Why Inspect a Small Sample?
A one-sample t test uses the sample mean and sample standard deviation to assess a claim about a population mean. For a small sample, the shape of the population distribution matters: the usual t reference distribution is exact when observations are independent and come from a normal population. The earlier material on Conditions for a One-Sample T Test sets out the full conditions for the test. Here the focus is narrower: what can plots of a small sample reveal about skewness and unusual observations?
A plot does not prove that a population is normal. It shows the pattern in the observed data, which are only a sample from that population. With few observations, even a plot that looks reasonably balanced gives limited evidence about the population’s shape. Conversely, a very asymmetric pattern or a strikingly isolated value is a reason to pause before relying on a small-sample t test.
What to Look for in a Dotplot or Histogram
A dotplot places a dot for each observation on a number line, stacking dots when observations have the same value. It preserves the individual data values and is often useful for small samples. A histogram groups values into intervals and displays the count in each interval. It gives a compact view of distribution shape, but the apparent shape can depend on the interval widths and where the intervals begin.
When inspecting either plot, consider the overall pattern rather than searching for perfect symmetry. Ask whether the observations are fairly balanced around the center, whether one tail extends much farther than the other, and whether there are isolated values or large gaps. A longer tail on the high-value side suggests right skew; a longer tail on the low-value side suggests left skew. A possible outlier is an observation that lies unusually far from the rest of the data. “Possible” matters: a plot identifies a value for investigation, not an error or a reason for automatic removal.
There is no single sample size that separates “small” from “large” for every setting, and there is no universal visual threshold for an unacceptable departure from normality. The more pronounced the skewness or isolation, the greater the concern for a small-sample t test. A few observations cannot reveal every feature of the population distribution.
Worked Example: A Dotplot With a Balanced Pattern
Suppose a hypothetical sample records the completion time, in minutes, for eight independently selected practice tasks: 12, 13, 14, 15, 15, 16, 17, and 18. A dotplot would place one dot at each of 12, 13, 14, 16, 17, and 18, and two dots at 15.
The values pair around 15: 12 and 18, 13 and 17, 14 and 16, with two observations at 15. The dotplot therefore has a balanced appearance, with no conspicuously long tail, isolated point, or large gap. In this sample there is no obvious visual warning of substantial skewness or an outlier.
That is not proof that completion times in the population are normally distributed. The sample contains only eight observations, so it provides limited information about the population. The careful conclusion is that this dotplot does not show a strong departure, while the small sample leaves substantial uncertainty about the population shape.
Worked Example: A Histogram With a Long High-Value Tail
Consider a hypothetical sample of ten measurements, in seconds: 8, 9, 9, 10, 10, 10, 11, 12, 15, and 29. To make a histogram, use the intervals from 0 up to but not including 10, from 10 up to but not including 20, and from 20 up to but not including 30.
The interval counts are 3, 6, and 1, respectively. Most observations lie between 8 and 15 seconds, while 29 seconds falls by itself in the highest interval. The histogram would show a sparse extension toward higher values, suggesting right skew and flagging 29 seconds as a possible outlier.
The binning affects how much detail is visible: different interval boundaries could change the bars. A dotplot of the same data would show the individual observations directly and make the separation of 29 from the main group clear. Neither plot establishes why the value is high. It could be a genuine measurement, a recording mistake, or something else that needs investigation.
For a small-sample t test, this is a meaningful warning. The appropriate next step is to check the observation against its source and the data-collection process. If 29 is a valid observation, it should not be deleted merely because it makes the plot or a test result less convenient. The visible asymmetry should be reported when deciding whether a small-sample t analysis is well supported.
Using a Normal Probability Plot
A normal probability plot compares the ordered sample values with scores expected from a normal distribution. If the observations follow a normal pattern reasonably well, the plotted points should fall roughly along a straight line. Departures from a line can reveal curvature associated with skewness or an isolated point at an end of the distribution. The plot is also called a normal quantile plot; software may put the theoretical scores on either axis.
The straight-line pattern has a direct justification. If a variable \(X\) has a normal distribution with mean \(\mu\) and standard deviation \(\sigma\), its population quantile at probability \(p\) is
Thus, when the theoretical score is \(z=\Phi^{-1}(p)\), the corresponding population quantile is \(Q(p)=\mu+\sigma z\), a straight-line function of \(z\). In a sample, the ordered values are not exactly equal to their population quantiles, and the plotting positions are approximations. Therefore, even normal data will not produce a perfectly straight plot. The diagnostic question is whether deviations look like ordinary scatter around a line or instead form a clear, systematic pattern.
A smooth bend or curve suggests that the sample shape differs from a normal pattern; pronounced asymmetry may be consistent with skewness. A point far from the general line, particularly at one end, may indicate a possible outlier. These are clues, not automatic classifications. With very few points, one observation can strongly affect the apparent pattern, and the plot has limited power to distinguish modest departures from random variation.
Worked Example: An Isolated Upper Point on a Normal Probability Plot
Suppose a hypothetical sample of seven independent measurements, in units, is 9.6, 9.8, 10.0, 10.1, 10.2, 10.4, and 12.0. The values are already ordered. Using \(p_i=(i-0.5)/7\), the plotting positions are approximately 0.071, 0.214, 0.357, 0.500, 0.643, 0.786, and 0.929. Their standard normal scores are approximately \(-1.465\), \(-0.792\), \(-0.366\), 0, 0.366, 0.792, and 1.465.
A normal probability plot pairs these scores with the ordered measurements. The six measurements from 9.6 through 10.4 form a compact, gradually increasing sequence. The value 12.0 is separated from that group. To see the scale of the departure, a line through the central points at scores \(-0.792\) and 0.792 has slope
and intercept about 10.1. That line predicts a value near \(10.1+0.379(1.465)=10.66\) at the highest theoretical score, whereas the observed value there is 12.0. The upper point is therefore far above the trend suggested by the central observations and would appear as a potential departure from the line.
This plot raises a concern; it does not establish that 12.0 is erroneous or that the population is nonnormal. The value should be checked for data or measurement problems. If it is valid, its separation is relevant when judging whether a small-sample t test is adequately supported. The plot alone cannot settle that question.
A Practical Diagnostic Sequence
Use the plots to make and explain a judgment, rather than treating them as a pass-or-fail rule. A useful sequence is:
Use a dotplot for a very small sample when individual values and gaps matter. A histogram can summarize shape, but inspect the binning before interpreting a feature.
Look for substantial imbalance, a longer tail, clusters, gaps, or values far from the main body. Name the direction of skewness when it is apparent.
Verify questionable values against the records and measurement process. A valid unusual value remains part of the data unless a defensible analysis decision says otherwise.
Look for points that follow an approximate line, systematic curvature, or an isolated endpoint. Interpret the pattern in light of the small number of observations.
Describe what the sample plots show and whether they raise a concern for small-sample t inference. Do not claim that visual inspection proves population normality.
When the plots show no strong skewness or isolated values, they provide some reassurance about using the small-sample procedure, but not certainty. When they show serious asymmetry or a striking outlier, the usual small-sample justification is less convincing. The conclusion should reflect the strength of the evidence and the fact that a small sample may not reveal the population’s full shape.
Common Mistake: Deleting an Outlier Because It Changes the Result
A common but incorrect argument is: “This point is far from the others, so it is an outlier; removing it makes the data look more normal, so I will run the test without it.” A plot can flag an observation for scrutiny, but distance from the rest does not show that the observation is invalid. Removing a legitimate value solely because it changes the result can distort the analysis.
The correction is to investigate the observation’s origin. If there is a documented recording or measurement error, correct it when possible and explain the correction. If the value is valid, retain it in the data and acknowledge the diagnostic concern when interpreting a small-sample t analysis. A transparent report separates what the plot reveals from what is known about the measurement.
Check Your Understanding
Answer each question by referring to what a plot can establish and what it cannot.
- Why is a dotplot often useful for a very small sample, and what feature might a histogram hide?
- A histogram has a longer tail toward high values. What pattern does this suggest, and why is it not proof about the population?
- In a normal probability plot, what does a roughly straight pattern indicate? Why should the points not be expected to lie exactly on a line?
- A single measurement is far above the other observations. What should be checked before deciding how to handle it?
- Why is “the plot looks normal, so the population is normal” too strong a conclusion for a small sample?