Measures of Central Tendency and Dispersion

Measures of Central Tendency and Dispersion

1. Definition, role & why a pharmacologist needs descriptive statistics

• Summary (descriptive) statistics condense a set of observations into a few numbers that convey the essential information in a data set — typically a measure of location/central tendency (where the centre of the data lies) plus a measure of spread/dispersion (how widely the values scatter around it) (SaSO Ch.2, pp.12–13).
• Two numbers — a measure of location and a measure of variability — are usually the minimum needed to describe a continuous variable; quoting location without spread is uninformative (SaSO Ch.2, p.12).
• In pharmacology these underpin everything downstream: reporting a trial's baseline table, summarising a dose–response or pharmacokinetic parameter (Cmax, AUC, t½, clearance), describing adverse-event rates, and feeding the point estimate ± its precision into confidence intervals and hypothesis tests (the location/spread split is the foundation the inferential layer is built on) (SaSO Ch.2, p.12; Ch.3, p.33).
• Descriptive statistics are chosen to match (a) the type of data and (b) the shape of the distribution — the two governing axes for the whole topic (SaSO Ch.1, pp.1–2; Ch.2, pp.25–26).

1.1 Types of data (governs which summary is legitimate)

• Quantitative (numerical) data — measured or counted on a meaningful numeric scale (SaSO Ch.1, p.1):
  • Continuous — can take any value in a range, limited only by measurement precision (e.g. blood pressure, height, weight, serum drug concentration) (SaSO Ch.1, p.1).
  • Discrete (count) — whole-number counts (e.g. number of children in a family, number of asthma attacks, number of deaths) (SaSO Ch.1, p.1).
• Categorical (qualitative) data — observations fall into categories (SaSO Ch.1, p.1):
  • Ordinal — ordered categories with a natural ranking but no fixed numeric distance between them (e.g. grade of breast cancer; pain rated 1 = poor … 5 = excellent; better/same/worse) (SaSO Ch.1, p.1; Ch.2, p.26).
  • Nominal — unordered categories (e.g. sex; blood group A/B/AB/O; alive/dead) (SaSO Ch.1, p.1).
  • Binary (dichotomous) — a special nominal case with exactly two categories (e.g. male/female, dead/alive, responder/non-responder); summarised by a proportion rather than a mean and SD (SaSO Ch.1, p.1; Ch.2, p.21).
• A continuous variable can be down-converted into ordinal or binary categories (e.g. blood pressure → hypertensive/normotensive), but doing so discards information and is generally avoided unless there is a clinical reason (SaSO Ch.1, p.2).
• ⚠ Scale dictates the summary: mean and SD are properly used only for quantitative data that are roughly symmetric; ordinal data are best summarised by the median; nominal/binary data by proportions. Applying a mean to nominal data is meaningless (SaSO Ch.1, p.1; Ch.2, p.26).