ETF1100 Chap.3 Categorical Data and Probability
Categorical Data and Probability
Categorical Data and Probability builds contingency tables and reads joint, marginal and conditional information without losing the denominator. The chapter compares conditional distributions to identify association, applies complement, union and independence ideas, and explains why reversing a condition changes the probability. Business conclusions remain descriptive unless a design supports causal interpretation.
What this chapter covers
- 01
Joint and marginal counts
- 02
Conditional proportions
- 03
Contingency-table association
- 04
Complement probabilities
- 05
Union and overlap
- 06
Independence and causal limits
Worked example · free
Compare first-contact resolution by channel
- 1Use 90 as the chat denominator and compute 80.0%.
- 1Use 110 as the email denominator and compute 70.0%.
- 1Report a 10.0 percentage-point observed difference.
- 1State that channel and resolution are associated in the recorded sample.
- 1Avoid causation because ticket complexity and staffing may differ.
Key terms
- Joint Probability
- The probability that two stated events occur together.
- Marginal Probability
- The probability of one event using a table total and ignoring the other variable.
- Conditional Probability
- The probability of an event after restricting the denominator to a stated condition.
- Independence
- A relationship in which conditioning on one event does not change the probability of the other.
- Percentage Point
- The arithmetic difference between two percentages rather than their relative percent change.
- Contingency Table
- A table of joint counts for two categorical variables with row and column totals.
Categorical Data and Probability FAQ
What is the difference between joint and conditional probability?
Joint probability uses the whole sample as its denominator and asks whether two events occur together. Conditional probability restricts the denominator to the group named after the condition. The same cell count can therefore produce different probabilities depending on the question.
How do I check association in a contingency table?
Compare conditional distributions across groups using consistent denominators. If the outcome proportions differ meaningfully, the variables are associated in the recorded data. A large raw cell is not enough because it may simply come from a larger group.
Why does association not prove causation?
The groups may differ in other factors, selection into the data may be uneven and the direction of influence may be unclear. Causal language needs a design that rules out plausible alternatives, not merely different conditional percentages in an observational table.
What checksum helps detect a wrong conditional denominator?
Conditional proportions across all outcomes within the same conditioning group should sum to one, apart from rounding. You can also multiply a conditional probability by its conditioning probability to recover the corresponding joint probability. Failure of either check points to a reversed condition, omitted category or incorrect total.
Exam move
Draw every contingency table with totals and label the events before calculating. Practise three questions from the same cell: joint probability, the row-conditioned probability and the column-conditioned probability. Say the denominator aloud each time. Memorise the complement rule and the union rule with overlap subtracted once, but use multiplication for a joint probability only when independence is justified.
To assess association, compare conditional distributions rather than raw counts. Write differences in percentage points and reserve relative percent change for questions that explicitly request it. In interpretation, name the recorded population and state association without causal language. When an automatic question reverses the phrase after the vertical bar, rebuild the denominator rather than editing the previous decimal.
Link this chapter to later inference: observed conditional differences describe the sample, while intervals or tests quantify sampling uncertainty. Finish practice answers with one uncontrolled factor that could explain the pattern. Use a single two-way table to practise a full audit. Verify that interior cells sum to both sets of marginal totals and that the grand total matches the number of eligible observations.
Shade the denominator for each requested probability before touching a calculator. Compute one joint, two reversed conditional probabilities, a complement and a union, and label every decimal with its event statement. Next convert the conditional results to percentages and check that each conditioned distribution sums to roughly 100% after rounding.
To assess independence, compare a conditional proportion with its corresponding marginal proportion and explain the direction of any departure. Translate the gap twice: as percentage points and as a relative change from a named baseline. The two numbers should never share an unlabeled percent sign. Build a habit of asking how missing category values were treated, because excluding them can change every denominator.
If a pooled comparison reverses after separating groups, investigate a lurking factor rather than choosing the preferred version. For decision writing, give the observed association, its sample scope, a plausible uncontrolled explanation and the additional evidence needed for a causal claim.
During the final, draw a vertical bar in conditional notation and read it aloud as given; this simple cue prevents the most common reversal.
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