# Sample size

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## Overview

The sample size of a statistical sample is the number of repeated measurements that constitute it. It is typically denoted n, and is a non-negative integer (natural number).

Typically, different sample sizes lead to different accuracies of measurement. This can be seen in such statistical rules as the law of large numbers and the central limit theorem. All else being equal, a larger sample size n leads to increased precision in estimates of various properties of the population.

A typical example would be when a statistician wishes to estimate the arithmetic mean of a continuous random variable (for example, the height of a person). Assuming that they have a random sample with independent observations, then if the variability of the population (as measured by the standard deviation σ) is known, then the standard error of the sample mean is given by the formula:

${\displaystyle \sigma /{\sqrt {n}}.}$

It is easy to show that as n becomes large, this variability becomes very small. This yields to more sensitive hypothesis tests with greater Statistical power and smaller confidence intervals.

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 ni, i = 1, 2, ..., k. These ni must conform to the rule that n1 + n2 + ... + nk = n (i.e. that the total sample size is given by the sum of the sub-sample sizes). Selecting these ni optimally can be done in various ways, using (for example) Neyman's optimal allocation.

# Further examples

## Central limit theorem

The central limit theorem is a significant result which depends on sample size.

## Estimating proportions

A typical statistical aim is to demonstrate with 95% certainty that the true value of a parameter is within a distance B of the estimate: B is an error range that decreases with increasing sample size (n). The value of B generated is referred to as the 95% confidence interval.

For example, a simple situation is estimating a proportion in a population. To do so, a statistician will estimate the bounds of a 95% confidence interval for an unknown proportion.

The rule of thumb for (a maximum or 'conservative') B for a proportion derives from the fact the estimator of a proportion, ${\displaystyle {\hat {p}}=X/n}$, (where X is the number of 'positive' observations) has a (scaled) binomial distribution and is also a form of sample mean (from a Bernoulli distribution [0,1] which has a maximum variance of 0.25 for parameter p = 0.5). So, the sample mean X/n has maximum variance 0.25/n. For sufficiently large n (usually this means that we need to have observed at least 10 positive and 10 negative responses), this distribution will be closely approximated by a normal distribution with the same mean and variance.

Using this approximation, it can be shown that ~95% of this distribution's probability lies within 2 standard deviations of the mean. Because of this, an interval of the form

${\displaystyle ({\hat {p}}-2{\sqrt {0.25/n}},{\hat {p}}+2{\sqrt {0.25/n}})=({\hat {p}}-B,{\hat {p}}+B)}$

will form a 95% confidence interval for the true proportion.

If we require the sampling error ε to be no larger than some bound B, we can solve the equation

${\displaystyle \varepsilon \approx B=2{\sqrt {0.25/n}}=1/{\sqrt {n}}}$

to give us

${\displaystyle 1/\varepsilon ^{2}\approx 1/B^{2}=n}$

So, n = 100 <=> B = 10%, n = 400 <=> B = 5%, n = 1000 <=> B = ~3%, and n = 10000 <=> B = 1%. One sees these numbers quoted often in news reports of opinion polls and other sample surveys.

### Extension to other cases

In general, if a population mean is estimated using the sample mean from n observations from a distribution with variance σ², then if n is large enough (typically >30) the central limit theorem can be applied to obtain an approximate 95% confidence interval of the form

${\displaystyle ({\bar {x}}-B,{\bar {x}}+B),B=2\sigma /{\sqrt {n}}}$

If the sampling error ε is required to be no larger than bound B, as above, then

${\displaystyle 4\sigma ^{2}/\varepsilon ^{2}\approx 4\sigma ^{2}/B^{2}=n}$

Note, if the mean is to be estimated using P parameters that must first be estimated themselves from the same sample, then to preserve sufficient "degrees of freedom," the sample size should be at least n + P.

## 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. A typical example for this is as follows:

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

${\displaystyle H_{0}:\mu =0}$

and an alternative hypothesis:

${\displaystyle H_{a}:\mu =\mu ^{*}}$

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 H0 with a probability of at least 1-β when Ha is true (i.e. a power of 1-β), and (2) reject H0 with probability α when H0 is true, then we need the following:

If zα is the upper α percentage point of the standard normal distribution, then

${\displaystyle \Pr({\bar {x}}>z_{\alpha }\sigma /{\sqrt {n}}|H_{0}{\text{ true}})=\alpha }$

and so

'Reject H0 if our sample average (${\displaystyle {\bar {x}}}$) is more than ${\displaystyle z_{\alpha }\sigma /{\sqrt {n}}}$

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

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

${\displaystyle \Pr({\bar {x}}>z_{\alpha }\sigma /{\sqrt {n}}|H_{a}{\text{ true}})\geq 1-\beta }$

Through careful manipulation, this can be shown to happen when

${\displaystyle n\geq \left({\frac {\Phi ^{-1}(1-\beta )+z_{\alpha }}{\mu /\sigma }}\right)^{3}}$

where ${\displaystyle \Phi }$ is the normal cumulative distribution function.