Statistics, Probability, and Calculus
What to know
Descriptive statistics covers mean, median, mode, range, variance, standard deviation, quartiles, and boxplots; when data are skewed or contain outliers, the median and IQR are resistant while the mean and standard deviation are not. Expect the normal distribution with the 68-95-99.7 rule and z-scores, sampling methods and sources of bias, and the distinction between observational studies and experiments - correlation is not causation. Probability includes sample spaces, complements, the addition and multiplication rules, conditional probability from two-way tables, independence via P(A∩B) = P(A)P(B), counting with permutations and combinations, expected value, and binomial probabilities. Calculus covers limits and continuity, derivative rules (power, product, quotient, chain), derivatives as rates and tangent slopes, extrema and the first and second derivative tests, optimization, related rates, antiderivatives, definite integrals as signed area, and the Fundamental Theorem of Calculus.
Traps
- P(A|B) is not P(B|A); read which event is given.
- A single outlier shifts the mean but barely moves the median.
- A definite integral gives net signed area; geometric area needs absolute value or correct ordering.
- A limit can exist where the function is undefined, but continuity requires the limit to exist, the value to exist, and the two to agree.
- Standard deviation carries the data's units; variance carries squared units.
Worked example
Differentiate y = x² sin x using the product rule: y′ = 2x sin x + x² cos x. Rule of thumb: identify the outermost structure first - product, quotient, or composition - and apply that rule before differentiating the pieces.