Why Shape Matters With 12 Observations
In “Reading a Normal Probability Plot,” we looked for a roughly straight pattern as evidence that a small sample could be consistent with an approximately Normal population. This tutorial focuses on two particular warning signs in a sample of \(n=12\): strong skewness and possible outliers. Either can make a t procedure for a population mean unreliable.
As discussed in “Checking the Normal/Large Sample Condition,” a sample of 12 does not meet the \(n\geq30\) large-sample route. That means the sample’s shape deserves careful attention. A t procedure does not require every observation to be perfectly symmetric, but a small sample gives unusual values and strong asymmetry a greater opportunity to affect the sample mean and standard deviation.
Skewness describes the overall shape, while an outlier describes a value that is unusually separated from the rest. They can occur together, but they are not the same thing. A sample may have a long tail without one single isolated observation. Conversely, most of a sample may look fairly balanced while one value stands apart.
When you look at a graph, pay attention to the tail, not just where most of the observations are clustered. For example, if most values are relatively small and a few progressively larger values stretch away to the right, the distribution is right-skewed. The direction of skew refers to the direction of the extended tail, not the side where most observations lie.
A Practical Shape Check for a Small Sample
Begin with a dotplot, histogram, or boxplot, as described in “Graphing Sample Data to Check Normality.” With only 12 observations, a dotplot is often especially helpful because it shows the individual values and gaps directly. A histogram can suggest a tail, but its appearance can change with the bin choices. A boxplot gives a compact summary of the middle and tails, though it hides some detail about individual observations.
A boxplot may mark a point beyond its whiskers separately. Many boxplot conventions place whiskers at the most extreme observations within \(1.5\) times the interquartile range of the quartiles, and display more distant observations as separate points. Such a point is a flag for investigation, not proof that the value is wrong or that a t procedure must never be used. Check the data record and the context; do not delete a valid observation simply because a graph flags it.
A Normal probability plot can add another view. Strong, sustained curvature may support a concern about skewness, and a point that departs markedly from the overall pattern may signal a possible outlier. As in “Reading a Normal Probability Plot,” look at the whole pattern rather than demanding a perfect line.
This check is about whether the sample provides reasonable evidence for a population shape suitable for t inference. A sample graph cannot prove that the population is Normal, and a graph with no obvious warning signs cannot guarantee that the population has no unusual values. The conclusion should be appropriately cautious: describe what the sample shows and explain what that means for the proposed procedure.
Use a graph that shows the observations. Identify the main cluster, the direction of any extended tail, and any gaps.
Ask whether a value is separated from the rest of the sample, not simply whether it is the smallest or largest observation. If a graph flags it, investigate it.
A dotplot or boxplot can make a separated value visible, while a Normal probability plot can reveal curvature or an unusual departure from the overall pattern.
For \(n=12\), state whether the sample shows strong skewness or a pronounced outlier, and whether its shape provides reassuring support for t inference about a mean.
Worked Examples
Worked Example: A Long Right Tail in Water-Use Measurements
A town analyst randomly selects 12 households from a large population and records their daily outdoor water use, in gallons. The sorted measurements are:
\(4,\ 5,\ 5,\ 6,\ 6,\ 7,\ 7,\ 8,\ 9,\ 12,\ 19,\ 34\)
A dotplot shows many values grouped at the lower end and progressively larger gaps among the upper values.
State. Let \(\mu\) be the mean daily outdoor water use, in gallons, for households in the population represented by the sample. We are assessing whether the sample’s shape supports using a one-sample t procedure for \(\mu\).
Plan. The sample size is \(n=12<30\), so the large-sample route is not met. We will inspect the sample’s shape for strong skewness or a pronounced outlier. The analyst reports a random selection; the population is large enough that 12 households are less than 10% of it. These facts support the random and 10% checks, but the shape check still needs to be assessed.
Do. Most of the measurements are between 4 and 12 gallons, while the values 19 and 34 extend the pattern toward larger amounts. The gaps grow toward the high end, creating a long right tail. The value 34 is especially separated from the main cluster. This is evidence of strong right-skewness and a possible high outlier, rather than minor unevenness around a balanced shape.
Conclude. The sample of 12 shows strong right-skewness and a possible high outlier, so it does not provide reassuring shape evidence for a one-sample t procedure for the population mean. The value 34 should be checked against the original record and context, not automatically removed. If it is valid, it remains part of the sample and part of the reason for concern.
Worked Example: One Separated Value in Package Weights
A quality technician randomly selects 12 packages from a large production run and records their weights, in grams:
\(41,\ 43,\ 44,\ 45,\ 46,\ 47,\ 48,\ 49,\ 50,\ 51,\ 52,\ 79\)
A dotplot places the first 11 observations in a fairly compact sequence and shows a clear gap before 79. A boxplot marks 79 as a point beyond the upper whisker.
State. Let \(\mu\) be the mean weight, in grams, of packages in this production run. We are checking whether the sample’s shape supports t inference for \(\mu\).
Plan. Here \(n=12<30\), so we should look for evidence of skewness or an outlier rather than rely on the large-sample route. We will use the dotplot and boxplot together and treat the boxplot flag as a reason to investigate, not as a verdict about the value.
Do. The observation 79 is far above the other weights, with a clear gap separating it from the main group. The boxplot flag agrees with the dotplot’s indication of a possible high outlier. The other observations do not eliminate this concern: one isolated value can have a substantial effect on a sample mean and sample standard deviation when there are only 12 observations.
The technician should review the measurement and package record. If 79 resulted from a recording or measurement error, the error should be addressed using the appropriate data-handling process. If it is a valid package weight, it should not be discarded simply to make the graph look more regular.
Conclude. The pronounced high value makes the sample’s shape questionable for a t procedure for the mean. The shape evidence does not justify assuming that the population is approximately Normal, and the flagged value should be investigated before deciding how to proceed.
Worked Example: No Strong Shape Warning in Battery Lifetimes
A technician randomly selects 12 batteries from a large shipment and records operating lifetimes, in hours. The sorted sample is:
\(18,\ 19,\ 20,\ 20,\ 21,\ 21,\ 22,\ 22,\ 23,\ 23,\ 24,\ 25\)
A dotplot shows one main cluster, with observations spread on both sides of the center and no isolated point. A Normal probability plot shows a roughly straight pattern with modest scatter.
State. Let \(\mu\) be the mean operating lifetime, in hours, for batteries in the shipment. We are assessing the shape evidence for a t procedure for \(\mu\).
Plan. The sample size is \(n=12<30\), so we need to assess the sample’s shape. We will look for strong skewness and pronounced outliers in the dotplot, and use the Normal probability plot as additional evidence.
Do. The dotplot has one main cluster, no conspicuously long tail, and no observation separated from the rest. The Normal probability plot is roughly linear, with only modest scatter. These displays do not show strong skewness or a pronounced outlier.
Conclude. The sample provides reasonable shape evidence for using a t procedure, because it shows no strong skewness or pronounced outlier. This does not prove that the population distribution is Normal. Randomness and independence also need to be supported separately before making an inference about the shipment.
Common Mistakes and AP Exam Tips
- Calling the side with most data the skew direction. Skew is named for the longer tail. If most observations are small and the tail stretches toward larger values, the shape is right-skewed.
- Calling every extreme value an outlier. The minimum and maximum are always at the ends of a sample. Explain whether a value is separated from the rest or otherwise departs from the pattern.
- Treating a boxplot flag as proof. A point beyond a boxplot whisker is a signal to inspect the observation. It does not prove a recording error or settle the inference question on its own.
- Ignoring the sample size. With \(n=12\), do not claim the large-sample route applies. Explain how the observed shape bears on the Normal/Large Sample condition.
- Saying the graph proves the population’s shape. A graph describes the sample. A full-credit answer says the sample “provides evidence” or “does not provide reassuring evidence” about approximate Normality.
- Letting a favorable graph stand in for all conditions. A roughly symmetric sample without outliers addresses shape, not random selection or independence. Check and describe those separately.
- Deleting a valid observation to improve the graph. Investigate a possible outlier and correct a confirmed data error appropriately. Do not remove a valid value solely because it makes t inference less convenient.
A strong AP response names the sample size, describes the actual graphical evidence, and connects that evidence to t inference. For example: “Because \(n=12<30\), the large-sample route is not met. The dotplot shows a long right tail and a possible high outlier, so the sample does not provide reassuring shape evidence for a one-sample t procedure for the population mean.” If the displays show no strong warning signs, say so cautiously rather than claiming the population is definitely Normal.
Check Your Understanding
For each situation, describe the sample-shape evidence and what it means for t inference with a sample of 12.
- Most measurements cluster at low values, while a few increasingly large observations form a long upper tail. Which direction is the skew, and what shape concern should be reported?
- A boxplot marks one point beyond the whisker, but the data record has not been checked. What can you conclude, and what should happen next?
- A dotplot shows one balanced cluster with no long tail or isolated value, and a Normal probability plot is roughly linear. What does this support, and what does it not prove?
- Why is a sample’s largest observation not automatically an outlier?
- A sample looks roughly symmetric, but it was collected from volunteers. Which condition does the shape evidence address, and what separate concern remains?