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Statistics for finance · level 1

Standard error and the √n rule

The spread of a sample mean is the spread of the data divided by the square root of the sample size. Precision is expensive: four times the data for twice the precision.

Theory: Statistical Inference

Worked example: σ = 20, sample of 100. Standard error of the mean?

σ20
√n10
SE2

Step by step

  1. 1

    SE shrinks with the root of n

    SE = σ / √n

  2. 2

    Root

    √100 = 10

  3. 3

    Divide

    20 / 10 = 2

  4. 4

    The √n rule

    To halve the standard error you need four times the data.

σ = 20, sample of 100. Standard error of the mean? = 2

The theory behind it

Intuition

Precision improves with the square root of sample size: four times the data halves the error. That is why survey costs rise fast.

Common pitfalls

  • ×Confusing standard deviation (spread of data) with standard error (spread of the estimate).
  • ×Ignoring that the √n rule assumes independent observations.

In the interview

Market-research and sample-design rounds.