Harald Cramér#

Stage 2 · 🔗 Associations & Correlation · Lesson 14 of 56 · beginner

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A Swedish statistician#

Harald Cramér (1893–1985) was a Swedish mathematician and statistician whose name attaches to several cornerstones of the field. Where Pearson built the discipline’s foundations, Cramér gave much of it its rigorous mathematical footing, bridging probability theory and statistics. Two of his contributions appear directly in this course.

Cramér’s V#

The first is Cramér’s V, the standard measure of association between two categorical (nominal) variables — the categorical counterpart to Pearson’s \(r\). Built on Pearson’s chi-square statistic, it rescales it to a clean range of \(0\) (no association) to \(1\) (perfect):

\[V = \sqrt{\frac{\chi^2}{n \,\min(r - 1,\, c - 1)}},\]

where \(n\) is the sample size and \(r, c\) the numbers of rows and columns in the contingency table. It fills the “categorical ↔ categorical” slot of the association taxonomy.

The Cramér–Rao bound#

The second is the Cramér–Rao bound, a deep result in estimation theory: it sets a hard lower limit on the variance any unbiased estimator can achieve. An estimator that reaches it is as precise as possible — a benchmark that underlies the maximum-likelihood methods appearing later in this course.

A foundational text#

Cramér’s influence also came through teaching. His 1946 book “Mathematical Methods of Statistics” was among the first to put statistics on a firm measure-theoretic basis, and it trained a generation of statisticians. Alongside the V and the bound, his broader work spanned probability, actuarial mathematics and number theory.

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