- July 2009
- June 2009
- May 2009
- April 2009
- March 2009
- February 2009
- January 2009
- December 2008
- November 2008
- October 2008
- September 2008
- August 2008
- July 2008
- June 2008
- May 2008
- April 2008 114c
- March 2008
- February 2008
- January 2008
- December 2007
- November 2007
- October 2007
- September 2007
- August 2007
- July 2007
- June 2007
- May 2007
- April 2007
- March 2007
- February 2007
- January 2007
- December 2006
- November 2006
- October 2006
- September 2006
- August 2006
- Older Archives

## Resources & Links

- The Letters Prize
- Pre-2007 Victor Niederhoffer Posts
- Vic’s NYC Junto
- Reading List
- Programming in 60 Seconds
- The Objectivist Center
- Foundation for Economic Education
- Tigerchess
- Dick Sears' G.T. Index
- Pre-2007 Daily Speculations
- Laurel & Vics' Worldly Investor Articles