FoundationNumeric answerFree solution

You have 100 observations and a standard error of 0.40.4. How many observations do you need for a standard error of 0.10.1?

Answer with a number. Fractions, powers and expressions like 23/6 or C(52,5) are read correctly in practice.

Show the answer and worked solution

Answer

1,6001{,}600

Standard error falls as 1/n1/\sqrt{n}, so the sample ratio is the square of the error ratio: (0.4/0.1)2=16(0.4/0.1)^2 = 16, and 16×100=1,60016 \times 100 = 1{,}600. The shape is worth internalising because it sets the economics of every research programme: each extra digit of precision costs a hundredfold in data, and halving an error bar costs four times the sample. It is also why a short backtest proves so little and why the returns to collecting more of the same data fall away so quickly.

Worked solution

  1. Formula
    se∝1n ⇒ n2n1=(se1se2)2\mathrm{se} \propto \frac{1}{\sqrt{n}} \ \Rightarrow\ \frac{n_2}{n_1} = \left(\frac{\mathrm{se}_1}{\mathrm{se}_2}\right)^2
  2. Substitute
    =(0.40.1)2=16= \left(\frac{0.4}{0.1}\right)^2 = 16
  3. Solve
    n2=16×100n_2 = 16 \times 100
  4. Answer
    1,6001{,}600

Takeaway: Halving a standard error costs four times the data.

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You have 100 observations and a standard error of 0.4. · QuantMax