From a Summary Table to a Comparison Paragraph
A table can put two groups’ summary statistics side by side, but the reader still needs you to explain what the numbers show. A strong comparison paragraph does more than list values: it identifies the groups and variable, compares appropriate measures of center and spread, quantifies important differences, and interprets those differences in context.
In Comparing Groups of Different Sample Sizes, you practiced comparing relative frequencies with the correct group totals. Here, the summaries are already calculated. Your task is to choose which ones best answer the question and combine them into a clear comparison. The earlier tutorials on comparing centers, comparing spread, and matching measures provide the tools; this tutorial focuses on assembling them into a complete paragraph.
A summary table may include sample sizes, means, standard deviations, medians, IQRs, or five-number summaries. Before writing, identify what the question asks you to compare and which measures are appropriate. As explained in Matching Measures: Mean With Standard Deviation, Median With IQR, mean and standard deviation are a consistent pair for distributions that are reasonably symmetric without strong outliers. Median and IQR are generally more appropriate for skewed distributions or distributions with strong outliers.
A table of numerical summaries does not, by itself, show the full shape of either distribution. It may not establish whether a distribution is symmetric, skewed, or multimodal, and a table that omits individual values may not reveal every unusual feature. Use shape or outlier information only when it is supplied by a graph, description, or other evidence. A complete paragraph is not one that invents missing details; it is one that makes a thorough comparison using the evidence available.
A Reliable Structure for Your Paragraph
Use the following sequence as a planning tool. The final response should usually read as one connected paragraph, not as a list of calculations. You do not need to report every statistic in the table: include the measures that directly support a useful comparison.
Make clear what was measured and which two groups are being compared. Include units when they are available.
Use mean and standard deviation, or median and IQR, as appropriate to the distribution information. Do not mix measures from different pairs without a reason.
Subtract matching values in a stated order. Say which group has the greater center or spread and by how much, including units.
Use any provided information about shape or unusual values, and close by describing what the comparison says about these groups. Do not claim a cause from summaries alone.
Sample sizes help readers understand how many observations are represented in each group. They are useful context, but a larger sample size does not automatically mean a larger center or greater spread. Keep the comparison focused on the variable and measures relevant to the question.
Worked Example: Comparing Mean Sleep Time
Worked Example: Comparing Mean Sleep Time
A fictional school survey records the number of hours of sleep students report on a typical school night. Students are grouped according to whether they usually put their phones away at least 30 minutes before bed. The table gives summaries for 36 students who do so and 44 students who do not. The accompanying description says both distributions are roughly symmetric with no pronounced outliers.
| Phone routine | Sample size | Mean sleep (hours) | Standard deviation (hours) | Median (hours) | IQR (hours) |
|---|---|---|---|---|---|
| Puts phone away | 36 | 7.1 | 0.8 | 7.2 | 1.0 |
| Does not put phone away | 44 | 6.6 | 1.0 | 6.7 | 1.4 |
Choose the measures. Since the distributions are described as roughly symmetric with no pronounced outliers, compare the means and standard deviations. The sample sizes and medians and IQRs are available, but listing every number would make the paragraph less focused. The mean difference is \(7.1-6.6=0.5\) hour. The standard deviation difference is \(1.0-0.8=0.2\) hour.
Write the comparison. “Among the surveyed students, those who usually put their phones away at least 30 minutes before bed reported a greater mean amount of school-night sleep than those who did not: 7.1 hours compared with 6.6 hours, a difference of 0.5 hour. The reported sleep times also had less spread in the phone-away group, with a standard deviation of 0.8 hour compared with 1.0 hour, a difference of 0.2 hour. Both distributions were described as roughly symmetric with no pronounced outliers. These summaries describe an association in the surveyed students; they do not establish that putting a phone away causes students to sleep longer.”
This paragraph names the groups and variable, compares matched measures, includes units, and uses the supplied shape information. It also limits its conclusion to the surveyed students and avoids turning an observed association into a causal claim.
Worked Example: Using Median and IQR for Skewed Data
Worked Example: Using Median and IQR for Skewed Data
A fictional recreation center summarizes the number of minutes visitors wait to check in during two time periods. There are 40 morning visitors and 55 evening visitors in the samples. A description accompanying the table says both distributions are right-skewed, with some especially long waits.
| Check-in period | Sample size | Mean wait (minutes) | Standard deviation (minutes) | Median wait (minutes) | IQR (minutes) |
|---|---|---|---|---|---|
| Morning | 40 | 15.4 | 10.2 | 12 | 9 |
| Evening | 55 | 24.1 | 17.6 | 18 | 16 |
Choose the measures. Because the wait-time distributions are right-skewed and include some long waits, compare the medians and IQRs. The median difference is \(18-12=6\) minutes, so the evening median is higher. The IQR difference is \(16-9=7\) minutes, so the middle half of evening waits is more spread out by 7 minutes.
Write the comparison. “The sampled evening visitors generally waited longer to check in than the sampled morning visitors: the median wait was 18 minutes in the evening and 12 minutes in the morning, a difference of 6 minutes. The middle half of evening waits was also more variable, with an IQR of 16 minutes compared with 9 minutes in the morning, a difference of 7 minutes. Both distributions were described as right-skewed, so the median and IQR provide useful comparisons of their centers and middle-half spreads. These results summarize the visitors in the two samples.”
Notice that this paragraph does not describe the mean difference, even though the means are in the table. The mean and standard deviation are more affected by long waits, so the median and IQR are the more suitable pair for this comparison. The point is not to hide statistics; it is to select the statistics that best represent the distributions.
Worked Example: Comparing a Difference in Context
Worked Example: Comparing a Difference in Context
A fictional environmental club measures the mass of recyclable material collected during one weekly pickup on two routes. The table summarizes 25 pickups on Route Pine and 40 pickups on Route Lake. The accompanying description says both distributions are approximately symmetric and have no strong outliers.
| Route | Sample size | Mean collected (kg) | Standard deviation (kg) |
|---|---|---|---|
| Pine | 25 | 18.6 | 3.2 |
| Lake | 40 | 21.3 | 4.0 |
Calculate matching differences. Route Lake has the greater mean. In the order Lake minus Pine, the mean difference is \(21.3-18.6=2.7\) kg. Route Lake also has the greater standard deviation, by \(4.0-3.2=0.8\) kg.
Write the comparison. “For the sampled weekly pickups, Route Lake collected a greater mean mass of recyclable material than Route Pine: 21.3 kg compared with 18.6 kg, a difference of 2.7 kg per pickup. The collected masses also showed more variability on Route Lake, with a standard deviation of 4.0 kg compared with 3.2 kg on Route Pine, a difference of 0.8 kg. The two distributions were described as approximately symmetric without strong outliers, supporting a comparison using means and standard deviations. These summaries describe the observed pickups; they do not show that the route itself caused the difference.”
The phrase “in the order Lake minus Pine” makes the subtraction direction clear. Naming the variable and its units prevents the figures from becoming detached from their meaning. In this example, the variable is mass per pickup, so “2.7 kg per pickup” is clearer than “2.7 more.”
What the Table Does Not Tell You
Summary statistics compress information. Two groups can have the same mean and standard deviation while having different shapes, clusters, or gaps. A table containing only \(n\), mean, and standard deviation cannot establish that the distributions are symmetric or that they have no outliers. Likewise, median and IQR alone do not identify every feature of a distribution.
If a graph or written description is supplied, use it to discuss shape and unusual features, as in the examples above. If it is not supplied, limit the paragraph to what the table supports. For example, “The table does not provide enough information to compare the distributions’ shapes” is more accurate than guessing that both groups are symmetric. The earlier tutorial Comparing Distributions Needs All Four Features emphasizes considering shape, center, spread, and unusual values; a summary table may support some of these features better than others.
A numerical difference is also not, by itself, an explanation. If one group has a higher mean, the summary tells you how the observed centers compare, not why they differ. Unless the study design and evidence justify a causal conclusion, describe the difference between the groups rather than saying one group’s characteristic caused the other outcome.
Common Mistakes and AP Exam Tips
- Listing values without making a comparison. “The means are 7.1 and 6.6” gives evidence but leaves the reader to interpret it. State which group has the greater mean and by how much.
- Subtracting unmatched measures. Comparing one group’s mean with the other group’s median does not produce a standard difference in centers. Use the same measure for both groups.
- Choosing a measure without considering distribution shape. For skewed data or strong outliers, the median and IQR are often more useful than the mean and standard deviation. Use any shape information provided to justify the choice.
- Giving a difference without units or context. “The difference is 6” is incomplete. Say that the evening median wait is 6 minutes greater than the morning median wait.
- Claiming more than the table shows. Do not invent shape, outliers, or a cause. If those details are absent, state only what the summaries support.
- Reporting every available statistic. A long list can obscure the comparison. Select the measures that answer the question, and include other statistics only when they add useful information.
For a strong response, a grader should be able to identify the groups, the variable, the measures being compared, the direction and size of each difference, and the meaning of the comparison in context. Use exact values from the table, report units, and make sure every statement about shape or unusual values has evidence behind it.
Check Your Understanding
Use the summary information to plan or write a comparison. When shape information is missing, do not assume it.
- Group A has a mean of 14.2 points and standard deviation of 2.1 points. Group B has a mean of 12.8 points and standard deviation of 2.7 points. Find both differences and write one sentence comparing the groups.
- Two right-skewed distributions have medians of 9 minutes and 13 minutes, and IQRs of 5 minutes and 8 minutes. Which group has the greater median, and which has the greater middle-half spread? Quantify both differences.
- A table gives sample sizes, means, and standard deviations but no graph or shape description. Can you conclude that the groups are symmetric? Explain what the table does and does not support.
- A paragraph says, “Route X has a mean of 20 kg and Route Y has a mean of 17 kg, so Route X caused more material to be collected.” Identify the unsupported part and rewrite the comparison without making a causal claim.
- A table provides means, standard deviations, medians, and IQRs. What information about the distributions should guide your choice of which pair to compare?