Chapter 2: descriptive statistics, the normal distribution and hypothesis tests

Measurement scales, graphs, central tendency and dispersion, the normal distribution and the z table, confidence intervals, hypotheses and the comparison of means.

Slides (99, in French) · PDF in French (8.2 MB) · Questions

The chapter in 10 points

Key point. A p-value of 0.03 does not say that the tested hypothesis has a 3% chance of being true: this chapter teaches how to read what a test really says.

  1. Population, sample, variable, parameter and statistic; the type of scale (nominal, ordinal, interval, ratio) dictates which statistics are allowed.
  2. Graphical displays and measures of central tendency: mean, median, mode; the mean is sensitive to extreme values (one very rich person makes a “mean of millionaires”).
  3. Dispersion: sample variance s² = Σ(xi − x̄)²/(n − 1), where xi are the values, x̄ their mean and n their number; standard deviation s, range, quantiles.
  4. Notation used here: N(μ, σ²), where μ is the mean, σ the standard deviation and σ² the variance. Caution: the slides write N(m, s), with the standard deviation second. Skewness and kurtosis describe other aspects of the shape of the distribution.
  5. Jarque-Bera: JB = (n/6)[S² + (K − 3)²/4], where n is the number of values, S the skewness and K the raw kurtosis (centered on 3). Under the asymptotic conditions of the test, the 5% threshold is about 5.99 (χ² with 2 degrees of freedom).
  6. Standardized variable z = (x − μ)/σ and the table of N(0, 1); about 95% of the values lie between μ − 1.96σ and μ + 1.96σ.
  7. 95% confidence interval of the mean: x̄ ± 1.96 · s/√n for large n.
  8. Central limit theorem: for independent observations with the same distribution and a finite variance, the standardized mean tends to a normal distribution; its standard error is σ/√n. Dependence requires additional conditions.
  9. Hypothesis testing: the null hypothesis H₀ (no effect) against the alternative H₁, one- or two-sided test, risk α of wrongly rejecting H₀, test statistic and p-value, the probability under H₀ of a result at least as extreme; in the worked example, only one of three reading scores is significantly below the reference.
  10. Two independent samples, with means x̄1 and x̄2, standard deviations s1 and s2 and sizes n1 and n2: t = (x̄1 − x̄2)/√(s1²/n1 + s2²/n2) is the Welch statistic, with approximate Welch-Satterthwaite degrees of freedom (df). The paired test works on the differences; Student's test with equal variances uses a pooled variance.

To check the conditions of the models: Checking the Student formulas: NIST.

Open this experiment on its own page

Test a difference and read its uncertainty

Student t-test with a pooled variance for two treatments. Read together the gap between the means, the t statistic (that gap divided by its uncertainty), the degrees of freedom (df), the p-value and the rejection regions. Open this experiment on its own page.

These experiments extend the notions of the biostatistics course without adding a new method. Checking the Student formulas: NIST.

The slides (in French)

The 99 slides of this chapter are in French. Their text is written out and searchable in the French chapter, which shows one slide at a time (arrow keys, deep links). The summary, the experiments and the FAQ above are the English reading of the same chapter; formulas and numbers are identical.

Open the slides → · Download the chapter (PDF in French, 8.2 MB)

Outline of the chapter

  1. Introduction (slides 1–4, in French)
  2. I. Basic concepts (slides 5–19, in French)
  3. II. Describing data (slides 20–47, in French)
  4. III. The normal distribution (slides 48–66, in French)
  5. IV. Hypotheses (slides 67–81, in French)
  6. V. Hypothesis testing (means) (slides 82–99, in French)

Going further: questions and answers

Why divide by n − 1 and not by n?

The deviations are measured from the sample mean, which is as close to them as possible: dividing by n systematically underestimates the variance. Dividing by n − 1 corrects this bias, because one degree of freedom was used to estimate the mean.

Does a p-value of 0.03 give the probability that H0 is true?

No. Under H0 and the assumptions of the model, it is the probability of a statistic at least as extreme as the one observed, for the chosen test. It is neither P(H0 | data) nor a measure of the practical importance of the effect.

When is Student's distribution used for a mean?

For independent normal observations with an unknown variance, (x̄ − μ0)/(s/√n) follows exactly a Student distribution with n − 1 df, even for large n. Without normality, the approximation depends on the size and on the distribution; two-group and paired tests have their own conditions.

Is a normality test enough to choose a method?

No: a large sample detects small departures and a small sample lacks power. One must look at the graphs, the extreme values, the independence and the inference question; the central limit theorem does not repair biased data collection or any dependence.

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These slides are not a handout: they accompany the lectures and cannot replace them (disclaimer of the course). Teaching material, not clinical advice. The slides and the PDF are in French. License CC BY 4.0: free reuse, including modification and commercial use, with credit to the author (Cyril Voyant).