Central limit theorem would suggest N =25-30 for a random sample

I disagree, central limit theorem would suggest an infinite sample ;)

But then, in practice, a random sample of 25-30 individuals may (or may not) yield data that are normally distributed conditionally on the levels of an IV. To know whether it is or not the case, one tests with the appropriate test (e.g., Shapiro-Wilk).

However, Shaozhoong Shi, is this what you meant by your question?

If you rather asked the question of how large a sample must be to be able to detect a difference between conditions (e.g., a medicine vs no medicine), then you might want to read some literature on the size of the effect and then compute the minimum sample size you need in order to have good chances to evidence an effect if there is such an affect. I suggest this book:

Data Analysis: A Model Comparison Approach, by Charles M. Judd, Gary H. McClelland, and Carey S. Ryan. Published by Routledge, 2008.

Best of luck with your research!

cheers,

SCM

http://sphweb.bumc.bu.edu/otlt/MPH-Modules/BS/BS704_Probability/BS704_Probability11.html

is a good reference on all of this

Monte Carlo simulations clearly show that if you randomly sample from some distribution, 30 values, then those 30 values will be distributed normally 99% of the time.

You might be asking a different question which is

What is the sample size needed for a certain level of statistical precision (or confidence interval).

In that case, you would need some a priori estimate of what the intrinsic dispersion is, in the phenomena your measuring.

@G.Bothun: I respectfully but nonetheless strongly disagree with what you say, sir.

I suspect you confuse distribution of a random variable and distribution of the means of a random variable.

For one thing, if I draw 30 observation from, say, a binomial distribution of probability value 0.01, the distribution I'll get is by no means symmetrical, and by no means normal.

You can check this in R:

a

Shaozhong Shi, you may download the following and try it

https://www.researchgate.net/publication/262564897_Easy-to-use_Sampling_Formulae?ev=prf_pub

Ed

Data Easy-to-use Sampling Formulae

Shaozhong Shi, you may download the following and try it

https://www.researchgate.net/publication/262564897_Easy-to-use_Sampling_Formulae?ev=prf_pub

Ed

Data Easy-to-use Sampling Formulae

Dear Shaozhong Shi -

Sample size needs depend upon type of data, collection/survey design, tradeoffs with nonsampling error, and goals/subject matter for particular applications, but behind that there is a basic strategy:

You should limit bias as much as possible, and then base sample size needs on acceptable standard errors (which can be used in confidence intervals, and/or for hypothesis testing, which needs to include type II error analyses/effect size/sensitivity analysis). 

The salient factors are then the following (by stratum or subpopulation if appropriate):

(1) Population standard deviation (a fixed value)

(2) Sample size

(3) Standard error - say of a mean, proportion, or total

You need a basic estimate, or "guess" (see, for example, Cochran(1977), Sampling Techniques, Wiley) for (1), perhaps from a pilot study, or similar study, or basic idea of the distribution.  Perhaps several "guesses" could be used as a study. 

Then using (2) for your application, you will find the estimated value of (3) that may be useful to you. 

Simplified summary: estimate or guess standard deviation, and see what sample size gives you an acceptable standard error. 

(For qualitative data, I suppose the principle is basically the same: After stratification and/or whatever else may be necessary for design, and limiting bias and nonsampling error, see when variability is acceptably accounted for, and that will indicate your sample size.)

Beware of online 'calculators' for sample size.  They usually are only for proportions, for simple random sampling, ignoring any finite population correction factor, and for a worst case (p=q=0.5).  The last thing avoids estimating standard error, because with proportions you have a special case where there is a limit to how 'bad' it can be, and then you assume the worst.  You can see that these calculators are often irrelevant to applications people may have.

Cheers - Jim

Most researchers should consider Slovin's formula first: n = N / 1+N(e)2.

Here is a link to interesting website which describes the formula: 

Source [http://prudencexd.weebly.com/]

Sampling adequacy for more complex methods, such as factor analysis, requires different techniques, i.e. Kaiser–Meyer–Olkin measure of sampling adequacy test. 

Reference:

Yamane, T. (1967). Statistics: An introductory analysis. New York: Harper and Row.

Dear Shaozhong Shi, 

Sample size usually depends on the type of study, the data and research design and method of study. For instance,  Stoker (1989:130) notes: "The total sample size is usually determined by the sample size required for the smallest subgroup, which should be 50 to 100 respondents. In addition, for quantitative analysis, a sample of at least 100 should be obtained--even to calculate only percentages (Kent,1993). I have also shared an important and famous table regarding population and sampling.

Last but not least, I suggest you may want to have a look at the famous book of Krejcie, R.V. & Morgan. Hope you find it beneficial. 

Krejcie, R.V. & Morgan, D.W. (1970). Determining sample size for research activities. Educational and Psychological Measurement. 

You really should not use an arbitrary number like n=30 or 100 or 10 or 1000.  And there is no "one size fits all" kind of formula.  

Standard deviation is key for quantitative statistics, whether design-based or model-based, finite populations or infinite ones, within strata or not, for continuous data or proportions, etc.  See my previous comments above on standard deviation, sample size, and standard error (by subpopulation or stratum). 

..... 

There are a number of good sampling books.  There is a chapter on sample size requirements in Cochran, W.G(1977), Sampling Techniques, 3rd ed., John Wiley & Sons for simple random sampling.  Then later chapters provide sample size information for more complex designs.

....

Here is a paper on sample size requirements for model-based finite population sampling with an often useful degree of heteroscedasticity:

 https://www.researchgate.net/publication/261947825_Projected_Variance_for_the_Model-based_Classical_Ratio_Estimator_Estimating_Sample_Size_Requirements

.....

There is a huge variety of situations, but the key is estimated standard deviation of the population or subpopulation, or for regression (i.e., "prediction") the estimated standard deviation (of the random factors) of the estimated residuals, by model application. 

.................

Examples of some other sampling books follow:

Blair, E. and Blair, J(2015), Applied Survey Sampling, Sage Publications.

Lohr, S.L(2010), Sampling: Design and Analysis, 2nd ed., Brooks/Cole.

Särndal, CE, Swensson, B. and Wretman, J. (1992), Model Assisted Survey Sampling, Springer-Verlang.     

Brewer, KRW (2002), Combined survey sampling inference: Weighing Basu's elephants, Arnold: London and Oxford University Press.  

And also,

An Introduction to Model-Based Survey Sampling with Applications, 2012,

Ray Chambers and Robert Clark,  Oxford Statistical Science Series

Finite Population Sampling and Inference: A Prediction Approach,  2000,

Richard Valliant, Alan H. Dorfman, Richard M. Royall,

Wiley Series in Probability and Statistics.

Survey Sampling: Theory and Methods, Second Edition

Arijit Chaudhuri, Horst Stenger

March 29, 2005 by CRC Press

(First Ed 1992, Marcel Dekker)

.....

But I think you would best try a classic such as

Cochran, W.G(1977), Sampling Techniques, 3rd ed., John Wiley & Sons

or perhaps Hansen, Hurwitz, and Madow  (c 1953). 

............................................. 

Remember -

Key: estimated standard deviation

Conference Paper Projected Variance for the Model-based Classical Ratio Estim...