Statistics and Probability
Statistics and Probability tests how data are summarized, displayed, and used to estimate chance. Expect mean, median, mode, range, weighted averages, interpretation of graphs and tables, basic probability rules, expected value, and sampling ideas.
Core rules
- Mean = sum of values ÷ number of values. Median is the middle value after ordering; with an even count, average the two middle values. Mode is most frequent; range is maximum minus minimum.
- Outliers pull the mean more than the median, so the median better describes a skewed data set.
- Weighted mean = Σ(value × weight) ÷ Σ(weights).
- Probability of an event = favorable outcomes ÷ total equally likely outcomes. Complement: P(not A) = 1 − P(A).
- Independent events: P(A and B) = P(A) × P(B). Mutually exclusive events: P(A or B) = P(A) + P(B).
- Expected value = Σ(value × probability).
- Random sampling reduces bias; larger samples generally give more stable estimates. Correlation does not prove causation.
Worked example
Find the mean and median of 4, 7, 7, 9, 13. Sum = 40, count = 5, mean = 40 ÷ 5 = 8. Ordered values: 4, 7, 7, 9, 13; the middle value is 7, so median = 7. The mode is 7 and the range is 13 − 4 = 9. The mean exceeds the median because 13 pulls it upward.
Traps that cost marks
- Finding the median without ordering the data first.
- Forgetting to average the two middle values when the count is even.
- Using “at least one” directly instead of the complement 1 − P(none).
- Treating draws as independent when sampling without replacement changes the denominator.
- Computing an unweighted mean when categories have different counts or weights.
- Reading a graph's axis scale carelessly and inferring causation from an observed association.