From Conditional Distributions to Association
A two-way table organizes counts for two categorical variables. To ask whether those variables appear related, compare how the distribution of one variable changes across categories of the other. For example, do different study groups have similar pass rates, or do their outcome percentages differ? The comparison is made using conditional distributions: percentages calculated within each group.
This is a descriptive question about the pattern in the table. If the conditional distributions differ, the observed table shows evidence that the outcome’s distribution varies across groups. If they are similar, the table shows little or no apparent difference. Neither conclusion, by itself, establishes what would happen in a larger population or whether one variable causes the other.
The denominator matters. If comparing outcomes among categories of a group variable, calculate each outcome percentage using the total for that group. The percentages within each group should add to 100% (apart from possible rounding). Comparing counts alone can be misleading when group sizes differ: a larger group may have more observations in every outcome category even if its percentages are much like those of a smaller group.
For a table with more than two outcome categories, compare the whole distribution, not just one percentage. A noticeable change in how observations are spread across several categories can indicate an association even when no single percentage tells the whole story.
Worked Examples of Different Conditional Distributions
Worked Example: Pass Rates Differ Between Training Groups
Suppose, hypothetically, that 80 trainees in an online course and 70 trainees in an in-person workshop take the same assessment. The table records whether each trainee passes.
| Training group | Pass | Does not pass | Total |
|---|---|---|---|
| Online | 48 | 32 | 80 |
| In-person | 56 | 14 | 70 |
First calculate the conditional distribution of assessment outcome within each training group. For the online group, the pass percentage is
The percentage who do not pass is
For the in-person group, the corresponding percentages are
Both distributions are calculated within their own group, and each sums to 100%. The online group’s observed outcomes are 60% pass and 40% do not pass; the in-person group’s are 80% pass and 20% do not pass. These distributions differ, so this table shows a descriptive association between training group and assessment outcome. In particular, the observed pass percentage is higher in the in-person group. This comparison alone does not establish that the training format caused the difference.
Worked Example: Comparing Three Outcome Categories
Imagine a hypothetical survey of appointment respondents at two clinics. Each respondent rates a visit as poor, acceptable, or excellent. Clinic A has 100 responses: 10 poor, 30 acceptable, and 60 excellent. Clinic B has 80 responses: 24 poor, 32 acceptable, and 24 excellent.
Calculate percentages within each clinic. For Clinic A, the percentages are
For Clinic B, they are
| Clinic | Poor | Acceptable | Excellent |
|---|---|---|---|
| A | 10% | 30% | 60% |
| B | 30% | 40% | 30% |
The distributions differ across all three categories. Excellent ratings make up 60% of Clinic A’s responses but 30% of Clinic B’s; poor ratings make up 10% at Clinic A and 30% at Clinic B. Thus the observed rating distribution varies across clinics, which is evidence of a descriptive association between clinic and rating in these data. Because the outcome has three categories, looking only at the excellent-rating percentages would leave out part of the pattern. The full conditional distributions make the comparison clearer.
When the Conditional Distributions Are Similar
Association is not established merely because a table contains different counts. When group sizes differ, counts can differ even if the within-group percentages are the same. The relevant question is whether the conditional distributions differ, not whether the group totals or individual cell counts differ.
Worked Example: Completion Percentages Are Similar
Suppose a hypothetical program records whether participants complete an online form. Of 80 participants who receive a reminder, 40 complete it and 40 do not. Of 60 participants who receive no reminder, 30 complete it and 30 do not.
| Reminder group | Complete | Do not complete | Total |
|---|---|---|---|
| Reminder | 40 | 40 | 80 |
| No reminder | 30 | 30 | 60 |
For participants who receive a reminder, the completion and non-completion percentages are
For participants who receive no reminder, they are
The conditional distributions are identical: 50% complete and 50% do not complete in each group. Although the counts differ (40 completions versus 30), the group sizes differ too. This table shows no apparent association between reminder group and form completion based on the observed conditional distributions. It does not prove that the variables have no association in a broader population.
Why the Comparison Works
A conditional distribution describes the composition of a group in terms of the outcome categories. Comparing those distributions asks whether the outcome composition changes from one group to another. If it changes, the group category helps distinguish which outcome distribution is observed in this table. If the distributions match, knowing the group category does not distinguish the observed outcome percentages in this table.
This reasoning can be stated directly. Suppose the outcome has categories \(C_1,\ldots,C_k\), and the group variable has categories \(G_1,\ldots,G_m\). For a given group \(G_j\), the conditional percentage in outcome category \(C_i\) is the count in cell \((G_j,C_i)\) divided by the total count for group \(G_j\), then multiplied by 100%. For each group, these percentages sum to 100% because the outcome categories account for every observation in that group. Comparing the resulting rows (or columns, depending on table layout) compares like with like: the percentage in each outcome category within each group.
For example, in the training table, the online group’s pass percentage is \(48/80=0.60\), while the in-person group’s is \(56/70=0.80\). The denominators are the sizes of the corresponding training groups. Comparing these within-group percentages gives a meaningful description of how pass outcomes vary across the two groups; comparing 48 and 56 alone would not account for the different group sizes.
Identify which variable defines the groups and which variable’s distribution will be compared.
For each group, divide every outcome count by that group’s total, not by the overall total.
Look across all outcome categories for noticeable differences in percentages.
Describe whether the observed outcome distribution differs across groups, without claiming a cause or a population result that the table alone cannot establish.
What the Table Can and Cannot Show
A two-way table is a description of the observations it contains. Saying that variables “appear associated” here means that their observed conditional distributions differ. It is not a claim that the difference is statistically significant, and it is not a claim that the same pattern must hold in every sample or population. Deciding whether a pattern is more than could reasonably be attributed to sampling variation requires inferential methods, which are a separate question from describing a table.
The table also does not, on its own, show that changing one variable would change the other. For instance, a different pass percentage across training groups is an observed association. It does not establish that training format caused the difference. The table may describe a useful pattern, but the design and context are needed to justify stronger conclusions.
Common Mistakes
- Comparing counts instead of percentages. Seeing 48 passes in one group and 56 in another does not show which group has the higher pass rate. Divide each count by its own group total.
- Using the overall total as the denominator. That calculates a joint percentage, not the outcome distribution within a group. To compare outcome distributions across groups, use the group total for each conditional percentage.
- Checking only one category when the outcome has several. Compare the full set of conditional percentages so that changes across all outcome categories are represented.
- Treating similar sample percentages as proof of no association. The table supports a description of these observations. Similar percentages do not establish that the population distributions are exactly the same.
- Turning association into causation. A difference between group distributions describes a relationship in the data; it does not show that group membership caused the outcome difference.
Check Your Understanding
Use conditional distributions and explain your conclusions in context.
- A table shows 18 successes among 30 people in Group A and 24 successes among 60 people in Group B. Calculate the success percentage within each group. Do these observed percentages suggest an association between group and success?
- Why is it misleading to compare 40 completions in a group of 80 with 30 completions in a group of 60 without calculating percentages?
- Two groups have outcome percentages of 20% poor, 50% acceptable, and 30% excellent in each group. What does this say about the observed conditional distributions, and what does it not prove?
- A three-category outcome has different percentages across two groups. Why should you compare the full distribution rather than only the percentage in the most common category?
- A table shows different outcome percentages across two types of instruction. Give an appropriate descriptive conclusion and explain why it does not by itself establish causation.