Constructing a Normal Probability Plot

Why Construct a Normal Probability Plot?

A normal probability plot places observed data values alongside the values expected at corresponding positions in a normal distribution. Constructing one involves a sequence of choices and calculations: sort the observations, assign each rank a probability, convert each probability to a standard normal quantile, and plot the resulting pairs. The plot is a way to compare data with a normal model; this tutorial focuses on how to build it, not on interpreting its overall shape.

A plot must use the same rank for both members of each pair. The smallest observation is paired with the lowest theoretical normal quantile, the next-smallest with the next-lowest, and so on. Mixing ranks, or plotting the data in their original unsorted order, does not produce the intended comparison.

Plotting Positions and Theoretical Quantiles

For the examples in this tutorial, use the midpoint plotting-position convention \(p_i=(i-0.5)/n\), for \(i=1,2,\ldots,n\). These positions stay strictly between 0 and 1, so their normal quantiles are finite. Other conventions are also used. For example, a calculation may assign \(p_i=i/(n+1)\); software can use different conventions as well. As a result, the exact coordinates can differ slightly across methods, especially in small samples. State or check the convention when reproducing a plot.

The inverse standard normal quantile \(\Phi^{-1}(p_i)\) can be obtained from a normal table or software. For instance, if \(p_i=0.10\), the corresponding standard normal quantile is approximately \(-1.282\): 10% of the standard normal distribution lies below that value. This is a quantile, not a data observation.

Why the Points Are Paired This Way

The pairing comes from the quantiles of a normal model. Suppose a variable follows a normal distribution with mean \(\mu\) and standard deviation \(\sigma>0\), so \(X\sim N(\mu,\sigma^2)\). Its \(p\) quantile is the value \(q_p\) for which \(P(X\le q_p)=p\). Standardizing a normal variable gives \(Z=(X-\mu)/\sigma\sim N(0,1)\). If \(z_p=\Phi^{-1}(p)\), then \(P(Z\le z_p)=p\). Reversing the standardization gives the corresponding quantile on the original measurement scale:

Thus, under a normal model, a theoretical standard normal quantile \(z_i\) corresponds to a value \(\mu+\sigma z_i\) on the measurement scale. The sample observation at rank \(i\), \(x_{(i)}\), is paired with that theoretical position. This explains why the plot uses normal quantiles on one axis and the ordered data on the other.

The plotting positions are a practical way to assign probabilities to ranks; they are not a claim that a particular observation has exactly that cumulative probability. In the midpoint convention, rank \(i\) is assigned the midpoint of the \(i\)th of \(n\) equal-width probability intervals. The interval is \(((i-1)/n,i/n)\), and its midpoint is

This choice also avoids assigning probability 0 to the smallest observation or probability 1 to the largest, both of which would require infinite standard normal quantiles.

Axes and the Reference Line

With theoretical quantiles on the horizontal axis and ordered observations on the vertical axis, the normal-model relationship \(q_p=\mu+\sigma z_p\) has the form of a straight line. The theoretical line has intercept \(\mu\) and slope \(\sigma\). Since population values \(\mu\) and \(\sigma\) are generally unknown, a practical reference line estimates them from the sample.

This is one common, straightforward reference-line choice. Some software uses a line based on sample quartiles or another estimation rule. The plotted pairs and axis meanings still follow the construction above, but the reference line can move slightly because its estimation method differs. When a task or software output specifies a particular reference-line method, follow that method rather than assuming all reference lines are identical.

Constructing the Plot Step by Step

Common Mistakes and Why Construction Details Matter

• Plotting unsorted observations in collection order. The theoretical quantiles represent ranks, so the values must first be sorted. Pairing the first collected value with the lowest quantile is only correct if it is the smallest observation.
• Reversing the axes without noticing. A normal probability plot may be drawn with axes switched by some software, but the convention here is theoretical standard normal quantiles horizontally and ordered data vertically. Check axis labels before reading or reproducing a plot.
• Pairing an observation with the wrong rank. Each \(x_{(i)}\) must be paired with \(z_i\) calculated from the same \(i\). Ties still occupy consecutive ranks, even though their vertical coordinates are equal.
• Using plotting positions of 0 or 1. The standard normal quantiles at probabilities 0 and 1 are not finite. Use an interior plotting-position convention such as the midpoint method described here.
• Treating every reference line as the same calculation. The model relationship is \(y=\mu+\sigma z\), while a plotted reference line estimates its location and scale. Different estimation conventions can give different lines; report the chosen rule.
• Confusing standardized quantiles with standardized observations. The horizontal coordinates are standard normal quantiles. Under this construction, the vertical coordinates are the original ordered measurements, not their z-scores.

Construction details matter because the plot is a collection of rank-matched comparisons, not simply a scatterplot of arbitrary pairs. Clear axis labels, units, a stated plotting-position convention, and an identified reference-line method make the construction reproducible. How the arrangement of points is used to assess the normal model is a separate question.