Statistics, Probability, and Data Analysis
Statistics, Probability, and Data Analysis
This domain covers summarizing data, interpreting displays, designing samples, and computing probabilities. Questions often ask you to compare mean and median, identify bias, or decide whether two events are independent.
Core rules
- Mean = sum of values ÷ number of values. Median is the middle value after ordering; with an even number of values, average the two middle values. Mode is the most frequent value.
- Range = maximum − minimum. Interquartile range (IQR) = Q3 − Q1 and describes the middle 50% of data. The median and IQR are resistant to outliers; the mean and range are not.
- A random sample supports generalization to the population only when selection is unbiased. Voluntary-response and convenience samples often overrepresent strong opinions.
- Probability of an event is favorable outcomes ÷ total equally likely outcomes. For independent events, P(A and B) = P(A)P(B). For mutually exclusive events, P(A or B) = P(A) + P(B).
- Conditional probability is P(A|B) = P(A and B)/P(B). Without replacement, the denominator changes after each draw.
- Correlation does not imply causation; a lurking variable may explain an association.
Worked example
A bag contains 4 red and 6 blue marbles. Two are drawn without replacement. Find P(both red). First draw: P(red) = 4/10 = 2/5. After removing one red, 3 red remain out of 9: P(second red | first red) = 3/9 = 1/3. Multiply: 2/5 × 1/3 = 2/15. The events are dependent because the first draw changes the composition of the bag.
Traps
Do not average medians or add probabilities of overlapping events without subtracting the overlap. With replacement, denominators stay the same; without replacement, they do not. A high correlation is not proof of cause. Also check whether a display’s vertical axis starts at zero before calling a difference “large.”