Normal Probability Plots With Small Samples

Why Sample Size Changes What a Normal Plot Can Show

A normal probability plot places ordered observations against theoretical quantiles from a normal distribution. If the data are reasonably consistent with a normal model, the plotted points tend to follow a straight-line pattern. Earlier material on constructing and interpreting normal probability plots explains how to read the general shape. Here the key issue is how much confidence to place in that shape when the sample is small.

With only a few observations, there are only a few plotted points. The graph cannot show the fine detail of the distribution, and each point represents a large fraction of the sample. One unusually large or small value can therefore change the visible pattern substantially. Even data drawn from a normal population will not generally fall exactly on a line: random samples have ordinary variation, and that variation is especially conspicuous when there are few observations.

Key idea: A small-sample normal probability plot can reveal a striking departure from straightness, but it gives limited information about the distribution’s shape. A small bend, gap, or isolated point is a feature to investigate, not by itself a conclusion that the data are nonnormal.

A plot is a diagnostic, not a formal test with a universal cutoff. Its interpretation depends on the pattern, the sample size, the measurement context, and what decision is being made. In particular, the plot alone does not establish why an unusual value occurred, whether observations are independent, or whether a normal model is appropriate for a later analysis.

Plotting Positions and Sparse Points

To make the small-sample issue concrete, consider a common way to choose theoretical normal scores. Sort the observations from smallest to largest, and assign the observation with rank \(i\) the plotting position \(p_i=(i-0.5)/n\). The corresponding theoretical score is the standard normal quantile \(z_i=\Phi^{-1}(p_i)\), where \(\Phi^{-1}(p)\) is the value with cumulative standard normal probability \(p\). Plot each observed value against its assigned score. Different software or textbooks may use slightly different plotting positions, so their scores may differ a little, especially in small samples.

p_i=\frac{i-0.5}{n}, \qquad z_i=\Phi^{-1}(p_i), \qquad i=1,\ldots,n

For example, if \(n=5\), the plotting positions are 0.1, 0.3, 0.5, 0.7, and 0.9. Their standard normal scores are approximately \(-1.282\), \(-0.524\), \(0\), \(0.524\), and \(1.282\). Five points can suggest a pattern, but cannot trace a detailed curve. The two endpoint observations are also the only observations that directly show the sample’s extremes.

There is a mathematical reason that the outer points can feel visually influential. The standard normal quantile \(z=\Phi^{-1}(p)\) changes more rapidly with \(p\) near the tails than near the center. Differentiating the relation \(\Phi(z)=p\) gives the rate of change:

\frac{dz}{dp}=\frac{1}{\phi(z)}

Here \(\phi(z)\) is the standard normal density. It is smaller in the tails than near the center, so \(1/\phi(z)\) is larger there. This helps explain why adjacent tail plotting positions can correspond to relatively widely separated theoretical scores. It does not mean that an endpoint observation is automatically an outlier or that the sample is nonnormal.

Worked Examples: Reading Small Samples Carefully

Worked Example: One Endpoint Looks a Little Irregular

Suppose five hypothetical measurements, in centimeters, are \(7, 9, 10, 11,\) and \(14\). Use the common plotting positions above, and compare the plotted observations with the illustrative straight line \(y=10+2z\), where \(z\) is the theoretical normal score. This line is used only to quantify the comparison; it is not a universal rule for fitting a normal plot.

For \(n=5\), the scores are approximately \(-1.282,-0.524,0,0.524,1.282\). Substituting them into the illustrative line gives predicted values \(7.436,8.952,10.000,11.048,\) and \(12.564\) cm.

\text{residual}=\text{observed value}-\text{value on line}

The residuals are therefore \(7-7.436=-0.436\), \(9-8.952=0.048\), \(10-10.000=0\), \(11-11.048=-0.048\), and \(14-12.564=1.436\) cm. The last point sits noticeably above this particular line, while the other four are close to it.

This is a visible endpoint irregularity, but the sample has only five observations, and that endpoint is one of only five data values. The plot does not provide enough detail to tell whether the pattern reflects ordinary sampling variation, a genuine feature of the population, or something specific to the measurement. It is reasonable to note and investigate the high value; it is not reasonable to declare the entire population nonnormal on this plot alone.

Worked Example: Small Deviations Can Still Look Nearly Linear

Consider six hypothetical completion times, in minutes: \(4,7,9,11,13,\) and \(16\). With \(n=6\), the plotting positions \((i-0.5)/6\) are approximately \(0.0833,0.2500,0.4167,0.5833,0.7500,\) and \(0.9167\). The corresponding theoretical normal scores are approximately \(-1.383,-0.674,-0.210,0.210,0.674,\) and \(1.383\).

Compare the data with the illustrative line \(y=10+4z\). Its predicted times are approximately \(4.468,7.304,9.160,10.840,12.696,\) and \(15.532\) minutes. Subtracting predicted from observed gives residuals of \(-0.468,-0.304,-0.160,0.160,0.304,\) and \(0.468\) minutes.

The deviations are small and change gradually across the ordered observations. A plot of these six points would look broadly linear, though it would not lie exactly on a line. This is a useful reminder that perfect straightness is not expected, even when a normal model is plausible. At the same time, six points cannot verify normality; they provide only a limited check for a conspicuous departure.

Worked Example: A Single Extreme Observation

Suppose eight hypothetical measurements, in millimeters, are \(1,2,3,4,5,6,7,\) and \(30\). The first seven values are closely spaced, while the last is much larger. With \(n=8\), the common plotting positions run from \(0.0625\) to \(0.9375\). The standard normal scores are approximately \(-1.534,-0.887,-0.489,-0.157,0.157,0.489,0.887,\) and \(1.534\).

For a numerical comparison, use the illustrative line \(y=4.5+2z\). It predicts approximately \(1.432,2.726,3.522,4.186,4.814,5.478,6.274,\) and \(7.568\) mm. For the largest measurement, the residual is \(30-7.568=22.432\) mm. This point is far above the line, unlike the small residuals among the first seven measurements.

That is a strong visual warning about the value \(30\): the data do not follow this straight-line pattern throughout. But the plot does not tell us whether the value is a recording error, a valid extreme measurement, or evidence that a normal model does not describe the population well. The next step is to check the original measurement and its context. If the value is verified, the unusually large departure should be reported and considered when choosing an analysis; deleting it merely to make the plot straighter would not be justified.

What Counts as Convincing Evidence?

In a small sample, a departure is more persuasive when it is substantial and coherent rather than a minor fluctuation among neighboring points. Examples include a pronounced curve across most of the points, a clear bend at one end, or one verified observation far from the overall linear pattern. Even then, the conclusion should be limited: the plot raises concern that a normal model may not describe these data well. It does not prove a particular alternative distribution.

A single point deserves a different kind of attention from a persistent pattern. A single extreme point may be important because it can affect summaries and analyses, but it gives little information about the shape of an entire tail. Several points forming a consistent bend or tail pattern provide a more sustained visual signal, though small samples may not contain enough points to show such a pattern clearly.

The comparison line is also not a magic boundary between normal and nonnormal data. Software may draw a fitted line, a line based on quartiles, or another reference line. The line helps the eye assess straightness; it does not turn a small sample into a precise test. A good interpretation states what is seen, how many observations support it, and what remains uncertain.

Why Caution Matters

Normality is often assessed because a later procedure uses a normal model or relies on an assumption about a distribution. A normal probability plot can contribute evidence, but the consequences of a small sample make overconfident readings risky. Calling data nonnormal because one of five points misses a line can discard a reasonable model. Calling data normal because six points appear roughly straight can overlook departures the sample was too small to reveal.

The relevant question is not “Are the points exactly on a line?” They will almost never be. Instead ask: Is there a clear, sustained pattern that is difficult to reconcile with a straight-line relationship, given how few observations there are? Then consider the measurement process, possible recording problems, and the purpose of the analysis. A plot can guide those questions, but it cannot answer all of them by itself.

Common Mistakes

Treating every bend as proof of nonnormality. Small samples naturally produce uneven point spacing and chance departures. Look for substantial, sustained structure rather than demanding exact straightness.
Treating a straight-looking plot as proof of normality. A handful of points cannot rule out features that the sample did not capture. Say that the plot shows no striking departure, not that normality has been established.
Calling one extreme point a whole-distribution pattern. An endpoint may be a genuine extreme or a data problem. Check it and describe its influence without claiming that it defines the population’s tail shape.
Deleting an inconvenient observation to improve the plot. First investigate its origin. Exclude a value only when there is a defensible reason, such as a documented recording or measurement error, and explain that decision.
Using a visual plot as a formal pass/fail test. There is no universal visual cutoff. Report the visible pattern and the uncertainty that comes with the sample size.

Key takeaway: With few observations, a normal probability plot is a low-resolution diagnostic. A conspicuous, coherent departure deserves attention, but small irregularities are not proof of nonnormality, and apparent straightness is not proof of normality.

Check Your Understanding

Use these questions to practise interpreting a normal probability plot without overstating what a small sample can show.

Why can one endpoint observation have a large visual effect when a plot contains only five points?
For \(n=5\), what plotting position is assigned to the third ordered observation using \(p_i=(i-0.5)/n\)? What is its standard normal score?
A plot with seven observations looks roughly linear, with one point slightly above the line. What is a cautious conclusion, and what would be an overconfident conclusion?
A small-sample plot has one extreme value far from the remaining points. Name two different explanations the plot itself cannot distinguish.
Why does a nearly straight plot with a small sample not establish that the population is normal?