A Focused Shape Check for 12 Observations
When a sample is small, a graph is an important part of deciding whether the Normal/Large Sample condition is supported for inference about a population mean. This tutorial focuses on what to examine when the sample contains exactly 12 observations: the overall pattern in a Normal probability plot and the shape of the observed data. The goal is not to prove that the population is Normal. It is to judge whether the sample shows a serious reason to doubt the Normal model used by a t procedure.
As explained in “Checking the Normal/Large Sample Condition,” a sample of 12 does not meet the large-sample route, which requires \(n\geq30\). The shape check therefore matters. Earlier tutorials, including “Graphing Sample Data to Check Normality,” “Reading a Normal Probability Plot,” and “Identifying Skewness and Outliers in Small Samples,” introduced the displays and warning signs. Here we bring those ideas together as a focused assessment: look for agreement across graphs, describe the evidence you actually see, and connect it to the proposed inference.
A Normal probability plot compares the ordered observations with expected values from a Normal distribution. For this check, focus on the overall pattern: do the plotted points follow a roughly straight path, or do they show a clear bend, curve, or departure at one end? A perfect line is not required. With only 12 points, some unevenness is natural, and one small deviation is not automatically a reason to reject a t procedure.
Then look at a display of the observations themselves, such as a dotplot. Ask whether the sample has one main cluster, whether one tail is much longer than the other, whether there are conspicuous gaps, and whether any value stands apart. These features give context for interpreting a probability plot. For instance, a probability plot that bends at the upper end together with a dotplot showing several increasingly large values is more concerning than a plot with one slightly uneven point and an otherwise balanced dotplot.
A Step-by-Step Graph Check
Use the following sequence when assessing the shape evidence for a sample of 12. It is a judgment based on displays, not a mechanical pass-or-fail test. Describe the features in context rather than relying on a vague statement such as “the graph looks okay.”
Record \(n=12\). Since \(12<30\), the large-sample route is not met, so assess the sample’s shape instead of relying on sample size alone.
Look for a roughly straight overall pattern. Note any sustained curvature, a tail that departs from the line, or a point unusually far from the rest of the pattern.
Use a dotplot or a similar graph to check for a long tail, strong skewness, a gap, multiple clusters, or a pronounced outlier. Identify the direction of any tail.
Say whether the displays show a clear warning, no strong warning, or an ambiguous pattern. Explain what that means for the shape support for t inference; do not claim that a graph proves population Normality.
The comparison across displays is useful because every graph emphasizes different details. A dotplot shows the actual spacing of the observations, while a Normal probability plot helps reveal a systematic departure from a Normal pattern. If both displays point to a long tail or an isolated value, the concern is clearer. If they are mostly reassuring but not perfectly regular, explain that the evidence is supportive rather than conclusive.
Keep this shape check separate from the other conditions for mean inference. As discussed in “Checking the Random Condition,” the way the data were collected matters for making a population claim. The 10% condition addresses independence when sampling without replacement from a finite population, as described in “Checking the 10% Condition for Independence.” A favorable graph does not establish either randomness or independence.
Worked Examples
Worked Example: A Roughly Straight Pattern in Sensor Readings
An environmental technician randomly selects 12 sensors from a large shipment and records each sensor’s calibration error, in units. The sorted observations are:
\(-2.4,\ -1.8,\ -1.3,\ -0.9,\ -0.5,\ -0.2,\ 0.1,\ 0.3,\ 0.6,\ 1.0,\ 1.4,\ 2.0\)
The technician’s Normal probability plot follows a roughly straight pattern, with mild scatter and no point far from the rest. A dotplot shows one main cluster spread on both sides of the center, without a long tail or isolated observation.
State. Let \(\mu\) be the mean calibration error, in units, for sensors in the shipment. 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 assess the Normal/Large Sample condition using the Normal probability plot and dotplot. The selection is random, and the shipment is large enough that 12 sensors are less than 10% of its size; these facts support the random and 10% checks separately.
Do. The probability plot is roughly linear overall, rather than showing sustained curvature or a marked departure at one end. The dotplot shows one main cluster, with observations on both sides and no pronounced skewness or outlier. The small irregularities in the probability plot do not form a clear pattern that contradicts the dotplot.
Conclude. The sample of 12 provides reasonable shape evidence for using a one-sample t procedure for the mean calibration error, because the displays show no strong skewness or pronounced outlier and the Normal probability plot is roughly straight. This does not prove that the population distribution is Normal; it describes the evidence available in this sample.
Worked Example: Matching Evidence of a Long Upper Tail
A community garden coordinator randomly selects 12 plots and records the number of minutes required to water each one. The sorted times are:
\(6,\ 7,\ 7,\ 8,\ 8,\ 9,\ 10,\ 11,\ 13,\ 17,\ 24,\ 38\)
The Normal probability plot bends away from a straight pattern toward the upper end. A dotplot shows most times grouped between 6 and 13 minutes, followed by increasingly large values extending to 38 minutes.
State. Let \(\mu\) be the mean watering time, in minutes, for the population of garden plots represented by the sample. We are checking whether the sample’s shape supports t inference for \(\mu\).
Plan. Here \(n=12<30\), so we cannot rely on the large-sample route. We will compare the Normal probability plot with the dotplot, looking for consistent evidence of a tail or an unusual observation. The coordinator reports a random selection, and the number of plots in the population is more than 120, so the sample is no more than 10% of that population.
Do. The upper-end bend in the probability plot suggests that the largest observations depart from the roughly linear pattern. The dotplot gives a matching signal: most observations are relatively small, while a sequence of larger times creates a long tail toward larger values. The value 38 is also separated from the main cluster. Together, these displays show more than a minor irregularity.
Conclude. The sample 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 watering time. The coordinator should check whether the value 38 is accurate and valid. A valid observation should not be removed simply to make the graphs look more Normal.
Worked Example: A Single Departure That Needs Cautious Judgment
A school technician randomly selects 12 classroom projectors from a large district inventory and records the time, in seconds, each takes to start. The sorted times are:
\(18,\ 19,\ 20,\ 20,\ 21,\ 21,\ 22,\ 22,\ 23,\ 23,\ 24,\ 31\)
The Normal probability plot is fairly straight for the first 11 points; the final point lies somewhat above that pattern. A dotplot shows the first 11 times in a compact cluster and the value 31 separated by a small gap, but not by a very large distance.
State. Let \(\mu\) be the mean startup time, in seconds, for projectors in the district inventory. We are assessing whether the sample provides shape support for a t procedure for \(\mu\).
Plan. Since \(n=12<30\), inspect the sample graphs for strong skewness or a pronounced outlier. The random selection supports the random condition. The district inventory has more than 120 projectors, so 12 is no more than 10% of the population; the 10% condition is also met. The remaining question here is the shape evidence.
Do. The probability plot is not perfectly straight, because its largest observation departs from the main pattern. However, the other points are fairly linear, and the dotplot does not show a long sequence of increasing gaps or a pronounced tail. The value 31 is flagged as a high outlier by the standard boxplot rule: using the median-of-halves quartiles, \(Q_1=20\) and \(Q_3=23\), so the upper fence is \(23+1.5(3)=27.5\), and \(31>27.5\). This gives substantial reason for caution with t inference for a sample of 12, although the graph alone does not prove that the population is non-Normal.
Conclude. The shape evidence is somewhat uncertain: the largest startup time departs from the probability plot’s pattern, but the rest of the displays do not show strong skewness or a clearly pronounced outlier. A careful response should report that feature rather than simply declaring the sample Normal or unusable. The value should be checked against the measurement record and context before relying on the t procedure.
Common Mistakes and AP Exam Tips
- Demanding a perfect straight line. With 12 observations, a probability plot can show modest scatter. A full-credit response describes the overall pattern and distinguishes mild irregularity from sustained curvature or a marked departure.
- Looking only at the probability plot. Use a dotplot or another display to see the observations’ spacing, tails, and possible isolated values. Explain whether the displays agree or whether the evidence is mixed.
- Treating one unusual point as an automatic decision. A point away from the probability-plot pattern or a boxplot flag is a reason to inspect the data. Describe how pronounced the departure is and check whether the observation is valid.
- Calling an ambiguous pattern definitely Normal. Say that the sample “provides reasonable shape evidence” or that the evidence is “somewhat uncertain.” A sample graph cannot establish the population’s exact shape.
- Forgetting the sample-size context. State that \(n=12<30\), so the large-sample route is not met. Then explain what the graphs show about the Normal/Large Sample condition.
- Using a favorable shape graph to claim the sample is representative. Shape evidence is not evidence of random selection. Check the data-collection method and independence separately.
A strong AP response gives a concrete description and a bounded conclusion. For example: “Because \(n=12<30\), the large-sample route is not met. The Normal probability plot is roughly linear, and the dotplot shows no strong skewness or pronounced outlier. Thus, the sample provides reasonable shape evidence for a t procedure for the population mean, although it does not prove that the population is Normal.” If the plots show a clear concern, name the concern and say that the sample does not provide reassuring shape evidence.
Check Your Understanding
For each situation, explain what the graphs suggest about the shape evidence for t inference with a sample of 12.
- A Normal probability plot is roughly straight, and a dotplot shows one balanced cluster with no separated point. What conclusion is supported, and what cannot be proved?
- A probability plot curves at its upper end, while a dotplot shows most observations low and a long tail toward larger values. What is the shape concern?
- A single point departs somewhat from an otherwise linear probability plot, but the dotplot shows no pronounced gap. How should the conclusion be phrased?
- Why is it useful to compare a Normal probability plot with a dotplot rather than relying on only one display?
- A sample of 12 looks reasonably balanced, but it was collected from volunteers. What does the graph address, and what separate issue remains?