Statistical Analysis
Core concepts
Statistical analysis in Methods centres on describing and comparing data, and on probability models. Know how to construct and interpret histograms, boxplots, dotplots and stem plots; compute mean, median, mode, range, interquartile range and standard deviation; and identify outliers using the 1.5×IQR rule. Understand the effect of skewness on the relative positions of mean and median. Compare datasets using parallel boxplots, discussing shape, centre, spread and outliers.
For probability, know sample spaces, the addition and multiplication rules, conditional probability, independence and tree diagrams. Recognise discrete random variables, their probability distributions, expected value and variance. For continuous variables, understand the normal distribution, standardise with $z=(x-\mu)/\sigma$, and use symmetry and tables or technology to find probabilities.
Traps
- Reading the median from a histogram without accounting for grouped data.
- Confusing standard deviation with variance, or sample with population formulas.
- Assuming independence when events are only mutually exclusive, or vice versa.
- Forgetting $P(A\mid B)\neq P(B\mid A)$.
- Using the normal model without checking that it is appropriate.
Worked example
Scores are normally distributed with $\mu=70$, $\sigma=8$. Find $P(X>82)$. Standardise: $z=(82-70)/8=1.5$. By symmetry and tables, $P(Z>1.5)\approx 0.067$, so about 6.7% of scores exceed 82.
Rule of thumb
Always state which measure of centre and spread suits the data's shape; for skewed data, prefer median and IQR.