Checking Normality with Probability Plots

A normal probability plot is simply a picture that tells you whether your data look like they came from a bell curve: if the points line up straight, they probably did. The formulas are a recipe, not a theory: $p_i = \frac{i - 0.5}{n}$ (what fraction of the population should sit below your $i$th smallest value), $z_i = \Phi^{-1}(p_i)$ (turn that fraction into a z-score with a z table read backward) and $x_{(i)} \approx \mu + \sigma z_i$ (if the data are normal, plotting values against those z-scores gives a straight line). The last formula is just least squares for that line. Sort, look up, plot, and look.

Is the normal model safe here? You have ten coating-thickness readings and a spec, and you're about to predict the fraction out of spec from a normal curve. Normal models are excellent in the middle and treacherous in the tails, which is exactly where spec limits live. Ten points make a useless histogram (the picture depends on where the bins start). A normal probability plot uses every point and gives a clear verdict: straight line or not.

The recipe:

1. Sort the data: $x_{(1)} \le x_{(2)} \le \dots \le x_{(n)}$.
2. Give the $i$th value the plotting position $p_i = (i - 0.5)/n$.
3. Convert each to a normal score $z_i = \Phi^{-1}(p_i)$ by reading a z table backward.
4. Plot $x_{(i)}$ against $z_i$. Straight means normal.

For $n = 10$ the scores are $\pm0.126, \pm0.385, \pm0.674, \pm1.036, \pm1.645$.

Why a line, and why $\mu$ and $\sigma$. The empirical CDF jumps from $(i - 1)/n$ to $i/n$ at $x_{(i)}$, so about a fraction $p_i$, the middle of the jump, of the population lies below $x_{(i)}$. If the population is $N(\mu, \sigma)$, the fraction below $x$ is $\Phi\left(\frac{x - \mu}{\sigma}\right)$, so

That's a straight line in $z_i$ that crosses $z = 0$ at $\mu$ with slope $\sigma$. One plot works for every normal at once because every normal is the standard one shifted by $\mu$ and stretched by $\sigma$. Read $\sigma$ off the plot as the rise from $z = 0$ to $z = 1$. The $-0.5$ matters: with $p_i = i/n$ the largest point would need $\Phi^{-1}(1) = \infty$.

Fitting the line. By eye is fine for a verdict. For numbers, use least squares. The normal scores are symmetric, so $\sum z_i = 0$: the line passes through $(0, \bar x)$, and the slope $\frac{\sum z_i(x_{(i)} - \bar x)}{\sum z_i^2}$ simplifies to $\hat\sigma = \frac{\sum z_ix_{(i)}}{\sum z_i^2}$. The correlation of the plotted points is close to 1 for normal data, and formal normality tests such as Shapiro–Wilk are built on that idea; goodness of fit has the testing logic.

Reading the shapes (data on the vertical axis):

• Straight: a normal model is plausible.
• Concave up (a smile, steep on the right): right skew. The upper tail stretches out, as for failure times, surface roughness, runout and anything bounded below by zero.
• Concave down: left skew.
• Steep at both ends (below the line on the left, above it on the right): heavy tails, more extreme values than a normal predicts.
• One point far off an otherwise straight line: an outlier. Check it before modeling.
• A kink or a step in the middle: two populations mixed, like two filling heads or two mold cavities.
• Flat stair steps: ties from coarse resolution (measurement error). The data are rounded, not abnormal.

A common mistake. With a small sample, it's tempting to call the data non-normal because the points wiggle around the line. Small normal samples always wiggle. With $n = 10$, expect points to snake around the line; judge the overall shape, especially at the ends, not single points. The second picture further down lets you draw truly normal samples and calibrate your eye.

Some software swaps the axes (data across, z up) or labels the z axis with percentages, the style of "normal probability paper". Then the slope is $1/\sigma$. Check the axes before reading off numbers.

Where this goes next. The same trick, transforming the axes until a model becomes a straight line, fits lifetimes in the Weibull distribution. And before quoting parts per million out of spec in process capability, a probability plot is the minimum due diligence.

Quality engineers make a probability plot before any capability study, because ppm-out-of-spec numbers come from the far tails of the normal curve, where small departures change the answer by factors of ten. Reliability engineers do the same with Weibull plots of failure times, and analysts check regression residuals on a normal probability plot before trusting confidence intervals. Statistical software puts the plot one click away, but reading it, straight or curved, skewed or heavy-tailed, is the engineer's job.