Check the Pairs Before Using a Paired t Procedure
A paired t procedure analyzes one difference for each pair. Before using it, you need evidence that the study design supports inference and that the differences are suitable for a t procedure. This tutorial focuses on three checks: random selection or an appropriate randomized process, the 10% condition when sampling without replacement, and the shape of the differences.
As in “Conditions for One-Sample Versus Paired Data,” the data are not treated as two unrelated samples. The design creates the pairs, and the analysis focuses on the list \(d_1,d_2,\ldots,d_n\). Here, \(n\) is the number of pairs—not the total number of measurements. This matters for both the 10% comparison and the Normality check.
Match Each Condition to the Right Evidence
Random selection or an appropriate randomized process. Look for a stated chance method used to select the individuals or objects that form the pairs. A random sample supports generalizing results to the population from which those pairs were selected. An appropriately randomized experiment can support conclusions about the treatment effect for the experimental units, but random assignment is not the same as random sampling. As discussed in “Random Assignment Versus Random Sampling Conditions,” the design determines which kind of conclusion is supported.
If the study says that participants volunteered, were conveniently available, or were chosen by staff judgment, that is not evidence of random selection. A large sample or a roughly Normal-looking set of differences cannot turn a convenience sample into a random sample. Be specific about what the design permits: without random selection, generalizing to a broader population may not be justified.
The 10% condition. When a random sample is selected without replacement from a finite population, observations can be dependent because selecting one unit changes which units remain available. The 10% condition is a practical check that this dependence is small enough to treat different sampled units’ differences as independent. Compare the number of pairs with the population size. Do not count the two measurements in each pair separately.
Here \(n\) is the number of pairs and \(N\) is the size of the population from which those pairs were sampled. Equivalently, show that \(n/N\leq0.10\). If the population is much larger than the sample, the condition is met. If the sample is more than 10% of the population, do not claim that the condition is met; explain the concern.
The condition does not prove independence by itself. The design should also make it reasonable that different pairs do not share a unit or otherwise influence one another. For example, two measurements on one person belong to the same pair; they are not two independent observations. Instead, assess whether the different people or objects contributing the pairs are independent, as explained in “Conditions for One-Sample Versus Paired Data.”
Normality of the differences. Assess the distribution of the \(d_i\) values—not the before measurements by themselves and not the after measurements by themselves. A paired t procedure uses the mean and standard deviation of the differences, so the shape condition applies to those differences. With a small sample, use a dotplot, stemplot, histogram, or Normal probability plot of the differences and look for strong skewness, pronounced outliers, or other substantial departures from an approximately Normal pattern.
For a larger sample, t procedures are more tolerant of departures from Normality. “Robustness of t Procedures” explains that this tolerance depends on the size and shape of the sample: a larger sample can often handle moderate skewness, but pronounced outliers remain a concern. Do not use a large-sample guideline to ignore an extreme value, and do not use the shape of either original measurement column as a substitute for examining the differences.
A Practical Condition-Check Sequence
Identify the repeated measurements or matched units. State the subtraction order so that every \(d_i\) has a clear meaning.
Identify how the pairs were selected or assigned. Explain what random selection or random assignment supports, and consider whether different pairs are independent.
If pairs were sampled without replacement, compare \(n\), the number of pairs, with \(0.10N\), where \(N\) is the population size.
Use a graph of the \(d_i\) values, paying attention to sample size, skewness, and outliers. Explain whether the evidence supports using a paired t procedure.
This sequence is an evidence audit, not a search for a single magic number. “Verifying Conditions From a Described Study” and “Writing Condition Checks in Full Sentences” emphasize the same habit: name each condition, cite the relevant study detail, and explain what that detail supports.
Worked Examples
Worked Example: Daily Energy Use in Sampled Buildings
A fictional energy team wants to estimate the mean change in daily electricity use after a building upgrade. The team randomly selects 12 buildings from 180 eligible buildings and records each building’s use before and after the upgrade. Define \(d_i=\text{after}_i-\text{before}_i\), in kilowatt-hours per day. The 12 observed differences are \(-8,-6,-5,-3,-2,-1,0,1,2,3,5,7\). A dotplot of these differences is roughly balanced around zero, with no pronounced outlier.
State. The question concerns the true mean after-minus-before daily electricity-use difference, in kilowatt-hours per day, for the eligible buildings. We are checking whether a paired t procedure is appropriate for these data; this condition check alone does not say whether the mean changed.
Plan. Use the differences because the same building was measured twice. Check the selection method, the 10% condition, independence between buildings, and the distribution of the differences.
Do. The buildings were randomly selected from the 180 eligible buildings, so the selection supports generalizing to that population. There are 12 pairs, and \(0.10(180)=18\); since \(12\leq18\), the 10% condition is met. Each pair comes from a different building, so independence between pairs is reasonable if the buildings’ energy-use changes do not influence one another. The sample has only 12 differences, so the shape check matters. The stated dotplot is roughly balanced and has no pronounced outlier, providing reasonable support for the shape condition.
Conclude. The random selection, 10% comparison, plausible independence between buildings, and graph of the differences support using a paired t procedure to study the mean change for eligible buildings. The conclusion is about mean energy-use change; this conditions check does not establish that the upgrade caused any change.
Worked Example: Growth Measurements for Seedlings
In a fictional greenhouse study, researchers randomly select 36 seedlings from a population of 400 seedlings. They measure each seedling’s height before and after a treatment period. Let \(d_i=\text{after height}-\text{before height}\), in centimeters. A histogram of the 36 differences shows a mild right skew and no isolated extreme values.
Check the design and independence. The 36 seedlings were randomly selected, supporting generalization to the stated population. Each seedling contributes one difference, and no seedling is counted in more than one pair. If the seedlings’ growth responses do not affect one another, treating the differences from different seedlings as independent is reasonable.
Check the 10% condition. Ten percent of the population is \(0.10(400)=40\). The sample has 36 pairs, and \(36\leq40\), so the condition is met. The count is 36 seedlings (pairs), not 72 height measurements.
Check the shape. The sample has 36 differences, so it is larger than 30. The differences show only mild skewness and no isolated extreme values. In light of the robustness of t procedures for a larger sample, this evidence is generally adequate for the shape condition. The mild skew should still be reported rather than described as perfect Normality.
Conclusion. The design supports inference to the greenhouse population, the 10% condition is met, and the sample size and graph provide reasonable support for a paired t procedure. If the graph had shown a pronounced outlier, the larger sample alone would not make that concern disappear.
Worked Example: Volunteer Runners and Recovery Time
A fictional running club has 70 members. Nine members volunteer for a study that records each person’s recovery time after a run, first without and then with a new routine. Define the difference as with-routine time minus without-routine time, in minutes. The nine differences are \(-5,-3,-2,-1,0,0,1,2,3\). A dotplot looks fairly balanced, with no pronounced outlier.
Check the selection condition. These nine runners volunteered; they were not randomly selected. The differences may describe the volunteers, but the design does not provide the usual support for generalizing to all 70 club members. The roughly balanced dotplot does not fix that limitation.
Check the 10% comparison. Ten percent of 70 is \(0.10(70)=7\). The study includes 9 runners, and \(9>7\), so the sample is more than 10% of the club. The 10% condition is not met for treating a sample selected without replacement from this finite population as independent. In this example, the selection was voluntary rather than a random sample, so the random-selection condition is already not met as well.
Check the differences’ shape. The dotplot is reasonably balanced and has no pronounced outlier, which is favorable shape evidence for these nine differences. That favorable feature does not override the two design concerns. A procedure’s conditions must be considered separately; meeting the shape condition does not make the selection random or satisfy the 10% condition.
Conclusion. A paired t procedure for generalizing to all club members is not justified by this design. The results can be described for the nine volunteers, but the condition check does not support the same population inference as a suitable random sample would.
Common Mistakes and Full-Credit Wording
Condition checks earn credit when they connect evidence to a specific condition. A statement such as “the conditions are met” is not enough by itself. In a full response, name the selection method, show the population comparison when it applies, and identify what graph or sample-size information supports the shape assessment.
- Using the two measurement columns to check Normality. The paired t procedure analyzes the differences. A before column or an after column may be skewed even when the differences are reasonably well behaved, or the reverse. A full-credit answer identifies a graph of the differences.
- Counting measurements instead of pairs. If 12 people each provide two measurements, there are 12 pairs and 12 differences. For the 10% condition, compare 12 with the population size—not 24.
- Assuming that “random” answers every condition. Random selection is relevant evidence, but still check the population comparison and the differences’ shape. Also consider whether different pairs are independent.
- Using the 10% condition to claim random selection. A sample can be smaller than 10% of a population and still be a convenience sample. The percentage concerns dependence when sampling without replacement; it does not establish how the sample was selected.
- Claiming the differences are Normal because \(n\geq30\). A larger sample supports using t procedures despite some departures from Normality, but it does not prove the differences are Normal or excuse a pronounced outlier. Describe the actual evidence and its limitations.
- Ignoring the target population. “The results generalize” is incomplete unless the population is named and the selection method supports that claim. Volunteers do not automatically represent all members of a group.
A concise but complete check might say: “The 12 pairs were randomly selected from 180 eligible buildings, supporting inference to that population. Since \(12\leq0.10(180)=18\), the 10% condition is met. The buildings contribute separate differences, and the dotplot of the 12 differences is roughly balanced with no pronounced outlier, so the design and shape provide reasonable support for a paired t procedure.”
Check Your Understanding
For each question, focus on the evidence relevant to the paired t conditions.
- A random sample of 15 pairs is selected without replacement from a population of 200. Calculate 10% of the population and decide whether the 10% condition is met.
- Why should a paired t procedure’s shape check use the differences rather than the two original measurement columns?
- A study includes 20 randomly selected people from a population of 500, with two measurements per person. What sample size should be used in the 10% comparison?
- A sample of volunteers has a roughly symmetric distribution of differences. Does that shape evidence establish that the sample was randomly selected? Explain.
- A paired study has 34 differences with mild skewness and no pronounced outliers. What feature of the sample supports using a t procedure, and what limitation should still be mentioned?