A Survey of 64 Families Yields the Following Data for the Number of Children Per Family.

Statistical way determining sample size of population

Sample size conclusion is the act of choosing the number of observations or replicates to include in a statistical sample. The sample size is an important feature of any empirical study in which the goal is to brand inferences most a population from a sample. In exercise, the sample size used in a study is usually determined based on the cost, time, or convenience of collecting the data, and the need for information technology to offering sufficient statistical power. In complicated studies there may be several unlike sample sizes: for example, in a stratified survey in that location would exist dissimilar sizes for each stratum. In a census, data is sought for an entire population, hence the intended sample size is equal to the population. In experimental pattern, where a report may be divided into unlike treatment groups, there may be different sample sizes for each grouping.

Sample sizes may be chosen in several ways:

• using experience – small samples, though sometimes unavoidable, can outcome in wide confidence intervals and adventure of errors in statistical hypothesis testing.
• using a target variance for an guess to be derived from the sample eventually obtained, i.due east. if a high precision is required (narrow confidence interval) this translates to a low target variance of the figurer.
• using a target for the ability of a statistical test to be practical once the sample is collected.
• using a conviction level, i.eastward. the larger the required conviction level, the larger the sample size (given a abiding precision requirement).

Introduction [edit]

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 precise gauge of this proportion if we sampled and examined 200 rather than 100 fish. Several cardinal facts of mathematical statistics depict this phenomenon, including the law of large numbers and the central limit theorem.

In some situations, the increase in precision 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 follows a heavy-tailed distribution.

Sample sizes may exist evaluated by the quality of the resulting estimates. For example, if a proportion is being estimated, one may wish to have the 95% conviction 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 comparison the support for a sure political candidate among women with the back up for that candidate among men, we may wish to take 80% ability to detect a departure in the back up levels of 0.04 units.

Interpretation [edit]

Estimation of a proportion [edit]

A relatively simple state of affairs is interpretation of a proportion. For case, nosotros may wish to estimate the proportion of residents in a customs who are at to the lowest degree 65 years old.

The computer of a proportion is ${\displaystyle {\chapeau {p}}=10/northward}$ , 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 onetime). When the observations are independent, this estimator has a (scaled) binomial distribution (and is too the sample hateful of data from a Bernoulli distribution). The maximum variance of this distribution is 0.25, which occurs when the truthful parameter is p = 0.5. In practice, since p is unknown, the maximum variance is often used for sample size assessments. If a reasonable estimate for p is known the quantity ${\displaystyle p(1-p)}$ may be used in identify of 0.25.

For sufficiently big due north, the distribution of ${\displaystyle {\hat {p}}}$ will exist closely approximated past a normal distribution.^[1] Using this and the Wald method for the binomial distribution, yields a confidence interval of the form

${\displaystyle \left({\widehat {p}}-Z{\sqrt {\frac {0.25}{n}}},\quad {\widehat {p}}+Z{\sqrt {\frac {0.25}{north}}}\right)}$ ,

where Z is a standard Z-score for the desired level of confidence (1.96 for a 95% confidence interval).

If nosotros wish to have a confidence interval that is W units full in width (W/2 on each side of the sample hateful), nosotros would solve

${\displaystyle Z{\sqrt {\frac {0.25}{n}}}=W/2}$

for n, yielding the sample size

sample sizes for binomial proportions given different conviction levels and margins of mistake

${\displaystyle n={\frac {Z^{2}}{W^{2}}}}$ , in the instance of using .5 as the most conservative approximate of the proportion. (Notation: W/ii = margin of error.)

In the figure beneath one can find how sample sizes for binomial proportions alter given different confidence levels and margins of error.

Otherwise, the formula would exist ${\displaystyle Z{\sqrt {\frac {p(one-p)}{n}}}=West/ii}$ , which yields ${\displaystyle north={\frac {4Z^{2}p(one-p)}{Westward^{2}}}}$ .

For example, if we are interested in estimating the proportion of the US population who supports a detail presidential candidate, and we desire the width of 95% confidence interval to be at nigh 2 percentage points (0.02), so nosotros would demand a sample size of (1.96ⁱⁱ)/(0.02ⁱⁱ) = 9604. It is reasonable to use the 0.5 estimate for p in this case because the presidential races are often close to 50/fifty, and information technology is also prudent to apply a conservative estimate. The margin of error in this case is 1 percentage point (half of 0.02).

The foregoing is commonly simplified...

${\displaystyle \left({\widehat {p}}-one.96{\sqrt {\frac {0.25}{n}}},{\widehat {p}}+ane.96{\sqrt {\frac {0.25}{n}}}\right)}$

will grade a 95% confidence interval for the true proportion. If this interval needs to be no more than Westward units broad, the equation

${\displaystyle four{\sqrt {\frac {0.25}{n}}}=Due west}$

can be solved for northward, yielding^{[2] [three]} due north = four/West ² = 1/B ² where B is the error bound on the approximate, i.e., the guess is usually given equally inside ± B. And so, for B = 10% ane requires northward = 100, for B = 5% i needs due north = 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 oftentimes in news reports of stance polls and other sample surveys. However, always call back that the results reported may not be the exact value as numbers are preferably rounded up. Knowing that the value of the n is the minimum number of sample points needed to acquire the desired result, the number of respondents then must lie on or in a higher place the minimum.

Estimation of a mean [edit]

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

${\displaystyle {\frac {\sigma }{\sqrt {north}}}.}$

This expression describes quantitatively how the approximate becomes more precise as the sample size increases. Using the central limit theorem to justify approximating the sample hateful with a normal distribution yields a confidence interval of the class

${\displaystyle \left({\bar {x}}-{\frac {Z\sigma }{\sqrt {n}}},\quad {\bar {x}}+{\frac {Z\sigma }{\sqrt {northward}}}\right)}$ ,

where Z is a standard Z-score for the desired level of confidence (1.96 for a 95% confidence interval).

If we wish to have a confidence interval that is Westward units full in width (W/2 on each side of the sample mean), we would solve

${\displaystyle {\frac {Z\sigma }{\sqrt {due north}}}=Due west/2}$

for due north, yielding the sample size

${\displaystyle n={\frac {4Z^{2}\sigma ^{ii}}{W^{2}}}}$ . (Note: W/2 = margin of error.)

For example, if we are interested in estimating the amount by which a drug lowers a subject's claret pressure with a 95% confidence interval that is six units wide, and nosotros know that the standard difference of blood pressure in the population is 15, then the required sample size is ${\displaystyle {\frac {4\times one.96^{2}\times fifteen^{2}}{six^{2}}}=96.04}$ , which would be rounded up to 97, considering the obtained value is the minimum sample size, and sample sizes must be integers and must lie on or to a higher place the calculated minimum.

Required sample sizes for hypothesis tests [edit]

A common problem faced by statisticians is calculating the sample size required to yield a certain power for a test, given a predetermined Type I error rate α. Every bit follows, this can exist estimated past pre-adamant tables for certain values, by Mead's resource equation, or, more generally, by the cumulative distribution function:

Tables [edit]

[4]   Power	Cohen's d
	0.two	0.v	0.viii
0.25	84	14	6
0.50	193	32	13
0.threescore	246	40	16
0.70	310	50	twenty
0.80	393	64	26
0.90	526	85	34
0.95	651	105	42
0.99	920	148	58

The table shown on the correct tin can be used in a two-sample t-test to judge the sample sizes of an experimental grouping and a control group that are of equal size, that is, the full number of individuals in the trial is twice that of the number given, and the desired significance level is 0.05.^[4] The parameters used are:

• The desired statistical ability of the trial, shown in column to the left.
• Cohen'south d (= outcome size), which is the expected departure between the means of the target values betwixt the experimental grouping and the control grouping, divided past the expected standard deviation.

Mead's resources equation [edit]

Mead'south resources equation is often used for estimating sample sizes of laboratory animals, every bit well as in many other laboratory experiments. It may non be every bit authentic as using other methods in estimating sample size, merely 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.^[5]

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 past 1 earlier insertion into the equation.

The equation is:^[5]

${\displaystyle Eastward=N-B-T,}$

where:

• Northward is the total number of individuals or units in the study (minus one)
• B is the blocking component, representing environmental effects allowed for in the design (minus 1)
• T is the handling component, corresponding to the number of treatment groups (including control group) existence used, or the number of questions beingness asked (minus i)
• E is the degrees of liberty of the error component, and should be somewhere between 10 and xx.

For example, if a written report using laboratory animals is planned with four handling groups (T=three), with eight animals per group, making 32 animals total (Northward=31), without any further stratification (B=0), then E would equal 28, which is in a higher place the cutoff of 20, indicating that sample size may be a fleck too large, and six animals per group might be more appropriate.^[half-dozen]

Cumulative distribution function [edit]

Allow Ten_i , i = 1, ii, ..., n exist independent observations taken from a normal distribution with unknown hateful μ and known variance σⁱⁱ. Consider 2 hypotheses, a nada hypothesis:

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

and an culling hypothesis:

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

for some 'smallest pregnant 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 (two) reject H ₀ with probability α when H ₀ is true, so we need the following:

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

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

then

'Reject H ₀ if our sample boilerplate ( ${\displaystyle {\bar {ten}}}$ ) is more than ${\displaystyle z_{\alpha }\sigma /{\sqrt {n}}}$ '

is a decision rule which satisfies (two). (This is a 1-tailed test.)

At present we wish for this to happen with a probability at to the lowest degree 1 −β when H _a is true. In this case, our sample average will come up from a Normal distribution with mean μ^*. Therefore, we require

${\displaystyle \Pr({\bar {10}}>z_{\blastoff }\sigma /{\sqrt {n}}\mid H_{a})\geq 1-\beta }$

Through conscientious manipulation, this can be shown (run across Statistical ability#Example) to happen when

${\displaystyle n\geq \left({\frac {z_{\blastoff }+\Phi ^{-1}(1-\beta )}{\mu ^{*}/\sigma }}\correct)^{two}}$

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

Stratified sample size [edit]

With more complicated sampling techniques, such as stratified sampling, the sample can oftentimes be split upwardly into sub-samples. Typically, if there are H such sub-samples (from H unlike strata) then each of them will have a sample size n_h , h = 1, 2, ..., H. These north_h must conform to the dominion that n _ane + n ₂ + ... + n _H = due north (i.e. that the full sample size is given by the sum of the sub-sample sizes). Selecting these north_h optimally can be done in various ways, using (for example) Neyman's optimal allocation.

There are many reasons to use stratified sampling:^[7] to decrease variances of sample estimates, to apply partly non-random methods, or to study strata individually. A useful, partly non-random method would be to sample individuals where hands attainable, merely, where not, sample clusters to save travel costs.^[8]

In general, for H strata, a weighted sample mean is

${\displaystyle {\bar {10}}_{w}=\sum _{h=1}^{H}W_{h}{\bar {x}}_{h},}$

with

${\displaystyle \operatorname {Var} ({\bar {x}}_{west})=\sum _{h=i}^{H}W_{h}^{ii}\operatorname {Var} ({\bar {10}}_{h}).}$ ^[9]

The weights, ${\displaystyle W_{h}}$ , frequently, just not always, represent the proportions of the population elements in the strata, and ${\displaystyle W_{h}=N_{h}/N}$ . For a fixed sample size, that is ${\displaystyle n=\sum n_{h}}$ ,

${\displaystyle \operatorname {Var} ({\bar {x}}_{w})=\sum _{h=1}^{H}W_{h}^{2}\operatorname {Var} ({\bar {x}}_{h})\left({\frac {1}{n_{h}}}-{\frac {1}{N_{h}}}\correct),}$ ^[10]

which can exist fabricated a minimum if the sampling rate within each stratum is made proportional to the standard deviation inside each stratum: ${\displaystyle n_{h}/N_{h}=kS_{h}}$ , where ${\displaystyle S_{h}={\sqrt {\operatorname {Var} ({\bar {x}}_{h})}}}$ and ${\displaystyle chiliad}$ is a abiding such that ${\displaystyle \sum {n_{h}}=north}$ .

An "optimum resource allotment" is reached when the sampling rates within the strata are made directly proportional to the standard deviations inside the strata and inversely proportional to the foursquare root of the sampling toll per element within the strata, ${\displaystyle C_{h}}$ :

${\displaystyle {\frac {n_{h}}{N_{h}}}={\frac {KS_{h}}{\sqrt {C_{h}}}},}$ ^[xi]

where ${\displaystyle Thou}$ is a constant such that ${\displaystyle \sum {n_{h}}=n}$ , or, more than by and large, when

${\displaystyle n_{h}={\frac {K'W_{h}S_{h}}{\sqrt {C_{h}}}}.}$ ^[12]

Qualitative research [edit]

Sample size determination in qualitative studies takes a unlike arroyo. Information technology is generally a subjective judgment, taken as the research proceeds.^[13] One arroyo is to continue to include farther participants or material until saturation is reached.^[fourteen] The number needed to reach saturation has been investigated empirically.^{[15] [sixteen] [17] [18]}

At that place is a paucity of reliable guidance on estimating sample sizes before starting the enquiry, with a range of suggestions given.^{[16] [19] [20] [21]} A tool alike to a quantitative ability calculation, based on the negative binomial distribution, has been suggested for thematic assay.^{[22] [21]}

See too [edit]

• Design of experiments
• Technology response surface example under Stepwise regression
• Cohen's h

Notes [edit]

1. ^ NIST/SEMATECH, "7.2.4.2. Sample sizes required", e-Handbook of Statistical Methods.
2. ^ "Inference for Regression". utdallas.edu.
3. ^ "Confidence Interval for a Proportion" Archived 2011-08-23 at the Wayback Automobile
4. ^ ^{a b} Chapter 13, page 215, in: Kenny, David A. (1987). Statistics for the social and behavioral sciences. Boston: Little, Dark-brown. ISBN978-0-316-48915-seven.
5. ^ ^{a b} Kirkwood, James; Robert Hubrecht (2010). The UFAW Handbook on the Care and Management of Laboratory and Other Research Animals. Wiley-Blackwell. p. 29. ISBN978-ane-4051-7523-4. online Page 29
6. ^ Isogenic.info > Resource equation by Michael FW Festing. Updated Sept. 2006
7. ^ Kish (1965, Section 3.one)
8. ^ Kish (1965), p. 148.
9. ^ Kish (1965), p. 78.
10. ^ Kish (1965), p. 81.
11. ^ Kish (1965), p. 93.
12. ^ Kish (1965), p. 94.
13. ^ Sandelowski, M. (1995). Sample size in qualitative enquiry. Research in Nursing & Wellness, 18, 179–183
14. ^ Glaser, B. (1965). The constant comparative method of qualitative analysis. Social Bug, 12, 436–445
15. ^ Francis, Jill J.; Johnston, Marie; Robertson, Clare; Glidewell, Liz; Entwistle, Vikki; Eccles, Martin P.; Grimshaw, Jeremy Yard. (2010). "What is an adequate sample size? Operationalising information saturation for theory-based interview studies" (PDF). Psychology & Health. 25 (10): 1229–1245. doi:ten.1080/08870440903194015. PMID 20204937. S2CID 28152749.
16. ^ ^{a b} Guest, Greg; Bunce, Arwen; Johnson, Laura (2006). "How Many Interviews Are Enough?". Field Methods. 18: 59–82. doi:x.1177/1525822X05279903. S2CID 62237589.
17. ^ Wright, Adam; Maloney, Francine L.; Feblowitz, Joshua C. (2011). "Clinician attitudes toward and apply of electronic problem lists: A thematic analysis". BMC Medical Informatics and Decision Making. 11: 36. doi:10.1186/1472-6947-11-36. PMC3120635. PMID 21612639.
18. ^ Mason, Mark (2010). "Sample Size and Saturation in PhD Studies Using Qualitative Interviews". Forum Qualitative Sozialforschung. 11 (3): viii.
19. ^ Emmel, Northward. (2013). Sampling and choosing cases in qualitative enquiry: A realist approach. London: Sage.
20. ^ Onwuegbuzie, Anthony J.; Leech, Nancy 50. (2007). "A Call for Qualitative Power Analyses". Quality & Quantity. 41: 105–121. doi:10.1007/s11135-005-1098-1. S2CID 62179911.
21. ^ ^{a b} Fugard AJB; Potts HWW (10 Feb 2015). "Supporting thinking on sample sizes for thematic analyses: A quantitative tool" (PDF). International Journal of Social Research Methodology. 18 (half dozen): 669–684. doi:10.1080/13645579.2015.1005453. S2CID 59047474.
22. ^ Galvin R (2015). How many interviews are plenty? Do qualitative interviews in edifice energy consumption inquiry produce reliable knowledge? Journal of Building Engineering, 1:2–12.

References [edit]

• Bartlett, J. E., II; Kotrlik, J. West.; Higgins, C. (2001). "Organizational inquiry: Determining appropriate sample size for survey enquiry" (PDF). It, Learning, and Performance Journal. 19 (1): 43–50.
• Kish, L. (1965). Survey Sampling . Wiley. ISBN978-0-471-48900-9.
• Smith, Scott (8 April 2013). "Determining Sample Size: How to Ensure You Get the Correct Sample Size". Qualtrics . Retrieved 19 September 2018.
• Israel, Glenn D. (1992). "Determining Sample Size". University of Florida, PEOD-6 . Retrieved 29 June 2019.
• Rens van de Schoot, Milica Miočević (eds.). 2020. Pocket-sized Sample Size Solutions (Open up Access): A Guide for Practical Researchers and Practitioners. Routledge.

Further reading [edit]

• NIST: Selecting Sample Sizes
• ASTM E122-07: Standard Practise for Computing Sample Size to Gauge, With Specified Precision, the Average for a Characteristic of a Lot or Process

External links [edit]

• A MATLAB script implementing Cochran'south sample size formula

connelltherstaid.blogspot.com

Source: https://en.wikipedia.org/wiki/Sample_size_determination

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