Mathematics and Statistics: Quantitative Reasoning
Arithmetic tools
Percent change = (new − old) ÷ old. Successive percent changes multiply, they do not add. Weighted averages require weighting by group size, not averaging the averages. Average speed is total distance over total time.
Probability
Independent events multiply. Mutually exclusive events add. The complement rule is often fastest. Permutations count arrangements where order matters; combinations count selections where it does not.
Statistics
Mean is sensitive to outliers; median is resistant. Standard deviation measures spread. In a normal distribution roughly 68% of values fall within one standard deviation, 95% within two, and 99.7% within three. Correlation is not causation. Random sampling reduces bias; margin of error shrinks with larger samples.
Finance
Compound interest: A = P(1 + r/n)^(nt).
Traps
Reporting percent change with the wrong base; averaging percentages across unequal groups; confusing percentage points with percent; misreading graph axes or per-capita versus total figures; assuming a large sample corrects a biased one.
Worked example
A price rises 20% and then falls 20%. Multiply: 1.20 × 0.80 = 0.96. The net result is a 4% decrease from the original, not zero change.
Rule of thumb
Before computing, label the base and the units. Most quantitative errors are labeling errors, not arithmetic errors.