For Paired Data, the Differences Are the Data
When each person or object contributes two related measurements, the two sets of measurements are not independent samples. The values belong together in pairs. A paired t procedure accounts for that connection by turning each pair into one difference. The resulting list of differences is the data used for inference about a mean.
This changes where you check the conditions. As explained in “Checking the Normal/Large Sample Condition” and “Robustness of t Procedures,” shape matters for mean inference. For a paired t procedure, the relevant shape is the shape of the differences, not the shape of the first measurements or the second measurements considered separately. The observations whose independence matters are also the pairs, rather than the individual measurements within each pair.
Recognize the Pairing and Define the Difference
Data are paired when there is a meaningful reason to match two measurements. Common examples include measurements taken from the same person before and after a treatment, measurements from the same object under two conditions, or measurements from two similar individuals deliberately matched in pairs. Having two groups, by itself, does not make data paired.
Choose an order for subtraction and use it consistently. For before-and-after measurements, for example, define the difference as “before minus after” or “after minus before.” Either choice is valid, but the sign and interpretation change with the choice. Each pair contributes exactly one difference, so if there are \(n\) pairs, the sample size for the paired t procedure is \(n\), not \(2n\).
The parameter is \(\mu_d\), the population mean of the differences defined in that same order. For example, if \(d_i\) is a participant’s resting heart rate before a program minus their resting heart rate afterward, then \(\mu_d\) is the mean before-minus-after change in resting heart rate for the population represented by the study.
Pairing matters because measurements from the same person or matched pair may be related. Treating all \(2n\) measurements as if they came from unrelated observations ignores that relationship. The paired method instead asks about the variation in the within-pair differences.
Check Conditions on the Difference List
The conditions have familiar names from earlier tutorials, but the unit of analysis is now the pair. “Checking the Random Condition” and “Checking the 10% Condition for Independence” explain the general ideas; apply them to how pairs were selected or assigned.
The connection within a pair is expected and is the reason for pairing. Independence is assessed between pairs. For example, two measurements from one participant need not be independent of each other; instead, the paired procedure combines them into one difference. The question is whether one participant’s difference can reasonably be treated as independent of another participant’s difference.
If pairs were randomly sampled without replacement from a finite population of pairs, apply the 10% condition to the number of pairs sampled and the number of pairs in that population. Do not count each measurement as a separate sampled unit for this check. In a randomized experiment, describe what was randomized and explain what that randomization supports, as in earlier tutorials on randomization.
The Normal/Large Sample condition also refers to the differences. If \(n\geq30\), the AP large-sample route is met for the differences. If \(n<30\), inspect a graph of the differences for strong skewness or pronounced outliers, using the graph-reading skills from “Graphing Sample Data to Check Normality” and “Identifying Skewness and Outliers in Small Samples.” A graph of the before measurements or after measurements alone does not answer this question.
A Condition-Checking Workflow
A clear paired-data response first establishes that pairing is appropriate and then checks the conditions for the differences. Use this sequence before making calculations or conclusions from a paired t procedure.
Identify what makes the two measurements in each pair related, such as being taken from the same person. If there is no defensible matching, the data are not paired.
State the subtraction order and identify \(\mu_d\), the population mean difference in that order. The sample size is the number of pairs.
Describe how the pairs were obtained or assigned. If sampling without replacement, apply the 10% condition to pairs. Consider whether differences from separate pairs are independent.
Use \(n\geq30\) for the large-sample route, or assess the distribution of the differences when \(n<30\). State what the graph shows rather than relying on the raw measurements.
The order is useful: first make sure there is real pairing, then define the quantity whose mean is being studied. Only after that can you correctly identify the sample size, the independence units, and the data whose shape must be assessed.
Worked Examples
Worked Example: Before-and-After Measurements With a Large Number of Pairs
A study team randomly selects 36 adults from a large community to record their resting pulse rate before and after a six-week walking program. Each adult has both measurements. Define \(d=\text{before}-\text{after}\), so a positive difference represents a lower pulse rate after the program. A graph of the 36 differences shows a slight right tail but no pronounced outliers. The community has more than 360 adults.
State. Let \(\mu_d\) be the mean before-minus-after change in resting pulse rate, in beats per minute, for the population of adults represented by the random sample. We are checking whether a paired t procedure for \(\mu_d\) is appropriate.
Plan. The data are paired because each adult contributes a before measurement and an after measurement. There are \(n=36\) differences, one for each adult. The adults were randomly selected, supporting generalization to the population from which they were selected. The 10% condition is met: 36 is no more than 10% of a population larger than 360, so it is reasonable to treat differences from different sampled adults as independent. The sample size meets the large-sample route because \(36\geq30\).
Do. The relevant graph is the graph of the 36 differences, not separate graphs of the before and after rates. It shows only a slight right tail and no pronounced outliers. Thus, the shape evidence raises no strong concern, and the large-sample route is met as well. The within-adult measurements are intentionally related; the independence condition concerns differences from separate adults.
Conclude. The paired t conditions are reasonably supported for inference about the mean before-minus-after change in resting pulse rate. The random sample supports generalizing to the represented adult population, the 10% condition supports independence between adults’ differences, and the differences’ shape and sample size support the t procedure. This conclusion is about the mean difference, not separate inference on the two raw measurement sets.
Worked Example: A Small Sample With Reassuring Differences
A technician randomly selects 10 rechargeable batteries from a large shipment and measures each battery’s operating time under two settings. Define \(d=\text{setting A time}-\text{setting B time}\), in hours. The 10 observed differences are:
\(-4,\ -3,\ -2,\ -1,\ 0,\ 0,\ 1,\ 2,\ 3,\ 4\)
A dotplot and Normal probability plot of the differences show a roughly symmetric, single-cluster pattern, with no gaps or pronounced outliers. The shipment contains more than 100 batteries.
State. Let \(\mu_d\) be the population mean difference in operating time, in hours, for setting A minus setting B. We are assessing whether a paired t procedure for \(\mu_d\) is reasonable.
Plan. Each battery was measured under both settings, so the measurements are naturally paired. The sample contains 10 batteries and therefore 10 differences. The random selection supports generalizing to the shipment. Since 10 is no more than 10% of a shipment containing more than 100 batteries, the 10% condition supports independence between sampled batteries’ differences.
Do. Here \(n=10<30\), so the large-sample route is not met. We must assess the small-sample shape condition from the differences. The stated dotplot and Normal probability plot show a roughly symmetric, single-cluster pattern without gaps or pronounced outliers. That evidence is reasonably consistent with an approximately Normal distribution of differences.
Conclude. The paired t conditions are reasonably supported for this sample. In particular, the shape condition is supported by the graphs of the 10 differences, not by a claim about the appearance of the measurements under setting A or setting B separately. The sample size for the paired procedure is 10, the number of batteries, rather than 20 measurements.
Worked Example: Raw Measurements Do Not Rule Out Problematic Differences
A small pilot project records a measurement under two conditions for each of eight objects. Each condition’s raw values are the same set: 100, 102, 104, 106, 108, 110, 112, and 114 units. The values are paired as follows. Define \(d=\text{condition A}-\text{condition B}\).
| Object | Condition A | Condition B | Difference \(d\) |
|---|---|---|---|
| 1 | 100 | 114 | -14 |
| 2 | 102 | 100 | 2 |
| 3 | 104 | 106 | -2 |
| 4 | 106 | 104 | 2 |
| 5 | 108 | 110 | -2 |
| 6 | 110 | 108 | 2 |
| 7 | 112 | 102 | 10 |
| 8 | 114 | 112 | 2 |
The separate raw-value displays contain the same evenly spaced values and have no isolated observations. The differences, however, include \(-14\) and \(10\), while the other values are between \(-2\) and \(2\). Suppose these eight objects were randomly selected from a large population.
State. Let \(\mu_d\) be the population mean condition-A-minus-condition-B difference, in units, for the population represented by the sample. We are assessing the paired t conditions.
Plan. The two measurements are paired by object, so a difference for each object is appropriate. There are eight pairs. The random selection supports generalization, and the population is large enough that eight sampled objects are no more than 10% of it, supporting independence between objects’ differences.
Do. Because \(n=8<30\), inspect the eight differences rather than relying on the raw-value displays. The two extreme differences are far from the main group of values, so the differences show pronounced outliers. The raw measurements’ lack of isolated values does not remove this concern: pairing determines which values are subtracted, and those pairings can produce a difference distribution with a different shape.
Conclude. The random and independence conditions are supported, but the small sample’s differences show pronounced outliers. The shape evidence does not support relying on a paired t procedure here. Separate graphs of the raw measurements cannot substitute for checking the distribution of the within-pair differences.
Worked Example: Two Groups Are Not Automatically Paired
A school compares the weekly study hours of 18 randomly selected students in an evening program with the weekly study hours of 18 different randomly selected students in a daytime program. Each group’s observations appear roughly symmetric, but no student is matched to a particular student in the other group.
State. The question concerns a comparison of mean weekly study hours for two student populations. We are deciding whether a paired t procedure is appropriate.
Plan. A paired t procedure requires a meaningful pair of observations for each difference. Here, each student contributes one measurement, and students in one group are not matched to students in the other group.
Do. There is no defensible way to subtract one evening-program student’s value from a particular daytime-program student’s value. The roughly symmetric appearance of each group’s raw values does not create pairs. The data are two separate samples, so there is no single list of meaningful within-pair differences to check.
Conclude. A paired t procedure is not appropriate for this comparison. The groups are not paired; a two-sample t procedure is the relevant approach, with its conditions considered separately in “Conditions for Two-Sample t Procedures.”
Common Mistakes and AP Exam Tips
- Checking the raw measurements for the paired shape condition. For paired inference, assess the distribution of the differences. A strong answer names the difference variable and describes its graph.
- Counting measurements instead of pairs. If 24 people each provide two measurements, the paired sample size is 24 differences, not 48 observations.
- Demanding independence within each pair. The paired measurements may be related; that is why they are paired. Check independence between the differences from separate pairs.
- Calling any two groups paired. Pairing needs a meaningful match, such as repeated measurements on the same person or deliberate matching. Two unrelated samples do not become paired because they have equal sample sizes.
- Using “before minus after” inconsistently. State the subtraction order and keep it fixed. The order determines the sign and meaning of each difference and of \(\mu_d\).
- Claiming the conditions hold without evidence. Name the random process, apply the 10% condition to pairs when relevant, and describe the shape of the differences. For a small sample, mention specific graph features such as skewness or outliers.
A full-credit condition statement is specific: “The 12 people were randomly selected, and the population contains more than 120 people, so the 10% condition supports independence between their differences. Because \(n=12<30\), I examined a graph of the differences; it shows no strong skewness or pronounced outliers, so the Normal/Large Sample condition is reasonably supported.” If the graph instead shows outliers, say so directly, even if the raw measurements look well behaved.
Check Your Understanding
Answer each question by identifying the relevant unit or data distribution.
- A random sample of 25 people each records a measurement before and after a program. What is the paired sample size?
- For paired data with \(n=14\), which values should you graph to assess the Normal/Large Sample condition?
- A study samples 40 matched pairs without replacement from a population of 500 pairs. Does the 10% condition hold? Explain using pairs as the units.
- Two independent classes each provide 30 measurements, and each group’s data look approximately Normal. Is a paired t procedure appropriate solely because the group sizes are equal? Why or why not?
- In a paired study, explain why measurements from the same person may be related while independence is checked between people’s differences.