What a Normal Probability Plot Shows
In “Graphing Sample Data to Check Normality,” we used dotplots, histograms, and boxplots to examine the shape of a small sample. A Normal probability plot provides another way to assess shape. Instead of showing how many observations fall in each interval, it compares the ordered sample values with the values we would expect at corresponding positions in a Normal distribution.
To make the comparison, a plotting method orders the observations from smallest to largest and pairs them with expected Normal values, sometimes called Normal scores. Each plotted point represents one observation and its corresponding expected value. The exact plotting positions can vary slightly among software, but the interpretation is the same: if the sample is consistent with a Normal distribution, the ordered observations should generally increase in step with the expected Normal values.
The axes are not arranged identically in every graphing tool. One version may put expected Normal values on the horizontal axis and observed values on the vertical axis; another may reverse them. Check the axis labels, but do not let the orientation distract from the main question: Do the points follow an approximately straight pattern?
“Straight” does not mean the points must lie exactly on a ruler-straight line. Real samples vary, and a small sample can look uneven even when its population is approximately Normal. Look for the overall pattern. A modest scatter around a straight trend is generally more reassuring than clear, sustained curvature or a point far from the rest of the pattern.
The line does not need to have a particular slope or pass through zero. The axes may use different scales, and the sample’s measurement units and center affect where the points fall. For this shape check, the important feature is approximate linearity, not the numerical values of a slope or intercept.
How to Read the Pattern
Start by following the points from the smallest observation to the largest. If the points rise at a fairly steady rate and remain reasonably close to an imagined straight line, the plot supports approximate Normality. It does not establish that the entire population is Normal; it shows that the sample’s ordered values do not display a strong contradiction to that model.
A curved pattern suggests that some parts of the sample increase differently from others. For example, the points may bend consistently rather than scatter around one straight trend. A pronounced bend is evidence that the sample may not have come from an approximately Normal population. A departure concentrated near one end may also deserve attention: an unusually distant point can disrupt an otherwise straight pattern. These are clues to describe, not automatic diagnoses of their cause.
- Roughly straight: the sample pattern is consistent with an approximately Normal population.
- Clear, sustained curvature: the sample pattern does not provide reassuring evidence for approximate Normality.
- A pronounced departure at one end: investigate the observation and the data context; do not assume it is an error.
- Minor irregularity: small deviations from a line are not, by themselves, proof of non-Normality, especially in a small sample.
A Normal probability plot complements, rather than replaces, the graphs discussed in “Graphing Sample Data to Check Normality.” A dotplot makes individual observations and gaps easy to see. A Normal probability plot emphasizes whether the ordered values follow a Normal pattern. Looking at both can make a judgment clearer, particularly when a small sample makes any one display look irregular.
Using the Plot for Mean Inference
As covered in “Checking the Normal/Large Sample Condition” and “Using the \(n\) at Least 30 Rule Correctly,” mean inference can use the large-sample route when \(n\geq30\). When \(n<30\), the sample’s shape matters more: a Normal probability plot can help assess whether the population might be approximately Normal. It provides evidence about shape, but it cannot prove the population’s distribution from a finite sample.
Keep this shape judgment separate from the other conditions. A roughly straight plot does not establish that the sample was random or that observations are independent. As in “Checking the Random Condition” and “Checking the 10% Condition for Independence,” describe the sampling method and check independence separately. A favorable plot cannot fix biased selection.
State what the observations measure, their units, and the population the sample is intended to represent.
Confirm which axis shows observations and which shows expected Normal values. Trace the points from the smallest value to the largest.
Say whether the points are roughly linear, show clear curvature, or include a pronounced departure. Give specific evidence rather than saying only that the plot “looks good.”
For \(n<30\), explain whether the plot supports approximate Normality. State that it is evidence about the population shape, not proof.
Worked Examples
Worked Example: A Roughly Linear Plot of Seedling Heights
A greenhouse manager randomly selects 16 seedlings from 240 seedlings of the same variety and records each height, in centimeters. The ordered heights are:
\(8.1,\ 8.4,\ 8.6,\ 8.8,\ 9.0,\ 9.1,\ 9.3,\ 9.5,\ 9.6,\ 9.8,\ 10.0,\ 10.2,\ 10.4,\ 10.7,\ 10.9,\ 11.2\)
A Normal probability plot of the heights shows the points rising in a roughly straight band, without a pronounced bend or a point separated from the rest.
State. Let \(\mu\) be the mean height of all seedlings of this variety in the greenhouse. We are assessing whether the sample provides shape evidence for a one-sample t procedure for \(\mu\).
Plan. The sample size is \(n=16<30\), so the large-sample route is not met. We will use the Normal probability plot to assess whether the sample pattern is consistent with an approximately Normal population. The random and independence conditions also need separate checks.
Do. The points follow a roughly straight pattern over the range of the ordered heights. There is no clear sustained curvature and no pronounced departure at either end. This is evidence consistent with an approximately Normal population; it is not proof that every seedling’s height follows a Normal distribution.
The manager selected seedlings at random, which supports the random condition. Because the selection was without replacement, check the 10% condition:
Since \(6.67\%<10\%\), the 10% condition is met.
Conclude. The sample is smaller than 30, but its Normal probability plot is roughly linear and has no pronounced departure, so it provides reasonable evidence that the population distribution is approximately Normal. The random selection and 10% check support the other stated conditions. The plot supports, but does not prove, the shape assumption for mean inference.
Worked Example: Curvature in a Plot of Delivery Times
A community center randomly samples 12 deliveries from a large set of deliveries and records each travel time, in minutes. The ordered sample is:
\(12,\ 13,\ 14,\ 15,\ 16,\ 18,\ 21,\ 25,\ 31,\ 40,\ 54,\ 76\)
The Normal probability plot bends away from a straight pattern as the points move toward the larger times.
State. Let \(\mu\) be the mean travel time for deliveries in the population represented by this sample. We are checking whether the sample plot supports the Normal/Large Sample condition for a t procedure.
Plan. Here \(n=12<30\), so we need shape evidence from the data. We will describe the plot’s overall pattern rather than treating any single point or visual feature as conclusive by itself.
Do. The ordered times do not increase along one roughly straight trend. The upper values become increasingly spread out: the step from 54 to 76 minutes is much larger than several steps among the smaller values. The plot’s reported bend toward the large times is clear curvature, not just small scatter around a line. This pattern does not provide reassuring evidence that the population is approximately Normal.
Conclude. Because the sample is small and its Normal probability plot shows clear curvature, the sample does not support the Normal/Large Sample condition for a t procedure by shape evidence. The manager should not describe the population as approximately Normal on the basis of this plot. The plot alone does not identify why the pattern occurs; the delivery process and the observations should be reviewed.
Worked Example: A Departure at One End of a Plot
A technician randomly selects 14 sealed containers from a production run of 400 and measures the amount of liquid in each, in milliliters. The ordered amounts are:
\(98,\ 99,\ 99,\ 100,\ 100,\ 100,\ 101,\ 101,\ 101,\ 102,\ 102,\ 103,\ 104,\ 116\)
The plot is close to a straight trend for most points, but the point for 116 milliliters lies well above the continuation of that trend.
State. Let \(\mu\) be the mean amount of liquid in all containers from the production run. We are assessing shape evidence for a one-sample t procedure for \(\mu\).
Plan. The sample size is \(n=14<30\), so we will use the plot to assess shape. Because one point departs from the main trend, we will note that feature and consider what can and cannot be concluded from it.
Do. Most points follow an approximately straight pattern, but the largest observation, 116 milliliters, departs markedly from that pattern. The plot is therefore not simply a straight band with minor scatter. The high point could reflect a real container, a measurement or recording issue, or some feature of the process; the plot does not determine which explanation is correct.
The flagged departure is a reason to investigate, not a reason to delete the value automatically. The technician should check the original record and measurement process. If 116 is a valid observation, it remains part of the sample and must be considered when judging whether the sample supports approximate Normality.
Conclude. Although most points are roughly linear, the pronounced departure at the upper end means the plot does not provide unqualified support for an approximately Normal population. The technician should investigate the observation and report the shape concern. A Normal probability plot cannot establish that the point is erroneous or prove that the population is non-Normal.
Common Mistakes and AP Exam Tips
- Expecting a perfect line. Real data rarely fall exactly on a line. State whether the overall pattern is roughly linear and distinguish modest scatter from a clear bend or pronounced departure.
- Using the slope or intercept as the Normality test. The location and scale of the data affect where a line sits and how steep it is. For this shape check, focus on whether the points follow a straight trend, not whether the trend has a particular slope or passes through zero.
- Ignoring the axis labels. Plotting tools can reverse the axes. Identify what each axis represents, then assess linearity; do not decide based on whether the observations happen to appear horizontal or vertical.
- Claiming the plot proves Normality. The plot displays a sample, not the entire population. A full-credit answer says the pattern is “consistent with” or “provides evidence for” approximate Normality, rather than saying it proves the population is Normal.
- Calling any departure decisive. Minor unevenness can occur in a small sample. Describe the size and location of the departure and explain whether it changes the overall impression.
- Treating one unusual point as an error. A point away from the trend deserves investigation. Check the context and data record; do not remove a valid observation simply because it weakens the desired pattern.
- Using shape evidence to claim other conditions are met. Linearity addresses the Normality part of the mean-inference conditions. Explain randomness and independence separately.
A strong AP response identifies the sample size, describes the plot’s actual pattern, and links that evidence to the Normal/Large Sample condition. For example: “Because \(n=16<30\), the large-sample route is not met. The Normal probability plot is roughly linear, with no pronounced departure, so it provides evidence that the population distribution is approximately Normal; it does not prove this.” If the plot bends or has a marked departure, name that feature and explain why it weakens the evidence.
Check Your Understanding
For each situation, describe what the Normal probability plot suggests and what conclusion would be appropriately cautious.
- A sample of 15 randomly selected tree leaves has points that follow a roughly straight band with modest scatter. What does the plot support, and what does it not prove?
- A plot’s points curve steadily away from a straight pattern, although no single point is especially far from the others. How should this affect the shape judgment for mean inference when \(n=18\)?
- A plot has a roughly linear middle, but its largest point is far from the continuation of the trend. What should be reported, and what should be investigated?
- Why is it important to check the axis labels before describing a Normal probability plot?
- A Normal probability plot looks roughly linear, but the sample came from volunteers. Which part of the mean-inference conditions does the plot address, and what concern remains?