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| '''Sample size determination''' is the act of choosing the number of observations or [[Replication (statistics)| replicates]] to include in a [[statistical sample]]. The sample size is an important feature of any empirical study in which the goal is to make [[statistical inference|inferences]] about a [[statistical population|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 a [[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 group]]s, there may be different sample sizes for each group.
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| Sample sizes may be chosen in several different ways:
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| *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 interval]]s or risks of errors in [[statistical hypothesis testing]].
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| *using a target variance for an estimate to be derived from the sample eventually obtained
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| *using a target for the power of a [[statistical hypothesis testing|statistical test]] to be applied once the sample is collected.
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| How samples are collected is discussed in [[sampling (statistics)]] and [[survey data collection]].
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| ==Introduction==
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| Larger sample sizes generally lead to increased [[Accuracy and precision|precision]] when [[statistical estimation|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]].
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| 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 error]]s or strong [[correlation and dependence|dependence]] in the data, or if the data follow a heavy-tailed distribution.
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| 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 [[statistical power|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.
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| ==Estimating proportions and means==
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| A relatively simple situation is estimation of a [[Proportionality (mathematics)|proportion]]. For example, we may wish to estimate the proportion of residents in a community who are at least 65 years old.
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| The [[estimator]] of a [[Proportionality (mathematics)|proportion]] is <math> \hat p = X/n</math>, 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 (statistics)|independent]], this estimator has a (scaled) [[binomial distribution]] (and is also the [[Sample (statistics)|sample]] [[arithmetic mean|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 for sample size assessments.
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| For sufficiently large ''n'', the distribution of <math>\hat{p}</math> will be closely approximated by a [[normal distribution]].<ref>[[NIST]]/[[SEMATECH]], [http://www.itl.nist.gov/div898/handbook/prc/section2/prc242.htm "7.2.4.2. Sample sizes required"], ''e-Handbook of Statistical Methods.''</ref> Using this approximation, it can be shown that around 95% of this distribution's probability lies within 2 standard deviations of the mean. Using the
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| [[Binomial_distribution#Confidence_intervals | Wald method for the binomial distribution]],
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| an interval of the form
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| :<math>(\hat p -2\sqrt{0.25/n}, \hat p +2\sqrt{0.25/n})</math>
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| will form a 95% confidence interval for the true proportion. If this interval needs to be no more than ''W'' units wide, the equation
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| :<math>4\sqrt{0.25/n} = W</math>
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| can be solved for ''n'', yielding<ref>[http://www.utdallas.edu/~ammann/stat3355/node25.html "Large Sample Estimation of a Population Proportion"]</ref><ref>[http://nebula.deanza.fhda.edu/~bloom/Math10/M10ConfIntNotes.pdf "Confidence Interval for a Proportion"]</ref> ''n'' = 4/''W''<sup>2</sup> = 1/''B''<sup>2</sup> 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 poll]]s and other [[sample survey]]s.
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| ===Estimation of means===
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| 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''<sup>2</sup>, the [[standard error (statistics)|standard error]] of the sample mean is:
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| ::<math>\sigma/\sqrt{n}.</math>
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| 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
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| :<math>(\bar x - 2\sigma/\sqrt{n},\bar x + 2\sigma/\sqrt{n}).</math>
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| If we wish to have a confidence interval that is ''W'' units in width, we would solve
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| :<math>
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| 4\sigma/\sqrt{n} = W
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| </math>
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| for ''n'', yielding the sample size ''n'' = 16''σ<sup>2</sup>/W<sup>2</sup>.
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| 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.
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| ==Required sample sizes for hypothesis tests {{anchor|Estimating sample sizes}}==
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| A common problem faced by the statisticians is calculating the sample size required to yield a certain [[Statistical power|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 generally, by the [[cumulative distribution function]]:
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| ===By tables===
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| {|class="wikitable" align="right"
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| !rowspan=2|<ref name=Kenny1987/><br> <br> [[statistical power|Power]] !!colspan=3| [[Cohen's d]]
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| |-
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| ! 0.2 !! 0.5 !! 0.8
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| |-
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| ! 0.25
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| | 84 || 14 || 6
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| |-
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| ! 0.50
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| | 193 || 32 || 13
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| |-
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| ! 0.60
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| | 246 || 40 || 16
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| |-
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| ! 0.70
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| | 310 || 50 || 20
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| |-
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| ! 0.80
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| | 393 || 64 || 26
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| |-
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| ! 0.90
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| | 526 || 85 || 34
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| |-
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| ! 0.95
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| | 651 || 105 || 42
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| |-
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| ! 0.99
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| | 920 || 148 || 58
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| |}
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| The table shown at right can be used 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.<ref name=Kenny1987>[http://davidakenny.net/doc/statbook/chapter_13.pdf Chapter 13], page 215, in: {{cite book |author=Kenny, David A. |title=Statistics for the social and behavioral sciences |publisher=Little, Brown |location=Boston |year=1987 |pages= |isbn=0-316-48915-8 |oclc= |doi= |accessdate=}}</ref> The parameters used are:
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| *The desired [[statistical power]] of the trial, shown in column to the left.
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| *[[Cohen's d]] (=effect size), which is the expected difference between the [[mean]]s of the target values between the experimental group and the [[control group]], divided by the expected [[standard deviation]].
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| ===Mead's resource equation===
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| Mead's resource equation is often used for estimating sample sizes of [[laboratory animal]]s, 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.<ref name=Hubrecht&Kirkwood2010>{{cite book |author=Kirkwood, James; Robert Hubrecht |title=The UFAW Handbook on the Care and Management of Laboratory and Other Research Animals |publisher=Wiley-Blackwell |location= |year=2010 |pages=29 |isbn=1-4051-7523-0 |oclc= |doi= |accessdate=}} [http://books.google.se/books?id=Wjr9u1AAhsdsdsdsdsdssdsdddddddddddddddddddst4C&pg=PA29 online Page 29]</ref>
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| All the parameters in the equation are in fact the [[Degrees of freedom (statistics)|degrees of freedom]] of the number of their concepts, and hence, their numbers are subtracted by 1 before insertion into the equation.
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| The equation is:<ref name=Hubrecht&Kirkwood2010/>
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| :<math> E = N - B - T,</math>
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| where:
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| *''N'' is the total number of individuals or units in the study (minus 1)
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| *''B'' is the ''blocking component'', representing environmental effects allowed for in the design (minus 1)
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| *''T'' is the ''treatment component'', corresponding to the number of [[treatment groups]] (including [[control group]]) being used, or the number of questions being asked (minus 1)
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| *''E'' is the degrees of freedom of the ''error component'', and should be somewhere between 10 and 20.
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| For example, if a study using laboratory animals is planned with four treatment groups (''T''=3), with eight animals per group, making 32 animals total (''N''=31), without any further [[Stratified sampling|stratification]] (''B''=0), then ''E'' would equal 28, which is above the cutoff of 20, indicating that sample size may be a bit too large, and six animals per group might be more appropriate.<ref>[http://www.isogenic.info/html/resource_equation.html Isogenic.info > Resource equation] by Michael FW Festing. Updated Sept. 2006</ref>
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| ===By cumulative distribution function===
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| Let ''X<sub>i</sub>'', ''i'' = 1, 2, ..., ''n'' be independent observations taken from a [[normal distribution]] with unknown mean μ and known variance σ<sup>2</sup>. Let us consider two hypotheses, a [[null hypothesis]]:
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| : <math> H_0:\mu=0 </math>
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| and an alternative hypothesis:
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| : <math> H_a:\mu=\mu^* </math>
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| for some 'smallest significant difference' μ<sup>*</sup> >0. This is the smallest value for which we care about observing a difference. Now, if we wish to (1) reject ''H''<sub>0</sub> with a probability of at least 1-β when
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| ''H''<sub>a</sub> is true (i.e. a [[Statistical power|power]] of 1-β), and (2) reject ''H''<sub>0</sub> with probability α when ''H''<sub>0</sub> is true, then we need the following:
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| If ''z''<sub>α</sub> is the upper α percentage point of the standard normal distribution, then
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| : <math> \Pr(\bar x >z_{\alpha}\sigma/\sqrt{n}|H_0 \text{ true})=\alpha </math>
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| and so
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| : 'Reject ''H''<sub>0</sub> if our sample average (<math>\bar x</math>) is more than <math>z_{\alpha}\sigma/\sqrt{n}</math>'
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| is a [[decision rule]] which satisfies (2). (Note, this is a 1-tailed test)
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| Now we wish for this to happen with a probability at least 1-β when
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| ''H''<sub>a</sub> is true. In this case, our sample average will come from a Normal distribution with mean μ<sup>*</sup>. Therefore we require
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| : <math> \Pr(\bar x >z_{\alpha}\sigma/\sqrt{n}|H_a \text{ true})\geq 1-\beta </math>
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| Through careful manipulation, this can be shown to happen when
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| : <math> n \geq \left(\frac{z_{\alpha}-\Phi^{-1}(1-\beta)}{\mu^{*}/\sigma}\right)^2 </math>
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| where <math>\Phi</math> is the normal [[cumulative distribution function]].
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| ==Stratified sample size==
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| With more complicated sampling techniques, such as [[stratified sampling]], the sample can often be split up into sub-samples. Typically, if there are ''H'' such sub-samples (from ''H'' different strata) then each of them will have a sample size ''n<sub>h</sub>'', ''h'' = 1, 2, ..., ''H''. These ''n<sub>h</sub>'' must conform to the rule that ''n''<sub>1</sub> + ''n''<sub>2</sub> + ... + ''n''<sub>''H''</sub> = ''n'' (i.e. that the total sample size is given by the sum of the sub-sample sizes). Selecting these ''n<sub>h</sub>'' optimally can be done in various ways, using (for example) Neyman's optimal allocation.
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| There are many reasons to use stratified sampling:<ref>Kish (1965, Section 3.1)</ref> to decrease variances of sample estimates, to use partly non-random methods, or to study strata individually.
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| A useful, partly non-random method would be to sample individuals where easily accessible, but, where not, sample clusters to save travel costs.<ref>Kish (1965), p.148.</ref>
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| In general, for ''H'' strata, a weighted sample mean is
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| : <math> \bar x_w = \sum_{h=1}^H W_h \bar x_h, </math>
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| with
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| : <math> \operatorname{Var}(\bar x_w) = \sum_{h=1}^H W_h^2 \,\operatorname{Var}(\bar x_h). </math><ref>Kish (1965), p.78.</ref>
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| The weights, <math>W_h</math>, frequently, but not always, represent the proportions of
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| the population elements in the strata, and <math>W_h=N_h/N</math>. For a fixed sample
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| size, that is <math> N = \sum{N_h} </math>,
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| : <math> \operatorname{Var}(\bar x_w) = \sum_{h=1}^H W_h^2 \,Var_h \left(\frac{1}{n_h} - \frac{1}{N_h}\right), </math><ref>Kish (1965), p.81.</ref>
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| which can be made a minimum if the [[sampling rate]] within each stratum is made
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| proportional to the standard deviation within each stratum: <math> n_h/N_h=k S_h </math>, where <math> S_h = \sqrt{Var_h} </math> and <math>k</math> is a constant such that <math> \sum{n_h} = n </math>.
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| An "optimum allocation" is reached when the sampling rates within the strata
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| are made directly proportional to the standard deviations within the strata
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| and inversely proportional to the square root of the sampling cost per element
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| within the strata, <math>C_h</math>:
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| : <math> \frac{n_h}{N_h} = \frac{K S_h}{\sqrt{C_h}}, </math><ref>Kish (1965), p.93.</ref>
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| where <math>K</math> is a constant such that <math> \sum{n_h} = n </math>, or, more generally, when
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| : <math> n_h = \frac{K' W_h S_h}{\sqrt{C_h}}. </math><ref>Kish (1965), p.94.</ref>
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| == Too-large sample size problem ==
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| A too large sample size may lead to the rejection of a [[null hypothesis]] even if the actual effect is so small, that it does not have practical importance. See [[Null_hypothesis#Too-large_sample_size_problem]].
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| ==Software of sample size calculations==
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| See [[Statistical power#Software for Power and Sample Size Calculations|Software for Power and Sample Size Calculations]].
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| ==See also==
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| {{Portal|Statistics}}
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| *[[Degrees of freedom (statistics)]]
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| *[[Design of experiments]]
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| *[[Replication (statistics)]]
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| *[[Sampling (statistics)]]
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| *[[Statistical power]]
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| *[[Stratified sampling]]
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| *Engineering response surface example under [[Stepwise regression]].
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| *{{cite journal |last=Bartlett |first=J. E., II |last2=Kotrlik |first2=J. W. |last3=Higgins |first3=C. |year=2001 |url=http://www.osra.org/itlpj/bartlettkotrlikhiggins.pdf |title=Organizational research: Determining appropriate sample size for survey research |journal=Information Technology, Learning, and Performance Journal |volume=19 |issue=1 |pages=43–50 |doi= }}
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| *{{cite book |authorlink=Leslie Kish |last=Kish |first=L. |year=1965 |title=Survey Sampling |publisher=Wiley |isbn=0-471-48900-X }}
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| ==Further reading==
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| * [http://www.itl.nist.gov/div898/handbook/ppc/section3/ppc333.htm NIST: Selecting Sample Sizes]
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| * [http://ravenanalytics.com/Articles/Sample_Size_Calculations.htm Raven Analytics: Sample Size Calculations]
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| * [[ASTM]] E122-07: Standard Practice for Calculating Sample Size to Estimate, With Specified Precision, the Average for a Characteristic of a Lot or Process
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| ==External links==
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| * [http://www.nss.gov.au/nss/home.NSF/pages/Sample+size+calculator Sample size calculator] from the Australian [[National Statistical Service]]
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| * [http://www.raosoft.com/samplesize.html Sample Size Calculator by Raosoft, Inc.]
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| {{Statistics|collection|state=expanded}}
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| {{DEFAULTSORT:Sample Size}}
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| [[Category:Sampling (statistics)]]
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| [[de:Zufallsstichprobe#Stichprobenumfang]]
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