Computation. Consider a normal sample of size $n = 10:$ small enough for easy computation by hand, but not necessarily large enough to make a useful normal probability plot. (Computations below are in R statistical software.)
set.seed(1028); x = round(rnorm(10, 100, 15), 1)
samp.q = sort(x); samp.q
[1] 75.9 78.7 83.8 101.9 105.0 107.8 116.3 123.7 128.8 140.5
One style of normal probability plot puts 'Theoretical Quantiles' on the horizontal axis and 'Sample Quantiles' on the vertical axis.
As discussed in comments, the theoretical quantiles are the normal
quantiles of $(i - .5)/n:$
i = 1:10; theor.q = qnorm((i-.5)/10); theor.q
[1] -1.6448536 -1.0364334 -0.6744898 -0.3853205 -0.1256613
[6] 0.1256613 0.3853205 0.6744898 1.0364334 1.6448536
The the normal probability plot is:
plot(theor.q, samp.q), pch=19)
abline(v = theor.q, col="green2")
The idea is that when normal data are plotted in this way, points will fall
'nearly' in a straight line. One method of illustrating this intended linearity
is to plot the line $y = \bar X + Sx,$ based on the sample mean and standard deviation,
through the plotted points. [Generally, we will not know the population mean and variance, so it would not be possible to plot a line based on population parameters. For such a small sample, these two lines may be quite different, so (even if population parameters were known)
the line based on sample mean and standard deviation is often a more useful reference line for judging normality.]
abline(a = mean(x), b = sd(x), col="red")
abline(a = 100, b = 15, col="blue", lty="dotted")
Here is the default normal probability plot (also called 'normal quantile-quantile plot') in R; the default reference
line passes through the first and third quantiles.
qqnorm(x); qqline(x)
Intuition. A normal probability plot with Sanple Quantiles on the horizontal
axis may be compared to the plot of the Empirical CDF of the sample. (The ECDF
jumps up by $1/n$ at each observed data value.) With a large enough sample the
ECDF of a sample approximates the CDF of the distribution (red curve).
The left-hand panel below shows an ECDF of a normal sample of size $n = 15.$ Notice that the values
plotted on the vertical axis are $1/n, 2/n, \cdots, n/n.$ The CDF of the population from which the sample was drawn is shown as a red curve.
The right-hand panel shows the normal probability plot of same sample. The
values on the vertical axis are theoretical quantiles corresponding to
$(i-.5)/n,\, i = 1, 2, \dots, n.$ The red reference line is $y = -\frac{\mu}{\sigma} + \frac{x}{\sigma}.$ Roughly speaking, this line may be considered as a "quantile transformation to linearity"
of the normal CDF.
Also for reference, grey points correspond to normal probability plots of 20 additional
samples from the same normal distribution. Although the points of the normal probability
plot of the sample (blue points) do not lie exactly on a straight line,
they lie well within the 'cloud' of the normal probability plots of the 20
other normal samples.
Note: R code for the last figure:
set.seed(2018); n = 15; mu = 100; sg = 15
x = rnorm(n, mu, sg)
par(mfrow = c(1,2))
plot(ecdf(x), col="blue")
curve(pnorm(x, 100, 15), add=T, col="red", lwd=2)
qqnorm(x, pch=19, col="blue", datax=T)
for (j in 1:20) {
i = 1:n; tq = qnorm((i-.5)/n); sq = sort(rnorm(n,mu,sg))
points(sq, tq, col="grey") }
abline(a = -mean(x)/sd(x), b = 1/sd(x), col="red")
par(mfrow = c(1,1))
