Graph the Differences, Not the Two Measurement Columns
In “Conditions for a Paired t Procedure,” you learned to assess the distribution of the paired differences. This tutorial focuses on how to make that assessment from graphs, especially when the number of pairs is small. A graph cannot prove that a population of differences is Normal, but it can reveal features that make a paired t procedure more or less reasonable.
Begin with the difference variable already defined for the study. If \(d_i\) is the measurement under one condition minus the measurement under another, graph the list \(d_1,d_2,\ldots,d_n\). The paired t procedure uses the mean and standard deviation of this list. Graphing the two original measurement columns instead would assess the wrong distributions.
With a small sample, the shape of the observed differences matters because there is limited information for the sampling distribution of \(\bar d\) to be approximately Normal. A roughly symmetric, mound-shaped distribution without pronounced outliers is generally compatible with using a paired t procedure. Strong skewness or a pronounced outlier is a warning, not a feature to overlook.
Choosing and Reading a Graph
A dotplot is often a useful first choice for a small data set. Put the difference values on a number line with equal spacing, then place one dot above each observed value; stack dots when values repeat. A dotplot preserves individual observations, so you can see clusters, gaps, and possible isolated values.
A stemplot also retains the individual values and can be helpful when differences have several digits. A histogram gives a broader picture of concentration and tails, but its appearance can depend on the chosen intervals, or bins. For a very small sample, avoid relying on a histogram alone: changing the bins can make the same data look more or less symmetric.
A Normal probability plot compares the ordered differences with values expected from a Normal distribution. If the points lie roughly along a straight line, that pattern is compatible with Normality. Strong systematic curvature can indicate skewness, while a point far from the general pattern may signal an unusual observation. With only a few differences, however, the plot has limited detail; a roughly straight pattern is not proof of Normality.
When reading any of these graphs, look at the overall pattern rather than demanding perfect symmetry. Ask whether the data have one main cluster, whether the two sides of that cluster are reasonably balanced, and whether one tail stretches much farther than the other. Also notice gaps and observations separated from the rest. A gap alone does not establish that a value is an outlier; consider how far apart the values are and whether the observation is unusual in context.
Make the graph from one difference per pair, using the subtraction order defined for the study.
Note the number of clusters, the balance of the distribution, and the direction of any tail.
Identify unusually separated observations and explain whether they raise concern; do not call every gap or endpoint an outlier.
For a small sample, decide whether the graph provides reasonable support for the shape condition, and state any limitation.
This is the shape part of the condition check, not the whole check. As discussed in “Conditions for a Paired t Procedure,” a suitable graph does not replace checking the study design, independence, or the 10% condition when it applies.
Worked Examples
Worked Example: Battery-Life Differences for Paired Devices
A fictional technology club tests a power-saving setting on 12 phones. Each phone is tested both with and without the setting under the same routine. Define \(d_i=\text{hours with setting}-\text{hours without setting}\). The 12 differences, in hours, are \(-6,-5,-4,-3,-2,-1,0,0,1,2,3,4\). A dotplot has one dot at each listed value and a second dot at zero.
State. We need to decide whether the shape of these 12 paired differences gives reasonable support for a paired t procedure. The question here is about the shape condition, not whether the setting increases battery life.
Plan. Since there are only 12 pairs, inspect a graph of the differences for strong skewness, multiple clusters, and pronounced outliers. The repeated measurements on each phone form the pairs; the 12 differences, not the 24 individual battery-life measurements, are the data for this shape check.
Do. The dotplot shows one main cluster around zero. The values extend in both directions, with no pronounced isolated point and no long tail on just one side. The two sides are not perfectly mirror images, but the pattern looks reasonably balanced overall. This is a visual judgment, not a calculation or a proof that the population distribution is Normal.
Conclude. The dotplot of the 12 differences shows no strong skewness or pronounced outlier, so the shape evidence is reasonably supportive of a paired t procedure. The full decision still requires the design and independence checks described in “Conditions for a Paired t Procedure.”
Worked Example: Sound-Level Changes After Installing Panels
In a fictional acoustics project, 12 practice rooms are measured before and after sound-dampening panels are installed. Define \(d_i=\text{after}-\text{before}\), in decibels. The observed differences are \(-7,-5,-4,-3,-2,-1,0,1,2,3,4,8\). A dotplot shows most values from \(-7\) through \(4\), with a gap between \(4\) and \(8\), and one dot at \(8\).
State. We are assessing whether the graph of these 12 differences supports the shape condition for a paired t procedure.
Plan. With a small sample, inspect the dotplot for overall balance and for observations that are separated from the main group. Describe the gap accurately, then consider whether the value at \(8\) looks unusually separated.
Do. The values do not fill in the range from \(-7\) to \(8\): there are no observations at \(5\), \(6\), or \(7\). The gap from \(4\) to \(8\) makes the value \(8\) stand apart from the rest, so it is a possible high outlier and a reason for caution. The dotplot does not, by itself, establish that \(8\) is an error or prove that it comes from a different process. It does show that this small sample has a potentially unusual high value; the rest of the data also are not perfectly balanced around a center.
Conclude. The graph raises a substantial shape concern because the sample is small and the value \(8\) is separated from the main group by the gap from \(4\) to \(8\). The observed pattern does not provide reassuring support for a paired t procedure. Before proceeding, investigate the measurement and study context if possible; do not simply omit the value or declare the differences approximately Normal.
Worked Example: Turbidity Changes in Paired Water Samples
A fictional environmental class collects eight water samples and measures each sample’s turbidity before and after a treatment. Define \(d_i=\text{after}-\text{before}\), in the study’s turbidity units. The differences are \(1,1,2,2,3,4,5,7\). A dotplot shows repeated values at \(1\) and \(2\), followed by progressively fewer observations at larger values.
State. Decide whether the distribution of these eight differences is suitable in shape for a paired t procedure.
Plan. Because the sample is very small, use a dotplot to examine the differences for symmetry, skewness, and possible extreme values. Do not substitute a graph of the before or after measurements.
Do. Most values are near the low end, and the observations extend toward larger values, forming a longer right tail. The distribution is not roughly symmetric; the pattern suggests right skewness. There is no need to label \(7\) a definite outlier to recognize the overall imbalance.
Conclude. The dotplot of these eight differences shows noticeable right skewness. Since the sample is small, this graph does not provide reasonable support for the shape condition for a paired t procedure. A different method or more data may be considered, but the graph alone does not justify treating the differences as approximately Normal.
Common Mistakes and AP Exam Tips
A strong condition check reports what the graph actually shows and connects that evidence to the small sample size. Avoid turning a visual judgment into an absolute claim: “the differences are Normal” overstates what a dotplot or histogram can establish. Prefer wording such as “the dotplot is roughly symmetric with no pronounced outliers, so the shape condition appears reasonable for this small sample.”
- Checking the original measurements. The paired t procedure uses the differences. A graph of before values or after values does not answer whether the differences have a suitable shape.
- Calling the graph proof. A graph can provide evidence compatible with Normality, but it cannot prove the population distribution is Normal. Say what you see and qualify the conclusion.
- Calling every endpoint an outlier. The largest or smallest observation is not automatically an outlier. Look at its separation from the other values and the overall pattern; describe a possible outlier when the evidence is suggestive.
- Ignoring an actual gap. If values jump from \(4\) to \(8\), report that gap. Do not describe the observations as filling the interval between those values. Then explain whether the separated value raises concern.
- Using the sample-size rule to dismiss a warning. A larger sample can make t procedures more tolerant of moderate departures from Normality, as covered earlier in the course. It does not make a pronounced outlier irrelevant. For a small sample, shape deserves particular attention.
- Overinterpreting a histogram. Bins can change the apparent shape, especially when there are few observations. Check individual values with a dotplot or stemplot when possible.
- Confusing a shape check with the complete conditions check. A favorable graph does not establish random selection, independence, or the 10% condition. Address those separately, as in “Conditions for a Paired t Procedure.”
A full-credit shape statement identifies the graph, the data shown, and the relevant visual evidence. For example: “The dotplot of the 12 after-minus-before differences is roughly balanced around its center, with no pronounced outlier. Although the sample is small, this pattern provides reasonable support for the shape condition.” If the graph instead shows a separated value or strong skewness, name that feature and explain why it weakens the case for using a paired t procedure.
Check Your Understanding
For each question, focus on what the graph of the paired differences can and cannot tell you.
- Why should a paired t shape check use one difference per pair rather than the two original measurement columns?
- A dotplot of 10 differences has one main cluster, roughly balanced sides, and no pronounced outlier. What cautious conclusion can you make about the shape condition?
- A small-sample dotplot has values through \(4\), then a value at \(8\), with no observations between. What should you say about the gap and the value at \(8\)?
- What is one advantage of a dotplot over a histogram when the sample of differences is small?
- A Normal probability plot has a curved pattern. What concern might that suggest, and why should you avoid saying that the plot proves the population distribution is non-Normal?