Standardized Version of the Regression Line

Predicting Relative Standing With a Regression Line

In “Properties of the Least-Squares Line,” you learned that the line passes through \((\bar{x},\bar{y})\), and in “Linking the Slope to Correlation” you saw that its slope is \(b=r(s_y/s_x)\). These facts lead to a useful way to express a regression prediction: instead of predicting the response in its original units, predict how many standard deviations above or below its mean the response is predicted to be.

To do this, standardize both variables. A z-score describes a value’s distance from its variable’s mean in standard-deviation units. The standardized regression equation is especially simple: the predicted response z-score is the correlation multiplied by the predictor z-score. This lets you predict relative standing directly, without first calculating the line’s intercept and slope in the original units.

Why the Standardized Equation Works

The equation follows from two properties of the least-squares line already established in this course. First, the line passes through \((\bar{x},\bar{y})\). Second, its slope is \(b=r(s_y/s_x)\). Therefore, the predicted response can be written as the mean response plus the slope times the predictor’s distance from its mean:

To express that prediction in response standard-deviation units, subtract \(\bar{y}\) and divide by \(s_y\). Then use \(b=r(s_y/s_x)\) and \(z_x=(x-\bar{x})/s_x\):

This is the standardized version of the regression line. Its intercept is \(0\), because a predictor at its mean has \(z_x=0\) and the line predicts a response at its mean, \(\hat{z}_y=0\). Its slope is \(r\). The correlation is unitless, and standardized values are measured in standard-deviation units, so the equation does not involve the original measurement units.

For example, if \(z_x=1\) and \(r=0.70\), the line predicts \(\hat{z}_y=0.70\): the response is predicted to be \(0.70\) standard deviations above its mean. This is a statement about relative standing, not about the response’s original units. To obtain a prediction in those units, multiply \(\hat{z}_y\) by \(s_y\) and add \(\bar{y}\).

Positive and Negative Relative Standing

The sign of \(r\) determines how the line relates the predictor’s relative standing to the response’s predicted relative standing. When \(r\) is positive, predictor values above their mean have positive z-scores and produce positive predicted response z-scores. Predictor values below their mean produce negative predicted response z-scores. When \(r\) is negative, the signs go in opposite directions: a predictor value above its mean leads to a prediction below the response mean, and vice versa.

The size of the correlation matters too. Since correlations have absolute value at most \(1\), multiplying by \(r\) cannot make the predicted response z-score larger in absolute value than the predictor z-score. For instance, if \(r=-0.60\) and \(z_x=1.5\), then \(\hat{z}_y=-0.90\). The line predicts a response below its mean, and the predicted distance is \(0.90\) response standard deviations. The multiplication accounts for both direction and size.

This rule concerns the predicted response. An actual response can lie above or below its prediction; as covered in “Prediction Versus Observed Values,” the difference between an observed response and its prediction is the residual. Thus, a predicted z-score is not a guarantee about the actual z-score for a particular case.

Converting a Predicted Z-Score Back to Original Units

The standardized equation is useful when a question asks for relative standing. If the question instead asks for a predicted response in its original units, convert the predicted z-score back. The definition \(\hat{z}_y=(\hat{y}-\bar{y})/s_y\) rearranges to:

You can also find the predictor z-score from its original value using \(z_x=(x-\bar{x})/s_x\). This gives a complete path from an original predictor value to a predicted response: standardize \(x\), multiply by \(r\), and convert the predicted response z-score back using \(\bar{y}\) and \(s_y\). Keep track of which mean and standard deviation belong to which variable.

Common Mistakes and AP Exam Tips

• Writing \(z_y=r z_x\) as though it describes every observed pair. The regression equation is \(\hat{z}_y=r z_x\). The hat matters: it marks the response value predicted by the line. An observed response can differ from that prediction.
• Using the response standard deviation to standardize \(x\). Calculate \(z_x\) with \(\bar{x}\) and \(s_x\), and convert a predicted response using \(\bar{y}\) and \(s_y\). Each variable has its own mean and standard deviation.
• Dropping the sign. Do not use \(|r|\) in the equation. A negative correlation reverses the direction of predicted relative standing. Keep both signs when multiplying.
• Calling a z-score a percentage or a percentile. A z-score is a number of standard deviations from the mean; it is not itself a percent or a percentile rank. State the predicted distance above or below the mean in standard-deviation units.
• Confusing the correlation with a predicted response z-score. The predicted response z-score equals \(r\) only when \(z_x=1\). In general, calculate \(r z_x\).
• Stopping at the z-score when the question asks for original units. Convert with \(\hat{y}=\bar{y}+\hat{z}_y s_y\). Include the response’s units in the final prediction.
• Overstating what the line predicts. A prediction is not a promise that an individual’s actual response will equal \(\hat{y}\). A complete interpretation says what the line predicts, in context, and does not claim that every case lies on the line.

For an AP-style response, state whether the predicted response is above or below its mean, give the predicted distance in standard-deviation units, and identify the relevant context. For example: “For a student whose weekly app time is \(1.5\) standard deviations above the sample mean, the regression line predicts a comprehension score \(1.20\) standard deviations above the sample mean score.” If original units are requested, also report the converted prediction and its units.