Recall that the third condition — the "N" condition — of the linear regression model is that the error terms are normally distributed. In this section, we learn how to use a "normal probability plot of the residuals" as a way of learning whether it is reasonable to assume that the error terms are normally distributed.

Here's the basic idea behind any normal probability plot: if the data follow a normal distribution with mean \(\mu\) and variance \(σ^{2}\), then a plot of the theoretical percentiles of the normal distribution versus the observed sample percentiles should be approximately linear. Since we are concerned about the normality of the error terms, we create a normal probability plot of the residuals. If the resulting plot is approximately linear, we proceed, assuming that the error terms are normally distributed.

The theoretical p^th percentile of any normal distribution is the value such that p% of the measurements fall below the value. Here's a screencast illustrating a theoretical p^th percentile.

The problem is that to determine the percentile value of a normal distribution, you need to know the mean \(\mu\) and the variance \(\sigma^2\). And, of course, the parameters \(\mu\) and \(σ^{2}\) are typically unknown. Statistical theory says its okay just to assume that \(\mu = 0\) and \(\sigma^2 = 1\). Once you do that, determining the percentiles of the standard normal curve is straightforward. The p^th percentile value reduces to just a "Z-score" (or "normal score"). Here's a screencast illustrating how the p-th percentile value reduces to just a normal score.

The sample p^th percentile of any data set is, roughly speaking, the value such that p% of the measurements fall below the value. For example, the median, which is just a special name for the 50th percentile, is the value so that 50%, or half, of your measurements, falls below the value. Now, if you are asked to determine the 27th percentile, you take your ordered data set, and you determine the value so that 27% of the data points in your dataset fall below the value. And so on.

Consider a simple linear regression model fit a simulated dataset with 9 observations so that we're considering the 10th, 20th, ..., and 90th percentiles. A normal probability plot of the residuals is a scatter plot with the theoretical percentiles of the normal distribution on the x-axis and the sample percentiles of the residuals on the y-axis, for example:

The diagonal line (which passes through the lower and upper quartiles of the theoretical distribution) provides a visual aid to help assess whether the relationship between the theoretical and sample percentiles is linear.

Note that the relationship between the theoretical percentiles and the sample percentiles is approximately linear. Therefore, the normal probability plot of the residuals suggests that the error terms are indeed normally distributed.

Statistical software sometimes provides normality tests to complement the visual assessment available in a normal probability plot (we'll revisit normality tests in Lesson 7). Different software packages sometimes switch the axes for this plot, but its interpretation remains the same.

Let's take a look at examples of the different kinds of normal probability plots we can obtain and learn what each tells us.