Statistics, Probability, and Data Analysis
Domain overview
Statistics, Probability, and Data Analysis tests how data are collected, displayed, summarized, and interpreted, plus the rules of probability and basic inference. Expect items on sampling methods, bias, summary statistics, probability models, conditional probability, and the distinction between association and causation.
Core rules
- Study design: random sampling supports generalization; random assignment supports causal claims; bias arises from convenience, voluntary-response, and undercoverage sampling.
- Center and spread: the mean is sensitive to outliers, the median is resistant; standard deviation measures typical distance from the mean; IQR = Q3 − Q1.
- Displays: histograms show distribution shape; box plots show median, quartiles, and outliers; scatterplots show association, direction, form, and strength.
- Probability: 0 ≤ P(A) ≤ 1; P(A or B) = P(A) + P(B) − P(A and B); P(A and B) = P(A)P(B|A).
- Conditional probability: P(A|B) = P(A and B)/P(B), with P(B) > 0.
- Expected value: E(X) = Σ x·P(x).
Worked example
A bag contains 3 red and 5 blue marbles. Two are drawn without replacement. Find P(red then blue). P(red first) = 3/8. After removing a red marble, 2 red and 5 blue remain, so P(blue second | red first) = 5/7. Multiply: (3/8)(5/7) = 15/56 ≈ 0.268. The reasoning is that the denominator changes because the first draw is not replaced; the events are dependent, so the multiplication rule uses the conditional probability.
Traps
- Adding probabilities of overlapping events without subtracting the intersection.
- Treating without replacement as independent.
- Using the wrong denominator in a conditional probability.
- Concluding causation from a strong correlation.
- Reporting mean and median as interchangeable for skewed data.