The formula to work out sample sizes for means are useful in many instances. There are several versions of the formula needed to cover one-sample and two-sample tests, one-tailed and two-tailed comparisons, and whether the standard deviation is known or needs to be estimated. In this article, the formula for each situation is given, along with the Microsoft Excel function that performs the calculations. Another article at this site shows the Excel sample size formula for proportions tests. A free sample size calculator is available, for various applications.
Sample Size Formula For Means: Terms Used and Applications
α = Level of acceptability of a false positive result (0.05 is typical)
β = Level of acceptability of a false negative result (0.10 is typical)
σ = Known standard deviation
δ = Amount of difference that matters (Practical difference)
The formulae described here are valid for testing the mean of a sample to a population of known mean and standard deviation. Figure 1 shows the equivalent formula for one-sample mean with unknown σ, and σ is estimated by s.
Sample Size Formula For One-Sample Mean
This type of test is used when a sample is to be taken from a population and compared to a known population "P".
In the case where the standard deviation of "P" is known, the formula is:
Sample Size = (Zα + Zβ)².σ² / δ²
It will be noted that this is the one-tail version of the formula. The two-tail version of the formula simply substitutes α/2 and β/2 for α and β:
Sample Size = (Zα/2 + Zβ/2)².σ² / δ²
Sample Size Examples For One-Sample Mean
In certain instances there are specialized protocols that must be strictly adhered to. Testing the efficacy of new drugs, for example, normally follows a protocol that includes a prescribed sample size selection method. Where there are no legal implications to the test, such as an engineering process improvement, it is possible to estimate sample sizes using certain assumptions. It is common, for instance, to use the following values in the sample size calculation:
α = 0.05
β = 0.10
δ / σ = 0.5
Using these values, the sample sizes may be estimated as 35 for a one-tailed test, and 52 for a two-tailed test. To detect a one-sigma shift, the one-tailed and two-tailed sample sizes needed are 9 and 13 respectively.
Excel Sample Size Formula For One-Sample Mean
In cell A1, enter the value of α
In cell A2, enter the value for β
In cell A3, enter the value for σ
In cell A4, enter the value of the critical difference δ
It is also possible to use sigma = 1 and enter the critical difference in terms of σ units.
The Excel formula to estimate sample size for a one-sided test is then
=( NORMSINV(1 - A1) + NORMSINV(1 - A2) )^2 * A3^2 / A4^2
For a two-sided test, the formula is
=( NORMSINV(1 - A1 / 2) + NORMSINV(1 - A2 / 2) )^2 * A3^2 / A4^2
Sample Size Formula For Means - Summary
The sample size calculation formula is easy to use, and the Excel functions to use it are described here. Also described are "ready-reckoner" answers to some typical values of α, β and σ / δ.
A free sample size calculator is available.
Sample Size Method References
Other relevant articles include those on the method of selecting sample sizes and the definitions of Confidence and Power.
Martin Bell - Martin holds a B.Sc. degree in chemical engineering, and an M.Sc. degree in electronics and computing. He has spent more than 25 years ...
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