Sample size determination is the act of choosing the number of observations to include in a statistical sample. The sample size is an important feature of any empirical study in which the goal is to make inferences about a population from a sample. In practice, the sample size used in a study is determined based on the expense of data collection, and the need to have sufficient statistical power. In complicated studies there may be several different sample sizes involved in the study: for example, in as survey sampling involving stratified sampling there would be different sample sizes for each population. In a census, data are collected on the entire population, hence the sample size is equal to the population size. In experimental design, where a study may be divided into different treatment groups, there may be different sample sizes for each group.

Sample sizes may be chosen in several different ways:

expedience - For example, include those items readily available or convenient to collect. A choice of small sample sizes, though sometimes necessary, can result in wide confidence intervals or risks of errors in statistical hypothesis testing.

using a target variance for an estimate to be derived from the sample eventually obtained

using a target for the power of a statistical test to be applied once the sample is collected.

How samples are collected is discussed in sampling (statistics) and survey data collection.

Introduction

Larger sample sizes generally lead to increased precision when estimating unknown parameters. For example, if we wish to know the proportion of a certain species of fish that is infected with a pathogen, we would generally have a more accurate estimate of this proportion if we sampled and examined 200, rather than 100 fish. Several fundamental facts of mathematical statistics describe this phenomenon, including the law of large numbers and the central limit theorem.

In some situations, the increase in accuracy for larger sample sizes is minimal, or even non-existent. This can result from the presence of systematic errors or strong dependence in the data, or if the data follow a heavy-tailed distribution.

Sample sizes are judged based on the quality of the resulting estimates. For example, if a proportion is being estimated, one may wish to have the 95% confidence interval be less than 0.06 units wide. Alternatively, sample size may be assessed based on the power of a hypothesis test. For example, if we are comparing the support for a certain political candidate among women with the support for that candidate among men, we may wish to have 80% power to detect a difference in the support levels of 0.04 units.

Estimating proportions and means

A relatively simple situation is estimation of a proportion. For example, we may wish to estimate the proportion of residents in a community who are at least 65 years old.

The estimator of a proportion is $\backslash hat\; p\; =\; X/n$, where ''X'' is the number of 'positive' observations (e.g. the number of people out of the ''n'' sampled people who are at least 65 years old). When the observations are independent, this estimator has a (scaled) binomial distribution (and is also the sample mean of data from a Bernoulli distribution). The maximum variance of this distribution is 0.25/''n'', which occurs when the true parameter is ''p'' = 0.5. In practice, since ''p'' is unknown, the maximum variance is often used to for sample size assessments.

For sufficiently large ''n'', the distribution of $\backslash hat\{p\}$ will be closely approximated by a normal distribution with the same mean and variance. Using this approximation, it can be shown that around 95% of this distribution's probability lies within 2 standard deviations of the mean. Because of this, an interval of the form

:$(\backslash hat\; p\; -2\backslash sqrt\{0.25/n\},\; \backslash hat\; p\; +2\backslash sqrt\{0.25/n\})$

will form a 95% confidence interval for the true proportion. If this interval needs to be no more than ''W'' units wide, the equation

:$4\backslash sqrt\{0.25/n\}\; =\; W$

can be solved for ''n'', yielding ''n'' = 4/''W''² = 1/''B''² where ''B'' is the error bound on the estimate, i.e., the estimate is usually given as ''within ± B''. So, for ''B'' = 10% one requires ''n'' = 100, for ''B'' = 5% one needs ''n'' = 400, for ''B'' = 3% the requirement approximates to ''n'' = 1000, while for ''B'' = 1% a sample size of ''n'' = 10000 is required. These numbers are quoted often in news reports of opinion polls and other sample surveys.

Estimation of means

A proportion is a special case of a mean. When estimating the population mean using an independent and identically distributed (iid) sample of size ''n'', where each data value has variance ''σ''², the standard error of the sample mean is:

::$\backslash sigma/\backslash sqrt\{n\}.$

This expression describes quantitatively how the estimate becomes more precise as the sample size increases. Using the central limit theorem to justify approximating the sample mean with a normal distribution yields an approximate 95% confidence interval of the form

:$(\backslash bar\; x\; -\; 2\backslash sigma/\backslash sqrt\{n\},\backslash bar\; x\; +\; 2\backslash sigma/\backslash sqrt\{n\}).$

If we wish to have a confidence interval that is ''W'' units in width, we would solve

:$4\backslash sigma/\backslash sqrt\{n\}\; =\; W$

for ''n'', yielding the sample size ''n'' = 16''σ²/W².

For example, if we are interested in estimating the amount by which a drug lowers a subject's blood pressure with a confidence interval that is six units wide, and we know that the standard deviation of blood pressure in the population is 15, then the required sample size is 100.

Required sample sizes for hypothesis tests

A common problem facing statisticians is calculating the sample size required to yield a certain power for a test, given a predetermined Type I error rate α. As follows, this can be estimated by pre-determined tables for certain values, by Mead's resource equation, or, more generalized, by the cumulative distribution function:

By tables

{|class="wikitable" align="right" !rowspan=2|  Power !!colspan=3| Cohen's d |- ! 0.2 !! 0.5 !! 0.8 |- ! 0.25 | 84 || 14 || 6 |- ! 0.50 | 193 || 32 || 13 |- ! 0.60 | 246 || 40 || 16 |- ! 0.70 | 310 || 50 || 20 |- ! 0.80 | 393 || 64 || 26 |- ! 0.90 | 526 || 85 || 34 |- ! 0.95 | 651 || 105 || 42 |- ! 0.99 | 920 || 148 || 58 |} The table shown at right can be used to in a two-sample t-test to estimate the sample sizes of an experimental group and a control group that are of equal size, that is, the total number of individuals in the trial is twice that of the number given, and the desired significance level is 0.05. The parameters used are:

The desired statistical power of the trial, shown in column to the left.

Cohen's d, which is the expected difference between the means of the target values between the experimental group and the control group, divided by the expected standard deviation.

Mead's resource equation

Mead's resource equation is often used for estimating sample sizes of laboratory animals, as well as in many other laboratory experiments. It may not be as accurate as using other methods in estimating sample size, but gives a hint of what is the appropriate sample size where parameters such as expected standard deviations or expected differences in values between groups are unknown or very hard to estimate.

All the parameters in the equation are in fact the degrees of freedom of the number of their concepts, and hence, their numbers are subtracted by 1 before insertion into the equation.

The equation is:

By cumulative distribution function

Let ''X_i'', ''i'' = 1, 2, ..., ''n'' be independent observations taken from a normal distribution with unknown mean μ and known variance σ². Let us consider two hypotheses, a null hypothesis:

: $H\_0:\backslash mu=0$

and an alternative hypothesis:

: $H\_a:\backslash mu=\backslash mu^*$

for some 'smallest significant difference' μ^* >0. This is the smallest value for which we care about observing a difference. Now, if we wish to (1) reject ''H''₀ with a probability of at least 1-β when ''H''_a is true (i.e. a power of 1-β), and (2) reject ''H''₀ with probability α when ''H''₀ is true, then we need the following:

If ''z''_α is the upper α percentage point of the standard normal distribution, then

: $\backslash Pr(\backslash bar\; x\; >z\_\{\backslash alpha\}\backslash sigma/\backslash sqrt\{n\}|H\_0\; \backslash text\{\; true\})=\backslash alpha$

and so

: 'Reject ''H''₀ if our sample average ($\backslash bar\; x$) is more than $z\_\{\backslash alpha\}\backslash sigma/\backslash sqrt\{n\}$'

is a decision rule which satisfies (2). (Note, this is a 1-tailed test)

Now we wish for this to happen with a probability at least 1-β when ''H''_a is true. In this case, our sample average will come from a Normal distribution with mean μ^*. Therefore we require

: $\backslash Pr(\backslash bar\; x\; >z\_\{\backslash alpha\}\backslash sigma/\backslash sqrt\{n\}|H\_a\; \backslash text\{\; true\})\backslash geq\; 1-\backslash beta$

Through careful manipulation, this can be shown to happen when

: $n\; \backslash geq\; \backslash left(\backslash frac\{\backslash Phi^\{-1\}(1-\backslash beta)+z\_\{\backslash alpha\}\}\{\backslash mu^\{*\}/\backslash sigma\}\backslash right)^2$

where $\backslash Phi$ is the normal cumulative distribution function.

Stratified sample size

With more complicated sampling techniques, such as stratified sampling, the sample can often be split up into sub-samples. Typically, if there are ''k'' such sub-samples (from ''k'' different strata) then each of them will have a sample size ''n_i'', ''i'' = 1, 2, ..., ''k''. These ''n_i'' must conform to the rule that ''n''₁ + ''n''₂ + ... + ''n''_''k'' = ''n'' (i.e. that the total sample size is given by the sum of the sub-sample sizes). Selecting these ''n_i'' optimally can be done in various ways, using (for example) Neyman's optimal allocation.

There are many reasons to use stratified sampling: to decrease variances of sample estimates, to use partly non-random methods, or to study strata individually. A useful, partly non-random method would be to sample individuals where easily accessible, but, where not, sample clusters to save travel costs.

In general, for ''H'' strata, a weighted sample mean is : $\backslash bar\; x\_w\; =\; \backslash sum\_\{h=1\}^H\; W\_h\; \backslash bar\; x\_h,$ with

: $\backslash operatorname\{Var\}(\backslash bar\; x\_w)\; =\; \backslash sum\_\{h=1\}^H\; W\_h^2\; \backslash ,\backslash operatorname\{Var\}(\backslash bar\; x\_h).$

The weights, ''W''(''h''), frequently, but not always, represent the proportions of the population elements in the strata, and ''W''(''h'')=''N''(''h'')/''N''. For a fixed sample size, that is n=Sum{n(h)},

: $\backslash operatorname\{Var\}(\backslash bar\; x\_w)\; =\; \backslash sum\_\{h=1\}^H\; W\_h^2\; \backslash ,\backslash operatorname\{Var\}(h)\; \backslash left(\backslash frac\{1\}\{n\_h\}\; -\; \backslash frac\{1\}\{N\_h\}\backslash right),$

which can be made a minimum if the sampling rate within each stratum is made proportional to the standard deviation within each stratum: $n\_h/N\_h=k\; S\_h$.

An "optimum allocation" is reached when the sampling rates within the strata are made directly proportional to the standard deviations within the strata and inversely proportional to the square roots of the costs per element within the strata: : $\backslash frac\{n(h)\}\{N(h)\}\; =\; \backslash frac\{K\; S(h)\}\{\backslash sqrt\{C(h)\}\},$

or, more generally, when

: $n(h)\; =\; \backslash frac\{K\text{'}\; W(h)\; S(h)\}\{\backslash sqrt\{C(h)\}\}.$

See also

Degrees of freedom (statistics)

Design of experiments

Replication (statistics)

Sampling (statistics)

Statistical power

Stratified Sampling

Engineering response surface example under Stepwise regression.

Notes

References

Bartlett, J. E., II, Kotrlik, J. W., & Higgins, C. (2001). "Organizational research: Determining appropriate sample size for survey research", ''Information Technology, Learning, and Performance Journal'', 19(1) 43-50.

Kish, L. (1965), ''Survey Sampling'', Wiley. ISBN 047148900x

Further reading

NIST: Selecting Sample Sizes

Raven Analytics: Sample Size Calculations

ASTM E122-07: Standard Practice for Calculating Sample Size to Estimate, With Specified Precision, the Average for a Characteristic of a Lot or Process

Category:Sampling (statistics)

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