Statistics and Probability
Statistics and Probability
This domain asks you to summarize data, judge how it was collected, and compute probabilities from a clearly defined sample space. Questions often hinge on interpretation: which measure of center is appropriate, what a graph does and does not show, and whether a probability should be conditional.
Core rules to hold
- Center and spread: the mean is sensitive to outliers and skew; the median is resistant. Range, interquartile range (Q3 - Q1), and mean absolute deviation describe spread. A box plot displays the five-number summary: minimum, Q1, median, Q3, maximum.
- Choosing a display: bar graphs compare categories, histograms and dot plots show distribution shape, box plots compare groups, scatterplots show association between two quantitative variables, and two-way tables support conditional reasoning.
- Sampling: a random sample supports generalization to the population it was drawn from. Convenience samples, voluntary-response samples, and leading questions introduce bias that no amount of data cleaning removes.
- Association: a line of best fit describes the direction and strength of a linear association. Correlation does not establish causation; a lurking variable may drive both.
- Probability rules: probabilities lie between 0 and 1; P(not E) = 1 - P(E); for mutually exclusive events P(A or B) = P(A) + P(B); for independent events P(A and B) = P(A) times P(B); for dependent events multiply by the conditional probability.
Worked example
A bag holds 4 red and 6 blue marbles. Two are drawn without replacement. P(first red) = 4/10 = 2/5. Given that a red marble was removed, 3 red and 6 blue remain among 9 marbles, so P(second red given first red) = 3/9 = 1/3. Multiply: P(both red) = 2/5 times 1/3 = 2/15, about 0.133. The complement gives P(at least one blue) = 1 - 2/15 = 13/15. Drawing with replacement would instead give 4/10 times 4/10 = 4/25, a visibly different answer.
Traps that cost marks
- Treating dependent draws as independent, or vice versa.
- Confusing the probability of A and B with the probability of A or B.
- Reading a median off a histogram by taking the tallest bar.
- Claiming a sample is representative merely because it is large.
- Interpreting a strong correlation as proof of cause and effect.