Central Limit Theorem

Larger samples also tend to bring the sample mean and standard deviation closer to their population values, μ and σ, respectively. That is a separate result: the law of large numbers.

The central limit theorem is about something else: the shape that sample means form around the population mean.

Is a Sample Size of 30 Enough?

Thirty is the figure most often linked to the central limit theorem. It is a rule of thumb, not part of the theorem itself.

The theorem describes a limit. The fit to the normal curve keeps improving as samples grow, and no sample size marks the point where it suddenly switches on.

Where the Figure of 30 Comes From

The figure of 30 is a textbook convention. A sample that size makes the t distribution close to the normal curve. It also makes the mean of moderately skewed scores roughly normal.

So 30 reflects a judgement about acceptable accuracy. It is not a property of the theorem.

Calling 30 a requirement is wrong. If the population is already normal, the sample mean is exactly normal at every sample size, even with one or two scores. No limit theorem is needed in that case.

For other populations, the size needed depends on three things. They are how far the population departs from normal, how accurate the approximation must be, and which part of the distribution matters.

The tails matter most, because a p-value is a tail area of the sampling distribution.

Skewness Sets the Speed

Skewness, a measure of how lopsided a distribution is, decides how quickly the approximation arrives. Boos and Hughes-Oliver (2000) identified it as the key quantity in deciding how large n must be for confidence intervals around a mean.

The skewness of a sample mean equals the population’s skewness divided by √n, the square root of the sample size. So quadrupling a sample only halves the skew.

The table shows how often the mean of 30 scores falls beyond each 5% cut-off of the normal curve. A perfect fit would give 5% in each tail.

Population	Skewness	Upper tail at n = 30	Lower tail at n = 30
Normal	0	5.0%	5.0%
Exponential (e.g. waiting times)	2	5.9%	3.7%
Lognormal (very skewed)	6.2	6.2%	1.5%

The errors are lopsided. In a right-skewed population, the upper tail is too heavy and the lower tail too light. Two-sided tests partly cancel these errors, but one-sided tests do not.

For the lognormal population, the lower tail needs about 1,000 observations to come within a percentage point of its 5% target.

The mathematics of this speed goes back to Berry (1941). He proved that the largest gap between the sample mean’s distribution and the normal curve can never exceed a ceiling that shrinks in proportion to 1/√n.

The ceiling is higher for populations with more skew and heavier tails. The result is now known as the Berry–Esseen theorem.

Berry’s ceiling is a worst-case guarantee, though. For the exponential population at n = 30, it is about 34 times the real gap. So it cannot tell you what sample size a given study needs.

How Skewed Are Real Psychological Data?

Real data are often far from the textbook bell curve. Cain and colleagues (2017) surveyed researchers who had published in Psychological Science and the American Educational Research Journal from 2010 to 2014.

They gathered 1,567 variables from 194 studies. Nearly three-quarters (74%) departed from normality, with skewness as high as 25.5.

Sample size was essentially unrelated to skewness. The team then simulated one-sample t-tests, using skewness levels and sample sizes taken from the survey. Their results showed:

• Two-sided tests: These stayed close to the intended 5% error rate except at extreme skew. At a skewness of 6.32, the rate was 17.7% with 18 cases.
• Larger samples: At that same skewness, it took about 555 cases to bring the two-sided rate down to 6.2%.
• One-sided tests: These were less forgiving. At a skewness of 2.77, the lower-tail rate was 9.2% with 18 cases and still 5.9% with 1,488.

The survey has limits. It relied on authors who chose to reply, from two journals, so it describes those studies rather than psychology as a whole.

Checking the Data You Have

The t statistic needs larger samples still. It divides by the sample’s own standard deviation, and in skewed data, the mean and standard deviation tend to rise together.

Take the exponential population at n = 30. A one-sided lower-tail t-test meant to be wrong 5% of the time is then wrong 9.7% of the time.

A methods paper by Hesterberg (2015) found the same problem in a sample far larger than 30.

The choice of method had financial consequences in this case. Comparing the two providers, a pooled t-test gave p = .0045. A permutation test, which reshuffles the group labels, gave .0171, about four times larger.

The groups were very unequal in size (1,664 against 23). That makes the t-test especially sensitive to skew. The paper shows the problem can happen, not how often it does in psychology.

So do not simply trust the figure of 30. Check the sampling distribution itself.

One way is to simulate the study’s design under a plausible population shape (Rousselet & Wilcox, 2020). Another is to bootstrap your sample and inspect the t statistic’s distribution (Hesterberg, 2015).

Why is the central limit theorem important?

The central limit theorem works whatever the population looks like. However the scores are distributed, the shape of the sampling distribution will approach normality as the sample size (N) increases.

Two conditions apply, though. The observations must be independent, and the population’s variance (its spread around the mean) must be finite.

A normal sampling distribution is useful because a researcher never knows exactly how close their sample mean is to the population mean. In practice, a study draws just one sample.

If many random samples were drawn, their means would cluster around the population mean in a predictable bell shape. That pattern tells the researcher how far one sample mean is likely to fall from the truth.

One large sample can then give a very good estimate.

Thus, the sampling error will decrease as the sample size (N) increases.

Sampling error is the gap between a sample’s mean and the population mean that arises purely by chance. It is not a mistake, and it is not bias. Its typical size is the standard error, σ/√N, so quadrupling a sample only halves it.

The predictable bell shape is the theorem’s real value. Because sample means behave this way whatever the population looks like, one set of methods works across many measures.

Z-tests, t-tests, confidence intervals, ANOVA and regression can all be used on reaction times, test scores and ratings without a separate theory for each.

It also clears up a common confusion. The normality assumption behind these tests concerns the sampling distribution of the statistic, not the raw scores. Every confidence interval and p-value rests on that sampling distribution.

How far this holds in a given study depends on sample size and skew.

Central Limit Theorem Example: Reaction Times

Reaction times show the theorem at work. Most responses are quick, but a few are very slow, so the scores have a long right tail.

Take a population of reaction times with a mean of 600 milliseconds (ms) and a skewness of 1.99. A computer simulation drew 200,000 random samples at each sample size and recorded each sample’s mean.

Sample size (n)	Average of the sample means (ms)	Spread of the means (SD, ms)	Skewness of the means
1	599.8	299.8	1.97
5	600.3	134.8	0.89
10	599.9	95.1	0.63
30	599.9	54.9	0.36
100	600.0	30.1	0.20

Three things stand out. The means stay centred on 600 ms at every size. Their spread shrinks in line with the standard error, σ/√n. And the skew fades smoothly, with no jump at 30.

Why does this happen? A single reaction time can land far out in the slow tail.

For the mean of 10 trials to land there, most of the trials must be slow at once, which is rare. Averaging thins the tails and smooths away the skew.

One catch remains in small samples. At n = 10, 54.3% of sample means fall below the true value. A typical sample mean is therefore slightly too low (Rousselet & Wilcox, 2020). Yet the average of all the sample means is exact.

Limitations of the Central Limit Theorem

Most misuses of the theorem come from reading more into it than it says. Six limits are worth knowing:

• Means, Not Data: It describes the distribution of sample means. A large sample from a skewed population is still a skewed sample.
• Not the LLN: The law of large numbers (LLN) says a sample mean homes in on μ as n grows. The theorem describes the shape of the variation around μ.
• No Minimum Size: It describes a limit, so it sets no required sample size. The figure of 30 is a convention.
• Independence: Observations must be independent. Clustered or repeated measures usually make the textbook standard error too small.
• Finite Variance: The population’s spread must be finite. With extremely heavy tails, sample means may never approach the normal curve.
• Bias: It describes chance variation only. A biased sample gives a tidy bell curve around the wrong value.

The last three limits deserve a closer look.

Dependent and Clustered Observations

The standard theorem assumes each observation is independent of the others. Repeated measures on one person, a time series, and pupils from the same class all break that assumption.

The sample mean may still look normal, since the theorem has versions for dependent data. What goes wrong is the standard error. More data does not fix it.

Take 100 observations where each correlates .5 with the next. The real spread of the mean is 1.72 times the textbook formula, which is like having only about 34 independent scores.

In a simulation, a nominal 95% confidence interval covered the true mean only about 74% of the time.

Clustering does the same. With 10 clusters of 20 people and an intraclass correlation of .1 (the share of variation lying between clusters), the spread was 1.70 times too large. Coverage fell to 75.1%.

Recruiting whole classes or families, as in cluster sampling, produces this kind of data. Multilevel models give correct standard errors by modelling the clusters.

Infinite Variance and Heavy Tails

The theorem also needs the population’s variance, its spread around the mean, to be finite. Some populations have none. The Cauchy distribution, a bell-like curve with very heavy tails, is the classic case.

Averaging Cauchy scores does not help at all. The mean of n observations has exactly the same distribution as one observation. Its spread (the interquartile range) stayed at about 2 from n = 1 to n = 10,000.

A Pareto population is a milder case. It has a finite mean but infinite variance.

In simulation, a nominal 95% t interval covered its true mean 62.5% of the time at n = 10, and only 78.7% at n = 10,000. Earthquake sizes and sales figures are standard examples of such heavy-tailed quantities.

Heavy tails with a finite variance cause less harm. In one simulation, 95% of scores came from an ordinary normal curve and 5% from a much wider one.

The mean of 30 scores still had an excess kurtosis (tail heaviness, zero for the normal curve) of 1.35, against 39.5 for the scores themselves.

Here the t interval erred on the safe side, covering the true mean 97.0% of the time. That costs power but does not inflate false positives.

Biased Samples

The theorem assumes random sampling from the population you want to study. Sampling bias produces a sampling distribution centred on the wrong value. The theorem then describes a very tidy bell curve around the wrong number.

A bigger sample does not cure this. A convenience sample that misses part of the population stays biased however large it grows.

Psychology has a well-known version of the problem. Much of its research relies on participants from Western, educated, industrialised, rich and democratic (WEIRD) societies (Henrich et al., 2010). Their results can differ markedly from those of other populations.

A small standard error around such a sample’s mean may precisely describe a population nobody was interested in. Polls face the same risk, since the sampling distribution says nothing about whether people were drawn at random from the right population.