Finding r From r-squared and the Slope Sign

Recovering the Sign That \(r^2\) Leaves Out

In “Finding \(r\)-squared From Computer Output,” you learned to identify the coefficient of determination in a regression summary. That value describes how much variation in the response is accounted for by its linear relationship with the explanatory variable, but it does not tell you whether the relationship is positive or negative. If a question gives \(r^2\) and the sign of the regression slope, you can recover the correlation \(r\).

The calculation has two parts. Taking the square root of \(r^2\) gives the magnitude of \(r\), meaning its distance from zero. Then use the sign of the slope to decide whether \(r\) is positive or negative. In simple linear regression, the slope and correlation have the same sign, as established in “Why Regressing Y on X Differs From X on Y.”

For example, if \(r^2=0.36\), then the correlation must be either \(+0.60\) or \(-0.60\), because both values square to \(0.36\). A positive fitted slope selects \(r=+0.60\); a negative fitted slope selects \(r=-0.60\). The value of \(r^2\) by itself cannot distinguish between these two possibilities.

This is a special case of undoing a square: both a positive and a negative number can have the same square. The square root symbol \(\sqrt{r^2}\) gives the nonnegative magnitude, not the signed correlation automatically. Be sure to use the slope sign after taking the square root.

A Reliable Two-Part Method

First, identify the coefficient of determination—not adjusted \(R\)-squared or another output statistic. In simple linear regression, the value labeled \(R\)-sq or \(R\)-squared is \(r^2\). If it is a decimal, take its square root; if it is a percentage, convert it to a decimal first. Next, read the sign of the fitted slope \(b\). A positive slope means \(r>0\), and a negative slope means \(r<0\).

The slope’s numerical size is not used in this calculation. A slope of \(0.2\) and a slope of \(20\) are both positive, so either indicates a positive correlation in simple linear regression. The intercept is not used either. These coefficients describe the fitted equation, while the task here is to combine the magnitude from \(r^2\) with the direction from the slope.

Notice that the units of the slope do not become units of \(r\). The slope is measured in satisfaction points per minute, while \(r\) is unitless. The slope tells us which sign to use here; its size does not set the size of the correlation.

Allow for Rounding in Computer Output

Computer output often rounds \(r^2\), so the square root you calculate may be approximate. Keep several digits during the calculation and round the final correlation sensibly. If you square a rounded value of \(r\), it may not exactly reproduce the displayed \(r^2\). That small difference can be due to rounding rather than a mistake.

For instance, suppose output shows \(R\)-sq \(=0.73\). Its square root is approximately \(0.854400\). If the slope is positive, report \(r\approx 0.8544\), or \(r\approx 0.854\) to three decimal places. Squaring \(0.8544\) gives \(0.72999936\), which is about \(0.7300\), consistent with an \(r^2\) value displayed as \(0.73\). If you instead square the rounded three-decimal value \(0.854\), you get \(0.729316\), or about \(0.7293\). The difference is expected because the displayed \(r\) was rounded.

If the software gives \(r^2\) as a percentage, first write it as a decimal. For example, \(49\%\) is \(0.49\), and \(\sqrt{0.49}=0.70\). Taking the square root of \(49\) would be a scale error: the coefficient of determination must be between 0 and 1 for this calculation.

This example also shows why it is good practice to say “approximately” when recovering \(r\) from rounded output. The printed \(0.73\) may itself stand for a more precise value, so the square root of the displayed number need not equal the software’s original correlation exactly.

What the Calculation Can—and Cannot—Tell You

The pair \(r^2\) and slope sign is enough to recover \(r\) in simple linear regression. The coefficient of determination gives the correlation’s magnitude, and the slope gives its direction. Neither piece alone does both jobs: \(r^2\) cannot distinguish positive from negative correlation, and the slope sign alone cannot tell you how strong the correlation is.

The intercept does not resolve the sign. Nor does a slope’s size tell you the strength of the correlation. The relationship between slope and correlation also depends on the variables’ standard deviations, as covered in “Why Regressing Y on X Differs From X on Y.” For this task, do not try to infer \(r\) from the numerical slope alone.

For ordinary simple linear regression with varying \(x\) and \(y\), \(r^2\) is between 0 and 1. If \(r^2=0\), then \(r=0\); the slope is zero in this setting, so there is no positive or negative sign to assign. If \(r^2=1\), then the magnitude is 1, and the slope sign determines whether \(r=+1\) or \(r=-1\). Most examples involve values strictly between 0 and 1.

Common Mistakes and AP Exam Tips

• Taking the square root and forgetting the sign. \(\sqrt{r^2}\) is nonnegative. A full-credit answer uses the slope sign to select \(+\sqrt{r^2}\) or \(-\sqrt{r^2}\).
• Assuming \(r^2\) shows direction. Squaring removes the sign. Explain that the slope, not \(r^2\), identifies whether the correlation is positive or negative.
• Using the intercept or slope magnitude to choose \(r\). Only the slope’s sign is needed for direction. The intercept and numerical size of the slope do not supply the correlation’s magnitude.
• Taking the square root of a percentage without converting it. Change \(49\%\) to \(0.49\) before calculating \(\sqrt{0.49}\).
• Calling a rounded check inconsistent too quickly. Square the value at the precision you actually used and allow for rounding in the original output and your answer. Keep extra digits until the final report.
• Reporting an unsigned answer when direction matters. A response of \(0.85\) is incomplete if the slope is negative. Report \(r\approx -0.85\) in that situation.

A concise AP-style response can show the key reasoning in one line: “\(\sqrt{0.64}=0.80\), and because the fitted slope is negative, \(r=-0.80\).” When output is rounded, use an approximation symbol or words such as “approximately.” If asked for an interpretation as well, describe the direction and strength of the linear relationship in context; do not interpret \(r\) as a percentage of variation.