Statistical Analysis: Data, Distributions and Inference
Core ideas
Statistical analysis in Advanced covers describing and comparing data, the normal distribution, and drawing inferences. You must compute and interpret measures of centre (mean, median, mode) and spread (range, interquartile range, standard deviation), read and construct box plots, histograms and stem-and-leaf plots, and identify outliers using the 1.5 × IQR rule.
The standard normal distribution has mean 0 and standard deviation 1; a raw score converts by z = (x − μ)/σ. For a general normal distribution, z-scores let you use tables or technology to find proportions, and the empirical rule gives roughly 68% within one standard deviation, 95% within two and 99.7% within three.
Inference includes sampling variability, the standard error of the mean σ/√n, confidence intervals of the form x̄ ± k·σ/√n, and the logic of hypothesis testing: a result is statistically significant when it is unlikely under the null assumption.
Traps
- The mean is sensitive to outliers; the median is not - say which is appropriate and why.
- Standard deviation uses n − 1 for a sample in some conventions; follow the one your course specifies.
- A z-score is not a probability; it is a position measured in standard deviations.
- Correlation does not imply causation, and extrapolation beyond the data is unreliable.
- A confidence interval describes the procedure, not the probability that a fixed parameter lies inside one computed interval.
Worked example
Scores are normal with μ = 60, σ = 8. A score of 76 gives z = (76 − 60)/8 = 2, so it lies two standard deviations above the mean - roughly the top 2.5% of scores.
Rule of thumb
Always state centre, spread and shape together when comparing two data sets, and quote the comparison in context.