Why Category Definitions Matter
A two-way table summarizes how two categorical variables occur together. Its patterns depend partly on how the categories are defined. Combining categories can make a table shorter and easier to read, but it can also hide differences that were visible before grouping. In some cases, the direction of a comparison changes because the grouped category represents a different outcome.
Grouping does not change anyone’s recorded response. It changes which responses are counted together. Before interpreting either version of a table, check that the categories answer the question of interest and that the percentages use the same denominator.
For example, a survey might record responses as “never,” “sometimes,” “often,” or “always.” Combining “never” and “sometimes” produces a broader category, but it no longer shows how many people chose each of those original responses. A grouped table is therefore not simply a clearer version of the same detail: it answers a more general question.
How Counts and Percentages Change
To combine categories, add their counts within each group. If the goal is to compare conditional distributions, calculate percentages using the total for each group as the denominator. These are row percentages when groups appear in rows. Comparing raw counts alone can be misleading when the groups have different sizes.
When two groups are combined, their conditional percentages do not usually have equal influence on the new percentage. The larger group contributes more observations, so its conditional percentage receives more weight. Suppose group 1 has \(n_1\) observations and a proportion \(p_1\) in a category, while group 2 has \(n_2\) observations and a proportion \(p_2\). The combined proportion is
This follows directly from the definition of a proportion. The number of observations in the category is \(n_1p_1\) in the first group and \(n_2p_2\) in the second. Add those counts, then divide by the combined group size \(n_1+n_2\). This is a weighted average, not generally the simple average of \(p_1\) and \(p_2\).
Worked Example: Combining Groups of Different Sizes
In a hypothetical survey, 20 of 50 people in Group A choose a particular response, while 45 of 150 people in Group B choose it. Find the percentage in that response after combining the two groups, and compare it with the unweighted average of their percentages.
In Group A, the proportion is \(20/50=0.40\), or 40%. In Group B, it is \(45/150=0.30\), or 30%. The combined count in the response is \(20+45=65\), and the combined sample size is \(50+150=200\). Thus,
The weighted-average calculation gives the same result:
The simple average, \((40\%+30\%)/2=35\%\), is not the combined percentage because the groups do not have equal sizes. Group B contributes three times as many observations as Group A.
Grouping Can Hide Differences
A grouped table can conceal a pattern when it merges categories that differ in ways relevant to the question. The combined percentages may match even though the original conditional distributions do not. The next example uses hypothetical survey counts, with 100 people in each group, so the counts are also the percentages within each group.
Worked Example: A Grouping That Hides a Difference
A hypothetical survey records how often people in Groups A and B take part in an activity. Compare the original response distributions with a table that combines “Never” and “Sometimes” into “Not often.”
| Group | Never | Sometimes | Often | Total |
|---|---|---|---|---|
| A | 20 | 60 | 20 | 100 |
| B | 40 | 40 | 20 | 100 |
In the original table, Group A has 20% “Never,” 60% “Sometimes,” and 20% “Often.” Group B has 40%, 40%, and 20%, respectively. The distributions differ: Group B has a larger percentage who never take part, while Group A has a larger percentage who sometimes take part. The percentage who take part often is the same.
After combining “Never” and “Sometimes,” the new counts are \(20+60=80\) for Group A and \(40+40=80\) for Group B. The grouped conditional distributions are therefore:
| Group | Not often | Often | Total |
|---|---|---|---|
| A | 80 (80%) | 20 (20%) | 100 |
| B | 80 (80%) | 20 (20%) | 100 |
The grouped table shows no difference between the groups in the percentage who take part often versus not often. That does not mean the original response patterns were identical. The grouping removed the distinction between “Never” and “Sometimes,” which was where the difference appeared.
A lack of visible difference in a grouped table should therefore be described narrowly. Here, the groups have the same percentage in the two broad categories; they do not have the same distribution across the original response categories. A useful check is to ask whether the categories being merged are interchangeable for the question being asked. If they are not, grouping may hide information readers need.
Grouping Can Strengthen a Selected Contrast
When several categories show differences in the same direction, combining them can make the difference for a broader outcome more prominent. The resulting contrast is legitimate if the combined outcome is meaningful, but it must not be presented as though it describes each original category separately. A broad category can focus attention on a chosen threshold while discarding distinctions within it.
Worked Example: A Larger Difference for a Combined Outcome
In a hypothetical survey, respondents rate a symptom as “None,” “Mild,” “Moderate,” or “Severe.” Each group contains 100 respondents.
| Group | None | Mild | Moderate | Severe | Total |
|---|---|---|---|---|---|
| A | 28 | 28 | 28 | 16 | 100 |
| B | 20 | 20 | 20 | 40 | 100 |
For each of the first three categories, Group A’s percentage is 8 percentage points higher than Group B’s: \(28\%-20\%=8\) percentage points. If the question is whether respondents report anything below “Severe,” combine “None,” “Mild,” and “Moderate.” Group A has \(28+28+28=84\) respondents in that combined category, compared with \(20+20+20=60\) in Group B. The conditional percentages are 84% and 60%, a difference of \(84\%-60\%=24\) percentage points.
The grouped comparison makes the difference for “below Severe” larger than the difference in any one of its three component categories. It is a valid description of that combined outcome, but it does not mean that the original table shows a 24-point difference in each response category. The full table also shows that Group B has a higher severe-symptom percentage: 40% compared with 16% in Group A. Reporting only the grouped contrast would leave out that important part of the pattern.
Grouping cannot add observations or reveal detail that was not collected. It can, however, make a particular contrast easier to see by adding together category differences that point in the same direction. To avoid overstating that contrast, state which categories were combined and, when the distinction matters, show the original distribution as well.
A Changed Threshold Can Reverse a Comparison
With ordered categories, a conclusion may depend on the threshold used to define an outcome. “High” and “at least moderate” are different outcomes. If the groups have different patterns across the categories, those outcomes can produce comparisons in opposite directions. This is not a contradiction: each comparison answers a different question.
Worked Example: The Direction Depends on the Grouping
A hypothetical questionnaire records a rating as “Low,” “Moderate,” or “High.” Each group has 100 respondents.
| Group | Low | Moderate | High | Total |
|---|---|---|---|---|
| A | 40 | 10 | 50 | 100 |
| B | 30 | 50 | 20 | 100 |
If the outcome of interest is a “High” rating, compare the High percentages: Group A has 50% and Group B has 20%. Group A’s percentage is 30 percentage points higher.
Now define the outcome as “at least Moderate,” combining Moderate and High. Group A has \(10+50=60\) respondents meeting that definition, while Group B has \(50+20=70\). The percentages are 60% and 70%, so Group B’s percentage is 10 percentage points higher.
The direction reverses because the threshold changed. Group A has more “High” ratings, while Group B has more ratings that are at least Moderate. Both statements follow from the same table. The correct comparison depends on which outcome definition is relevant to the question; neither grouped table replaces the information in the original categories.
A Practical Check Before Interpreting a Grouped Table
When a table has been simplified, trace each reported category back to the original categories and verify its count. Then check the denominator for each percentage. These steps help distinguish a real feature of the data from a feature created by a broad definition.
Identify what each response meant before any categories were combined.
For a combined category, sum its component counts separately in every row or column as appropriate.
Use the total for the relevant group as the denominator, and compare like with like.
Name the combined outcome or group, and avoid implying that the grouped result applies to its individual categories.
Grouping can be useful when categories are too detailed for a clear summary or when a broader threshold is substantively meaningful. It can also make a table misleading if categories are combined only because the resulting pattern looks stronger or simpler. The table should support the question, not determine it after the results are seen.
Common Mistake
A common mistake is to see equal percentages after grouping and conclude that the original categories showed no association. In the first example, the grouped percentages were identical, but the original distributions differed between “Never” and “Sometimes.” The correction is to state the conclusion at the level of the table being compared: the groups matched on the broad “Often” versus “Not often” outcome, but not on the finer response distribution.
A related mistake is to describe a selected combined outcome as if it represented the entire pattern. In the second example, “below Severe” was more common in Group A, but “Severe” itself was more common in Group B. A careful interpretation names the outcome definition and checks the remaining categories before summarizing the association.
Check Your Understanding
Use the definitions and examples above to answer these questions.
- Two groups have 30% and 50% in a category, with sample sizes 40 and 160. Why is the combined percentage not generally the simple average of 30% and 50%?
- In the first worked example, which difference in the original table disappears after “Never” and “Sometimes” are combined?
- In the symptom example, why is it accurate to say that the combined “below Severe” outcome differs by 24 percentage points, but inaccurate to say that each component category differs by 24 points?
- In the rating example, what two different questions are answered by comparing “High” and by comparing “at least Moderate”?
- Before interpreting a grouped table, name two checks that help ensure its percentages and categories are being read correctly.