Coefficient of Determination (r²)

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Coefficient of Determination (r²): meaning, method and solved example (AP Statistics) – Maths PDF

In short: Coefficient of determination, written r squared, reports what fraction of the ups and downs in the response variable can be traced to its straight-line link with the explanatory variable.

AP StatisticsStatisticsAP Statistics Unit 5

What is Coefficient of Determination?

The coefficient of determination, \(r^2\), tells you how much of the scatter in \(y\) the regression line accounts for. It is the correlation \(r\) multiplied by itself, so it runs from 0 to 1 and carries no sign.

An equivalent recipe is \(r^2 = 1 – \frac{SSE}{SST}\). SST measures the total variation of \(y\) around its mean, and SSE measures what remains around the line. With \(r^2 = 0.67\), roughly two thirds of the variation in \(y\) tracks the linear pattern with \(x\), while the other third stays unexplained.

How to spot it

  • Computer output shows “R-sq” as a percent beside a regression.
  • A question gives r and asks how much of the variation in y is accounted for by x.
  • You are asked to compare how well two models fit the same data.
  • The wording asks you to interpret the coefficient of determination in context.

The method

  1. To find the coefficient of determination, multiply the correlation coefficient by itself; or compute \(1 – \frac{SSE}{SST}\).
  2. Convert the result to a percent.
  3. Write a sentence naming both variables: the share of variation in the response that the line with the explanatory variable accounts for.
  4. Remember that a high value does not prove cause, and that it does not show a line is the right shape of model.
Total variation in free-throw percentage67.24%explained by training32.76%unexplainedr = 0.82, so r^2 = 0.82 x 0.82 = 0.6724
Blue part: the share of the variation tied to the line; amber part: scatter around it.

Coefficient of Determination: solved example

Problem

A coach finds a correlation \(r = 0.82\) between weekly shooting-drill hours and free-throw percentage for 30 players. What does \(r^2\) say? Then compute \(r^2\) for another model whose \(SST = 496\) and \(SSE = 6\).

Solution

First model: \(r^2 = 0.82^2 = 0.6724\). So about 67% of the differences in free-throw percentage line up with drill hours through the straight-line pattern, and the remaining 33% comes from other factors.

Second model: \(r^2 = 1 – \frac{6}{496} \approx 0.9879\), meaning about 98.8% of the variation is accounted for, an excellent linear fit.

Try it

Question 1

If \(r = -0.9\), find \(r^2\) and say what it means.

Show the answer

\(r^2 = 0.81\): 81% of the variation in \(y\) goes with the linear pattern in \(x\). The negative sign disappears when you square.

Question 2

A regression prints R-sq = 36% and has a positive slope. What is \(r\)?

Show the answer

\(r = +\sqrt{0.36} = 0.6\), positive because the slope is positive.

Coefficient of Determination vs. Correlation Coefficient

The correlation \(r\) shows direction and strength, from \(-1\) to 1. The coefficient of determination squares it, and squaring drops the direction and gives the share accounted for. An \(r\) of 0.5 sounds moderate, yet it covers only 25% of the variation.

Correlation Coefficient

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