Computing r-squared From r

Squaring a Correlation

In “Defining the Coefficient of Determination \(r^2\),” you learned that in simple linear regression the coefficient of determination is the square of the correlation. This tutorial focuses on the calculation: take a reported \(r\), multiply it by itself, and compare the result with the original correlation.

The sign of \(r\) can make the arithmetic easy to mishandle. A negative correlation squared gives a positive coefficient of determination, because multiplying two negative numbers gives a positive product. Use parentheses around the entire correlation when writing the calculation.

For example, if \(r=-0.8\), then \(r^2=(-0.8)(-0.8)=0.64\). The result is positive. If \(r=0.5\), then \(r^2=(0.5)(0.5)=0.25\). These calculations use the same operation, even though the first correlation is negative and the second is positive.

A useful check is to separate the sign from the magnitude. Squaring makes the result positive (unless \(r=0\)), and the result equals the square of the correlation’s absolute value: \(r^2=|r|^2\). Thus, for \(r=-0.8\), you can check the calculation by squaring \(0.8\): \(0.8\times0.8=0.64\). This check catches the common mistake of treating a negative correlation’s square as negative.

Because a correlation is between \(-1\) and \(1\), its square is between 0 and 1. A correlation of \(0\) squares to \(0\); a correlation of \(1\) or \(-1\) squares to \(1\). Values between those endpoints produce an \(r^2\) between 0 and 1.

Compare the Square With the Original Correlation

The correlation \(r\) and its square \(r^2\) are related, but they do not report the same feature. The sign of \(r\) tells the direction of the linear association: positive or negative. Squaring removes that sign. The coefficient of determination is nonnegative, so it cannot tell you whether the association is positive or negative.

To compare the strength represented by two correlations, compare their absolute values. For instance, \(|-0.8|=0.8\), which is greater than \(|0.5|=0.5\). Squaring these magnitudes gives \(0.64\) and \(0.25\), respectively. The larger absolute correlation produces the larger \(r^2\). In contrast, comparing the signed values \(-0.8\) and \(0.5\) as ordinary numbers would not tell you which association is stronger.

Correlations that have opposite signs but the same magnitude have the same coefficient of determination. For example, \(r=0.6\) and \(r=-0.6\) both give \(r^2=0.36\). Their linear associations have opposite directions, while their coefficients of determination are equal.

Keep the decimal calculation separate from any later contextual interpretation. First square \(r\) accurately. Then, if a question asks what the result means, describe the fraction of variation in the response accounted for by its linear regression on the explanatory variable, as in the earlier tutorial. Do not turn the calculation itself into a claim about direction, individual prediction accuracy, or causation.

Use Magnitude to Check Decimal Squares

For correlations written as decimals, multiplication is usually the most direct method. A second check can use a fraction: \(0.36=36/100\), so squaring it means squaring both the numerator and denominator. This method also helps track the decimal place. For a negative correlation, use its magnitude in this check, since squaring removes the sign.

For example, \(0.36^2=(36/100)^2=1296/10000=0.1296\). This is not \(0.1296\%\): it is the decimal coefficient of determination. If expressed as a percentage, it is \(12.96\%\). Keep the decimal form when a question asks only for \(r^2\), and convert to a percentage only when useful or requested.

Equal Squares and Boundary Values

Some comparisons are quickest when you first notice equal magnitudes. If \(r_1\) and \(r_2\) have the same absolute value, their squares must be equal, even if one correlation is negative and the other positive. For example, \(|-0.7|=|0.7|=0.7\), so both correlations have \(r^2=0.49\). Check by multiplication: \((-0.7)(-0.7)=0.49\) and \((0.7)(0.7)=0.49\).

The boundary values provide another quick check. If \(r=0\), then \(r^2=0\times0=0\). If \(r=-1\), then \(r^2=(-1)(-1)=1\); if \(r=1\), then \(r^2=(1)(1)=1\). These endpoint checks reinforce that \(r^2\) is never negative and never greater than 1 for a valid correlation.

Common Mistakes and AP Exam Tips

• Keeping a negative sign after squaring. A negative correlation is multiplied by itself, so \((-0.8)(-0.8)=0.64\), not \(-0.64\). Parentheses make the operation clear.
• Squaring only the digits and attaching the old sign. The whole number \(r\), including its sign, is squared. Equivalently, square \(|r|\) and report a nonnegative result.
• Comparing signed \(r\) values to compare strength. Compare absolute values instead. For example, \(-0.8\) is less than \(0.5\) on a number line, but its magnitude is greater than \(0.5\), so its squared value is larger.
• Assuming \(r^2\) tells direction. It cannot distinguish a positive correlation from a negative correlation of the same magnitude. State direction from the sign of the original \(r\).
• Moving the decimal incorrectly. Multiply the decimals or use a fraction check, and make sure the result is between 0 and 1. Do not attach a percent sign unless you have converted the decimal to a percentage.
• Calling the coefficient a correlation. \(r\) is the signed correlation; \(r^2\) is the nonnegative coefficient of determination. They are related, but they are not interchangeable.

For full-credit calculation work, show the substituted correlation with parentheses when needed, multiply it by itself, and report the nonnegative result. For a comparison, state which \(r^2\) is larger or that they are equal, then use the original correlations to describe direction. If asked to explain what a coefficient of determination means in context, identify variation in the response and use the interpretation pattern from the earlier tutorial; do not say it is a percentage of observations predicted correctly.