Stats Examples

Bayes’ rule

Suppose 1% of people have a condition. A test is 99% sensitive and 95% specific. If the test is positive,

$$P(\text{condition}\mid +)=\frac{0.99\cdot 0.01}{0.99\cdot 0.01+0.05\cdot 0.99}\approx \mathrm{0.167.}$$

Even a good test can have a low positive predictive value when the base rate is small.

Standardising a normal variable

If $X\sim \mathcal{N}(70,10^2)$, then $X=85$ corresponds to

$$z=\frac{85-70}{10}=\mathrm{1.5.}$$

Instead of memorising a new table for every mean and variance, convert to the standard Normal distribution.

Likelihood intuition

If data are assumed independent with density $p(x_i\mid \theta )$, then

$$L(\theta )=\prod _{i}p(x_i\mid \theta ).$$

The best parameter is not the one that makes the data “probable” in isolation; it is the one that makes the observed data most plausible relative to alternatives.

Correlation sanity check

A strong correlation between $X$ and $Y$ does not prove $X$ causes $Y$. It may reflect a hidden variable, selection effects, or reversed causation. In physics and ML, residual plots often reveal structure that a single correlation number hides.

These examples connect to Stats Key concepts, Stats Equations and definitions, and Stats Common pitfalls.