Change the denominator deliberately

Try

Two services report different numbers of successful deliveries. Service A completed 18 and service B completed 42. Decide whether those counts alone identify the more reliable service. You need the attempted-delivery denominators. A larger operation can have more successes and more failures without having a different success rate.

Predict

Predict whether “fraction of A deliveries that succeed” equals “fraction of successes supplied by A.” They refer to different restricted groups. Circle the phrase after “given” or “among” before choosing a denominator. Reversing that phrase is a common source of persuasive but incorrect claims.

Experiment

Build this original synthetic two-way table and complete its totals.

Service	Success	Failure	Total
A	18	12	30
B	42	28	70
Total	60	40	100

The joint proportion both A and successful is 18/100=0.18. The success proportion conditional on A is 18/30=0.60. The A proportion conditional on success is 18/60=0.30. Write a complete sentence for each number, naming its denominator group. Both services have success rate 0.60, although their raw success counts differ.

Understand

A two-way table cross-classifies two categorical variables for the same observational units. A joint count belongs to a particular row and column. Marginal totals summarize one variable, ignoring the other. Conditional proportions restrict attention to a row or column. Use consistent category definitions, mutually exclusive categories where required, and a clear treatment of missing observations.

Within this table, service and outcome show no association because the conditional success proportions are equal. In a probability model, equality of the relevant conditional distributions corresponds to independence. In a finite sample, however, equality or inequality is descriptive evidence. Deciding whether an observed difference supports a population conclusion requires an inferential method and design assumptions.

A segmented bar chart can display conditional percentages: give each service a bar of the same total length and divide it into success and failure portions. Label the original group sizes too. Equal-looking proportions based on five and five thousand observations do not have the same precision. A mosaic plot instead uses areas to encode joint proportions; its different geometry answers a different visual question.

Aggregation can conceal group composition. Suppose a service handles disproportionately difficult routes. Its pooled success rate mixes route difficulty with performance. Compare conditional rates within difficulty categories before interpreting pooled differences. In some datasets the pooled ordering reverses every within-group ordering, a phenomenon often called Simpson’s paradox. The arithmetic is consistent; the denominator mixture changed.

Stratifying a table does not automatically remove all confounding, establish causal direction or repair a biased sample. Retain the study design and any important omitted variables in the interpretation. A descriptive comparison is strongest when it states exactly what was counted and what conclusion remains unsupported.

Apply

Choose one table entry and deliberately write three different percentages using it. Check which sentence each percentage answers. Then sketch equal-length segmented bars for A and B. Explain why their internal divisions match while their underlying counts do not. The written practice asks you to challenge a measurement definition as well as compute rates.

Continue

Next choose displays for quantitative variables. The same discipline applies: a height, area or percentage must have a named reference and units.