ECON20003 · Quantitative Methods 2
Normality & Nonparametric One-Sample Tests
Normality & Nonparametric One-Sample Tests teaches you to check the assumption that underpins every t-test, then what to do when it fails. You diagnose normality with a histogram, a Normal Q-Q plot (points hugging the line ⇒ normal), the standardised skew and kurtosis ratios (|skew ÷ 2SE| or |kurt ÷ 2SE| > 1 ⇒ non-normal), and the Shapiro-Wilk test (small p ⇒ reject normality). If normality fails, you swap to a distribution-free test: the sign test for a median (skewed data) or the Wilcoxon signed-rank test (non-normal but roughly symmetric).
What this chapter covers
- 01Histogram and Normal Q-Q plot reading
- 02Standardised skew/kurtosis: |skew ÷ 2SE| > 1 ⇒ non-normal
- 03Shapiro-Wilk test (normtest.W, normtest.p): small p ⇒ reject normality
- 04Sign test for a median: #(+) ~ Binomial(n, 0.5); large-sample Z
- 05Wilcoxon signed-rank: E(T) = n(n+1)/4, Var(T) = n(n+1)(2n+1)/24
- 06Decision logic: normal ⇒ t; non-normal symmetric ⇒ signed-rank; non-normal skewed ⇒ sign test
Large-sample sign test for a median
- 1 markChoose the test. The data are skewed, so use the distribution-free sign test on the median. Under H₀ the number of plus signs follows Binomial(n, 0.5).
- 1 markState the hypotheses (right-tailed): H₀: median = 50 versus H₁: median > 50.
- 1 markCount plus signs: S = 11 ratings above 50 out of n = 14.
- 2 marksApply the large-sample normal approximation: Z = (S − 0.5n)/(0.5√n) = (11 − 7)/(0.5 × √14) = 4/(0.5 × 3.742) = 4/1.871 = 2.14.
- 1 markDecision rule: the right-tail critical value is z₀.₀₅ = 1.645. Since 2.14 > 1.645, reject H₀.
- 1 markConclude in context: there is significant evidence at the 5% level that the median product rating exceeds 50.
Key terms
- Normal Q-Q plot
- A plot of sample quantiles against theoretical normal quantiles. Points lying close to the straight reference line indicate approximate normality; systematic curves (S-shapes or bends at the ends) flag skew or heavy tails.
- Shapiro-Wilk test
- A formal test of normality. H₀ is that the data are normal, so a SMALL p-value (e.g. < 0.05) leads you to reject normality and switch to a nonparametric test.
- Sign test
- A distribution-free test of a median that counts how many observations fall above the hypothesised value; under H₀ that count is Binomial(n, 0.5), with a large-sample normal approximation Z = (S − 0.5n)/(0.5√n).
- Wilcoxon signed-rank test
- A nonparametric test of a median (or paired differences) that ranks the absolute deviations and sums the ranks of the positive side; it assumes a symmetric population and is more powerful than the sign test when that holds.
Normality & Nonparametric One-Sample Tests FAQ
How do I decide between the t-test, the Wilcoxon signed-rank test and the sign test?
First check normality (Q-Q plot, skew/kurt ratios, Shapiro-Wilk). If the data look normal, use the t-test. If they are non-normal but roughly symmetric, use the Wilcoxon signed-rank test. If they are non-normal and skewed, fall back to the sign test, which only assumes a median exists.
Does a small Shapiro-Wilk p mean my whole analysis is wrong?
It means the normality assumption behind a parametric t-test is in doubt, so you should report the matching nonparametric result instead. For large samples the CLT can rescue the mean's sampling distribution, but in the exam, if normality is explicitly rejected, switch tests and say why.
Exam move
Build a one-line decision rule in your head — normal → t, symmetric-but-not-normal → signed-rank, skewed → sign — and practise reading the three normality diagnostics (Q-Q, skew/kurt ÷ 2SE, Shapiro-Wilk) off an R printout. Know the large-sample Z formula for the sign test cold.
