Checking for a normal distribution

Theoretical distribution already gives $\Phi$ and the normal model used for confidence intervals. Many procedures that follow assume the random variable $X$ is normal. How do you check that assumption on a finished measurement series?

Draw the probability-paper line first

In the $(z,y)$-plane, mark the pairs $(z,\Phi (z))$ from the sample table below and join them by a straight line. The $y$-axis is deliberately irregular: its tick marks sit at the $\Phi$ values, spaced so those pairs fall on a line. That sheet is probability paper.

Drag the deviation $d$. Solid points mark the table on that warped scale. Blue points stand for a sample of empirical points. When $d$ is near $0$, the blue points sit on that line.

Standardize the measurements

From the series $x_1,\dots ,x_n$ form the empirical mean and empirical standard deviation $\bar{x}$ and $\Delta x$. The standardized values are

$$z_j:=\frac{x_j-\bar{x}}{\Delta x},j=1,\dots ,n.$$

Drag a raw height $x$, a mean $\bar{x}$, and a spread $\Delta x$. The green segment is the height of $x$. The blue segment is that same measurement after the shift-and-scale map $z=(x-\bar{x})/\Delta x$.

Plot the empirical ${\Phi }_{\ast }$ and read the line

For each standardized height, count how many of the $z_k$ are strictly smaller than $z_j$, and set

$${\Phi }_{\ast }(z_j)=\frac{1}{n}(\text{number of }{z}_k<z_j).$$

Mark the points $(z_j,{\Phi }_{\ast }(z_j))$ on the same probability paper. If those points stay very close to the straight line from the table, call $X$ approximately normally distributed. On the board above, that series is the blue cloud when $d$ is near $0$.

The same diagram is what probability paper is built to produce: the irregular $y$-scale turns the theoretical $\Phi$ into a line, so the eye only has to ask whether the empirical cloud hugs it.

A ${\chi }^2$ goodness-of-fit test for normality is stricter than this graphical check, and it matters more in practice.