6.7 Finding Residuals

Finding Residuals

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Find a Predicted Value and a Residual

A line fitted to paired data turns an x-value into a prediction. Your statistics text writes that line as $\hat{y} = a + b x$ , where $\hat{y}$ (read “y-hat”) is the predicted value, $a$ is the y-intercept, and $b$ is the slope. This is the same kind of line as $y = m x + b$ from Sections 6.5 and 6.6, written with different letters. Watch the letter $b$ : in $y = m x + b$ it is the y-intercept, but in the statistics form $\hat{y} = a + b x$ it is the slope. Real data points rarely sit exactly on the line. The gap between what was observed and what the line predicted is called a residual, and it tells you how far off the line was for that data point.

The sign of a residual tells you where the data point sits compared with the line.

You can also work backwards. The residual is the observed value minus the predicted value, so the observed value is the predicted value plus the residual.

$y = \hat{y} + residual$

Find Residuals for a Table of Data

Every data point has its own residual. Organizing the work in a table, one row per point, keeps the predicted values and the residuals lined up with the data.

Example 4

The table shows six students' practice problems and quiz scores. Use the line $\hat{y} = 12.5 + 0.8 x$ to find each student's residual.

Practice problems completed and quiz scores for six students
Problems completed, $x$	Observed score, $y$
$10$	$17$
$20$	$30$
$25$	$35$
$30$	$39$
$40$	$44$
$45$	$46$

Show solution

Find each predicted score, then subtract it from the observed score. For the first student, $\hat{y} = 12.5 + 0.8 (10) = 20.5$ and $residual = 17 - 20.5 = −3.5$ .

Predicted scores and residuals for the six students
Problems completed, $x$	Observed score, $y$	Predicted score, $\hat{y}$	Residual, $y - \hat{y}$
$10$	$17$	$20.5$	$−3.5$
$20$	$30$	$28.5$	$1.5$
$25$	$35$	$32.5$	$2.5$
$30$	$39$	$36.5$	$2.5$
$40$	$44$	$44.5$	$−0.5$
$45$	$46$	$48.5$	$−2.5$

Three students scored above the line's prediction (positive residuals) and three scored below it (negative residuals). This line happens to be the least-squares line for these six students, and the residuals of a least-squares line with an intercept always add to 0 (or very nearly 0 once the line's numbers have been rounded). Here, $−3.5 + 1.5 + 2.5 + 2.5 + (−0.5) + (−2.5) = 0$ .

Practice Makes Perfect

Find a Predicted Value and a Residual

In the following exercises, use the line $\hat{y} = 12.5 + 0.8 x$ to find the predicted value and the residual for the data point.

$open parenthesis 30, 40 close parenthesis$

Show solution

predicted value $36.5$ , residual $3.5$

$open parenthesis 25, 30 close parenthesis$

Show solution

predicted value $32.5$ , residual $−2.5$

$open parenthesis 40, 44.5 close parenthesis$

Show solution

predicted value $44.5$ , residual $0$

$open parenthesis 15, 26 close parenthesis$

Show solution

predicted value $24.5$ , residual $1.5$

In the following exercises, use the given line to find the predicted value and the residual for the data point.

$\hat{y} = 95 - 3.2 x$ ; data point $open parenthesis 10, 60 close parenthesis$

Show solution

predicted value $63$ , residual $−3$

$\hat{y} = −4.6 + 1.25 x$ ; data point $open parenthesis 12, 11.4 close parenthesis$

Show solution

predicted value $10.4$ , residual $1$

$\hat{y} = 2.1 + 0.45 x$ ; data point $open parenthesis 8, 5.9 close parenthesis$

Show solution

predicted value $5.7$ , residual $0.2$

$\hat{y} = 150 - 12 x$ ; data point $open parenthesis 6, 70 close parenthesis$

Show solution

predicted value $78$ , residual $−8$

In the following exercises, a data point's residual is given. Is the point above or below the line? Did the line predict too much or too little?

residual $−6.2$

Show solution

Below the line. The line predicted 6.2 too much.

residual $3.8$

Show solution

Above the line. The line predicted 3.8 too little.

Find Residuals for a Table of Data

In the following exercises, find the requested values.

Use the line $\hat{y} = 20 + 1.5 x$ to find the residual for each data point: $open parenthesis 4, 27 close parenthesis$ , $open parenthesis 10, 33 close parenthesis$ , $open parenthesis 16, 45 close parenthesis$ .

Show solution

$1$ , $−2$ , $1$

A line predicts $y hat equals 48.5$ for a data point, and the point's residual is $−3.5$ . What was the observed value $y$ ?

Show solution

observed value $45$

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