剩余误差

What are residuals?

A residual is the difference between the actual value of a data point and the predicted value suggested by a regression line or trend line.

Trend lines are approximations of the shape of the data and show the overall pattern the data takes. Lines of best fit are the most accurate trend lines.

Notice, though, that not all data points fall exactly on the best fit line. That's where we get residuals from.

Predicted Value vs. Actual Value

Residuals are calculated as the difference between a point's actual (measured) value and its predicted value.

This concept may seem familiar to you if you're a science nerd! If you know about percent error, then it's a very similar concept. If you haven't learned percent error yet, that's okay! You'll be familiar with it because you'll have learned about residual values.

Let's practice identifying predicted and actual values. We'll use the scatter plot and trend line below:

Calculating Residual Values

This brings us to an important rule:

Examples

So, now that we've got that in mind, let's try a few more examples. I'm going to put the original scatterplot picture below again, just so you don't have to keep scrolling up and down.

In the real world, you'll use this concept when you're looking at leasing cars and evaluate the value of different things.

Residual Values

A residual is a measurement to determine how well a scatter plot's data fits its trend line.

To find a point's residual, we subtract the predicted $y$-value from the actual $y$-value. $$\text{residual}=\text{actual }y\text{-value}-\text{predicted }y\text{-value}$$

The actual value, is represented by the dot on the scatter plot. The predicted value is given by the trend line or line of best fit.

Here's an example of a scatter plot with a trend line of $y=\frac12x+1$.

Let's look at an example. Suppose we want to find the residual of the purple point.

First, we look at the actual value. This is the $y$-coordinate of the data point itself. $$\text{actual}=1$$

Next, we find the predicted value. This is the $y$-value that the trend line guesses our a data point will appear at. Starting at our point (purple), move up or down until you reach the trend line. Look at the $y$-value at this point on the trend line.

As you can see, the predicted value of the purple point is: $$\text{predicted}=1.5$$

Finally, with these two values, we can calculate the residual. $$\begin{align} \text{residual}&=\text{actual}-\text{predicted}\\ \text{residual}&=1-1.5\\ \text{residual}&=-0.5 \end{align}$$

$$\begin{align} \text{residual}&=\text{actual}-\text{predicted}\\ \text{residual}&=4-3\\ \text{residual}&=1 \end{align}$$

In general, residuals will be positive when the data points are above the trend line. They will be negative when the data points are below the trend line.

(Video) Residual Values

by Anusha Rahman

This lesson by Anusha Rahman goes over how to find a residual value when given a point on a scatter plot and a line of best fit.

Summary

A residual value is the difference between an actual data point and the predicted value. The following equation determines the residual value:

Residual value = (actual value) - (predicted value)

In the video, we see a coordinate plane with random points and a line of best fit. If we want to find the residual value, we have to subtract the predicted value (a point on the line) from the actual value (a data point). We use $y$-values when solving for the residual value.

We are going to look at the orange point as the actual point.

The points:

Actual value: $(4,5)$

Predicted value: $(4, 2.4)$

When we plug it into the equation,

$$Residual value = 5-2.5$$ $$=2.5$$

Therefore, the residual is $2.5$.

High residual values mean the actual value is further away from the predicted value.