Statistics, Probability, and Data Analysis
What the Domain Covers
This domain tests summarizing and interpreting data, the normal distribution, study design and association, and the rules of probability including conditional probability, independence, and counting.
Describing Data
Mean, median, and mode locate center; range, interquartile range, and standard deviation describe spread. Skewed data drags the mean toward the tail, so the median better represents a typical value. By the 1.5·IQR rule, an outlier lies below Q1 − 1.5·IQR or above Q3 + 1.5·IQR. Standard deviation measures typical distance from the mean; adding a constant to every value shifts the mean but leaves the standard deviation unchanged.
The Normal Distribution
z = (x − μ)/σ gives the number of standard deviations from the mean. Roughly 68% of values fall within 1σ, 95% within 2σ, and 99.7% within 3σ.
Design and Association
Random sampling supports generalization to a population; random assignment supports causal conclusions. Watch for voluntary-response, convenience, and leading-question bias. Correlation r ranges from −1 to 1, measures only linear association, and does not establish causation; an outlier or lurking variable can create or conceal it.
Probability Rules
P(A or B) = P(A) + P(B) − P(A and B). For independent events, P(A and B) = P(A)·P(B). Conditional probability: P(A|B) = P(A and B)/P(B). Independence means P(A|B) = P(A). Expected value = Σ x·P(x). Use permutations when order matters and combinations when it does not.
Worked Example
Scores are normally distributed with μ = 500 and σ = 100. What proportion exceed 650? z = (650 − 500)/100 = 1.5. A standard normal table gives P(z < 1.5) ≈ 0.9332, so P(z > 1.5) ≈ 1 − 0.9332 = 0.0668. About 6.7% of scores exceed 650 - consistent with the empirical rule, which puts roughly 16% above +1σ and far less above +1.5σ.
Traps That Cost Marks
- Reversing conditional probabilities: P(A|B) ≠ P(B|A).
- Using P(A) + P(B) for "or" without subtracting the overlap.
- Reporting the mean for strongly skewed data.
- Sampling without replacement changes the denominator on the next draw.