Larger samples stabilize the mean, and repeated sample means approach a bell-shaped distribution.
Individual customer purchases may be strongly right-skewed: most are small, while a few are very large. Yet repeated averages of 100 customers look much more stable and regular.
The Law of Large Numbers and the Central Limit Theorem explain these two features. They are related, but they answer different questions.
Key question
Where does the sample mean go as n grows, and what distribution do repeated sample means create?
How do the two theorems differ?
Law of Large Numbers
Where does one sample mean go?
It approaches the population mean as the sample grows
Central Limit Theorem
What shape do repeated sample means make?
Their distribution approaches a normal shape
The Law of Large Numbers concerns one increasingly large sample and says that its mean approaches the population mean. The Central Limit Theorem concerns the distribution formed by sample means across repeated samples of the same size.
They are not the same statement
Convergence toward the true mean and convergence toward a normal sampling distribution are different properties.
Why does the sample mean approach the population mean?
A small sample can contain an unusual number of high or low values. In a larger sample, those chance imbalances tend to cancel, bringing the sample mean closer to the population mean.
| Sample size | Typical spread of the sample mean |
|---|---|
| n = 4 | σ/2 |
| n = 25 | σ/5 |
| n = 100 | σ/10 |
The rate matters
The standard error is σ/√n. Cutting the spread in half requires four times as many observations.
Why do sample means become normally distributed?
Repeated samples from the same population
The observations change, but the sample means form a stable pattern.
| Sample | Observations | Sample mean |
|---|---|---|
| 1 | 2 · 3 · 4 · 8 · 13 | 6.0 |
| 2 | 1 · 2 · 5 · 6 · 11 | 5.0 |
| 3 | 3 · 4 · 4 · 5 · 9 | 5.0 |
| 4 | 1 · 3 · 6 · 7 · 8 | 5.0 |
Even when the original population is not normal, averaging enough independent observations often produces a sampling distribution that is approximately normal.
The center of that distribution is μ and its standard deviation is σ/√n. Larger samples therefore produce a narrower distribution around the same center.
Does this work for strongly skewed populations?
A normal population produces normally distributed sample means even for small n. Strongly skewed populations, such as income or purchase value, may require much larger samples before the approximation becomes good.
| Population shape | Sample size needed for a useful approximation |
|---|---|
| Normal or nearly symmetric | often good with relatively small n |
| Moderately skewed | a moderate sample |
| Strong skew or extreme tails | larger samples and robustness checks |
The n ≥ 30 rule
Thirty is not a universal cutoff. The required sample size depends on skew, tail weight, and outliers.
Change sample size and population shape
Choose a skewed, uniform, or binary population and increase n to watch the sampling distribution change.
Population mean
1.000
Mean of sample means
1.004
Observed SD of means
0.454
Theoretical SE
0.447
Sampling distribution of the sample mean
Key lesson
The Law of Large Numbers explains stability of the mean; the Central Limit Theorem explains theshape and standard error of the sampling distribution.
Raw data need not be bell-shaped for sufficiently large sample means to form a predictable bell-shaped distribution.