Accuracy vs Precision

I had a hard time remembering this¹, almost as bad as Type 1 vs Type 2 (which are still the worst terminology I’ve ever seen).

Thinking of these by reframing them with actual statistical terms made them easy to remember.

Accuracy

A random variable (the estimator) is accurate if it has low bias.

Precision

A random variable is precise if it has low variance.

Examples

From these definitions, it’s finally clear to me why neither implies each other. Say your true distribution is a standard (\(\mu = 0, \sigma = 1\)) Gaussian. Consider these cases for your estimator, also a Gaussian with the following parameters:

(\(\mu = 0, \sigma = 1,000\))

Perfectly accurate because it’s unbiased. Imprecise as hell.

(\(\mu = 1,000, \sigma = 0.01\))

Inaccurate. Very precise.

(\(\mu = 1,000, \sigma = 1,000\))

Inaccurate and imprecise.

(\(\mu = 0, \sigma = 0.01\))

Perfectly accurate, more precise than the true distribution because its spread is lower.