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I have read/heard many times that the sample size of at least 30 units is considered as "large sample" (normality assumptions of means usually approximately holds due to the CLT, ...). Therefore, in my experiments, I usually generate samples of 30 units. Can you please give me some reference which should be cited when using sample size 30?

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    $\begingroup$ Without reference to the number of parameters you try to estimate, or equivalently the kind of model your are working with, it seems rather difficult to give you a clear answer. $\endgroup$
    – chl
    Sep 10, 2010 at 18:36
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    $\begingroup$ Acceptance of n=30 as boundary of small and large samples is not well supported by any statistical technique. $\endgroup$
    – Jibol
    Dec 4, 2017 at 9:14
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    $\begingroup$ One reference you can now cite (as of 2013) for why 30 units is not necessarily sufficient -- and showing that there can be no correct reference supporting your claim -- appears here on CV at stats.stackexchange.com/questions/69898. $\endgroup$
    – whuber
    May 27, 2021 at 15:43

5 Answers 5

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The choice of n = 30 for a boundary between small and large samples is a rule of thumb, only. There is a large number of books that quote (around) this value, for example, Hogg and Tanis' Probability and Statistical Inference (7e) says "greater than 25 or 30".

That said, the story told to me was that the only reason 30 was regarded as a good boundary was because it made for pretty Student's t tables in the back of textbooks to fit nicely on one page. That, and the critical values (between Student's t and Normal) are only off by approximately up to 0.25, anyway, from df = 30 to df = infinity. For hand computation the difference didn't really matter.

Nowadays it is easy to compute critical values for all sorts of things to 15 decimal places. On top of that we have resampling and permutation methods for which we aren't even restricted to parametric population distributions.

In practice I never rely on n = 30. Plot the data. Superimpose a normal distribution, if you like. Visually assess whether a normal approximation is appropriate (and ask whether an approximation is even really needed). If generating samples for research and an approximation is obligatory, generate enough of a sample size to make the approximation as close as desired (or as close as computationally feasible).

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Actually, the "magic number" 30 is a fallacy. See Jacob's Cohen's delightful paper, Things I Have Learned (So Far) (Am. Psych. December 1990 45 #12, pp 1304-1312). This myth is his first example of how "some things you learn aren't so".

[O]ne of my fellow doctoral candidates undertook a dissertation [with] a sample of only 20 cases per group. ... [L]ater I discovered ... that for a two-independent-group-mean comparison with $n = 30$ per group at the sanctified two-tailed $.05$ level, the probability that a medium-sized effect would be labeled as significant by ... a t test was only $.47$. Thus, it was approximately a coin flip whether one would get a significant result, even though, in reality, the effect size was meaningful. ... [My friend] ended up with nonsignificant results–with which he proceeded to demolish an important branch of psychoanalytic theory.

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    $\begingroup$ Beautiful reference--and spot on relevant. Thank you. $\endgroup$
    – whuber
    Sep 10, 2010 at 22:21
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    $\begingroup$ @whuber Do you remember which paper it was? The link is broken by now. Maybe this psych.colorado.edu/~willcutt/pdfs/Cohen_1990.pdf, "Things I Have Learned (So Far)"? The year matches to the one in the URL of the broken link. $\endgroup$
    – amoeba
    Dec 4, 2017 at 9:50
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    $\begingroup$ @Amoeba I saved this paper when I read it, so I can confirm what you found is the intended one. I have updated this answer to include a citation along with your link. $\endgroup$
    – whuber
    Dec 4, 2017 at 15:22
  • $\begingroup$ @Carlos Accioly I've updated it with the new link as the previous one was broken. $\endgroup$ May 16, 2019 at 10:36
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IMO, it all depends on what you want to use your sample for. Two "silly" examples to illustrate what I mean: If you need to estimate a mean, 30 observations is more than enough. If you need to estimate a linear regression with 100 predictors, 30 observations will not be close to enough.

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Mostly arbitrary rule of thumb. This statement depends on a number of factor to be true. For example on the distribution of the data. If the data comes from a Cauchy for example, even 30^30 observations are not enough to estimate the mean (in that case even an infinite number of observations would not be enough to cause $\bar{\mu}^{(n)}$ to converge). This number (30) is also false if the values you draw are not independent from one another (again, you may have that there are no convergence at all, regardless of sample size).

More generally, the CLT needs essentially two pillars to hold:

  1. That the random variables are independent: that you can re-order your observations without losing any information*.
  2. That the r.v. come from a distribution with finite second moments: meaning that the classical estimators of mean and s.d. tend to converge as sample size increases.

(Both these condition can be somewhat weakened, but the differences are largely of theoretical nature)

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    $\begingroup$ Your example illustrates the value of robust statistics. The sample median estimates the location parameter of a Cauchy distribution well. One could argue that the weakest link in using a t-test with 30 samples is the t-test, not the 30 samples. $\endgroup$ Sep 11, 2010 at 14:35
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    $\begingroup$ John:> "One could argue that the weakest link in using a t-test with 30 samples is the t-test, not the 30 samples". Very true, and also the assumption that the data is iid. Also, the median is MLE for Cauchy distributed random variables (and hence efficient), but in general you could need more than 30 observations. $\endgroup$
    – user603
    Sep 12, 2010 at 23:54
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    $\begingroup$ Not all versions of the CLT rely on being identically distributed, nor even independence. The basic ones taught to undergrads often do, but there are versions that don't make both assumptions e.g. the Lyapunov CLT assumes independence but not identical distributions, and the independence condition can also be relaxed, for example see here. That 'reordering' thing is also not the same as independence. Some forms of dependence don't rely on order. $\endgroup$
    – Glen_b
    Jun 16, 2013 at 23:40
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    $\begingroup$ A sample size 50,000 is insufficient for the CLT to work well enough to compute a confidence interval for the mean of a log-normal distribution. $\endgroup$ May 16, 2019 at 11:46
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This is meant to supplement user1108's answer stating that:

That said, the story told to me was that the only reason 30 was regarded as a good boundary was because it made for pretty Student's t tables in the back of textbooks to fit nicely on one page. That, and the critical values (between Student's t and Normal) are only off by approximately up to 0.25, anyway, from df = 30 to df = infinity. For hand computation the difference didn't really matter.

I did some investigation on this issue, and the earliest source I can find is Fisher's Statistical Methods for Research Workers (1925). I remember examining a copy of this text (you can see http://psychclassics.yorku.ca/Fisher/Methods/, for example) and noticing that the following table neatly fit on one page.

enter image description here

From what I recall reading in the text, there is nothing justifying why Fisher chose to stop at $n = 30$. So as far as I know, the only justification for this is that such tables can fit neatly on one page back in the day.

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