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Sample size calculations


At the end of the lecture and having completed the exercises students should be able to:

  1. Outline the necessity of a priori sample size calculations
  2. Explain the concepts of Type 1 errors, Type 2 errors and statistical power
  3. Describe the relationships between sample size, power, significance level and standard deviation


Altman, D.G., Statistics and ethics in medical research: III How large a sample? BMJ 281: 1336-8

Written for medics, good brief introduction to sample size calculations.
(Also appears in: D.G. Altman & S.M. Gore, 1982. Statistics in Practice, pp. 6-8. BMA, London)

Altman, D.G., 1991. Practical statistics for medical research, pp. 455-460. Chapman & Hall, London.

Written for medics, contains a good introduction to sample size calculations

Bland M., 1995. An Introduction to Medical Statistics, 2nd ed., pp. 331-341. Oxford Medical Publications, Oxford.

Written for medics, gives simple formulae and basic tables. Similar to, but slightly more mathematical than Altman's book.

Campbell M.J. & D. Machin,1993. Medical Statistics, A Commonsense Approach, 2nd ed., pp. 156-157. John Wiley & Sons, Chichester.

Written for medics, contains a brief summary, equations and examples.

Machin, D., M.J. Campbell, P.M. Fayers & A.P.Y. Pinol 1997. Sample Size Tables for Clinical Studies, 2nd ed. Blackwell Science, Oxford.

Written for statisticians. Tables and equations for most situations, lots of examples. Very mathematical.

Why bother with sample size calculations?

Sample size calculations are sometimes referred to as power calculations, for reasons that should be obvious when you have finished reading this document.


Possible results from a clinical study
Difference No difference
True situation Difference True positive False negative
No difference False positive True negative

Type 1 error

The risk of a false positive result (α).

i.e. the chance of detecting a statistically significant difference when there is no real difference between treatments.

Type 2 error

The risk of a false negative result (β).

i.e. the chance of not detecting a significant difference when there really is a difference.

Power (1-β)

The chance of not getting a false negative result.

i.e. the chance of spotting a difference as being statistically significant if there really is a difference.

Higher power is better, aim for at least 90% power. 80% power is the minimum acceptable.


There will be a bigger sample size with:

There will be a smaller sample size with:

Calculating sample sizes




An easier method

A simple way of calculating approximate sample sizes is given another page in this section.

Standardised difference (Δ)

Formula for standardised difference