• Understand the basic idea of the central limit theorem
Deeper study
• Compute probabilities using the central limit theorem
Real-world practice
• Apply the central limit theorem in practice
At a glance
The central limit theorem (CLT) states that when independent samples of sufficiently large size n are drawn from a population with mean μ and variance σ², the distribution of the sample mean X̄ converges to the normal distribution N(μ, σ²/n), regardless of the shape of the population distribution.
The standard error of the sample mean is σ/√n, so the larger the sample size n, the more tightly the sample mean clusters around the population mean
As a rule of thumb, a sample size of n≥30 is considered large enough for the normal approximation to hold
It is the theoretical basis of confidence intervals and hypothesis tests; for example, the roughly ±2.8% margin of error at 95% confidence for a poll of 1,200 people comes from this theorem
The central limit theorem's conclusion is a normal distribution, so you should understand the normal distribution first
Basic: What Is the Central Limit Theorem?
Difficulty 3/5
About 25 min
The central limit theorem is a key result stating that when the sample size is large enough, the distribution of the sample mean approaches a normal distribution, no matter what shape the population distribution has.
Think of a dice experiment. Roll a single die and the outcomes 1 through 6 are equally likely, a flat distribution. But roll several dice and take their average, and the distribution of those averages becomes bell-shaped. The more dice you average, the closer that shape gets to a perfect bell curve (a normal distribution).
Key points
•
The larger the sample, the closer the sample mean is to normally distributed
•
Applies regardless of the shape of the population distribution
•
As a rule of thumb, n≥30 is enough
•
The theoretical foundation of statistical inference
A simple example
Flip a coin 10 times and record the proportion of heads. Repeat this experiment 100 times, and those 100 proportions form a normal distribution centered on 0.5.
Check your understanding
Answer:
Because it lets you do normal-distribution-based statistical inference using the sample mean even when you do not know the population's distribution.
The central limit theorem says that as the sample size grows, the distribution of the sample mean converges to a normal distribution.
Mathematical statement:
Let Loading... be independent draws from the same distribution with mean Loading... and variance Loading.... Then the distribution of the sample mean Loading... converges to a normal distribution as n becomes large.
Derivation:
1) Expected value and variance of the sample mean
- Loading...
- Loading...
2) Standardization
- Standardized variable: Loading...
- By the central limit theorem: Loading... as Loading...
3) Conclusion
- Loading... holds approximately (for large n)
Mathematical formula
X̄ is the sample mean, μ the population mean, σ² the population variance, and n the sample size. The standard deviation of the sample mean (the standard error) is σ/√n, so it shrinks as the sample size grows.
Learn through examples
Example 1: Quality Control
Products with mean weight 500 g and standard deviation 20 g, sampled 25 at a time
Given:
• Population mean μ = 500 g, population standard deviation σ = 20 g, sample size n = 25
• Distribution of the sample mean: Loading...
• Standard error: Loading...
Calculation:
Loading...
Answer:
The probability that the sample mean is 495 g or less is about 10.6%.
Even if individual product weights vary a lot, the mean of a 25-unit sample stays tightly distributed around the population mean.
Example 2: Confidence Interval for a Poll
A candidate with 45% true support is polled using a sample of 1000 people
Applying the CLT to a Bernoulli distribution:
• Population proportion p = 0.45, sample size n = 1000
• Distribution of the sample proportion: Loading...
• Standard error: Loading...
95% confidence interval:
Loading...
Answer:
The 95% confidence interval is [41.9%, 48.1%], a margin of error of ±3.1%.
The larger the sample, the narrower the confidence interval and the more precise the estimate.
Example 3: Portfolio Risk Management
Computing the monthly risk of a stock portfolio
Portfolio return analysis:
• Individual stock monthly return: mean 1.2%, standard deviation 8%
• Portfolio: 100 stocks, equally weighted
• Distribution of the portfolio return: Loading...
Risk measure (VaR):
• 5% VaR: Loading...
• There is a 5% chance the monthly loss exceeds 0.116%
Answer:
Diversification cuts the risk of an individual stock (8%) down to a portfolio risk of 0.8%.
The central limit theorem provides the mathematical justification for the benefits of diversification.
Common mistakes
Mistake:
Applying the central limit theorem to a small sample
Why is it wrong?
The CLT is an asymptotic result; it holds when the sample size is large enough (usually n≥30)
Correct approach:
For small samples, take the population distribution into account or use the t-distribution
Mistake:
Confusing the distribution of the sample mean with the distribution of individual observations
Why is it wrong?
The sample mean has a different distribution from individual observations; its standard deviation shrinks by a factor of 1/√n
Correct approach:
Remember that the standard error of the sample mean = σ/√n
Case study: Analyzing the Accuracy of an Election Poll
Background
P, a major Korean polling firm, needs to measure candidate support accurately ahead of the 2024 presidential election. Past polls have sometimes missed the mark, so the firm wants to determine a statistically defensible sample size and margin of error.
The problem
Within a limited budget and timeline, what sample size yields a meaningful margin of error? And how should the results be interpreted and published so that the public can trust them? In a close race, even a small error can matter a great deal.
The data
Three main candidates (A, B, C). A random telephone survey of 1,200 adult voters nationwide. True support is 42% for A, 38% for B, and 20% for C, but this is unknown at the time of the survey. Response rate 65%, three-day fieldwork period, 95% confidence required.
Method
Sample-proportion analysis using the central limit theorem: model each candidate's support as a Bernoulli distribution and use the fact that, for a large enough sample, the sample proportion converges to a normal distribution. Margin of error: Loading.... Analyze each candidate separately, then assess the overall accuracy of the poll.
Solution and results
Candidate A (true support 42%):
• Sample proportion: Loading...
• Standard error: Loading...
• 95% confidence interval: Loading...
• Margin of error: ±2.8%
Candidate B (true support 38%):
• Sample proportion: Loading...
• Standard error: Loading...
• 95% confidence interval: Loading...
• Margin of error: ±2.7%
Overall accuracy of the poll:
• Maximum margin of error: ±2.8% (largest when p=0.5)
• The 4-point gap between the candidates is statistically significant
Published results:
• Candidate A: 42% (margin of error ±2.8%)
• Candidate B: 38% (margin of error ±2.7%)
• Candidate C: 20% (margin of error ±2.3%)
Statistical interpretation:
• The 4-point gap between A and B remains statistically significant even after accounting for the margin of error
• Probability of leading: about 85% for A, about 15% for B
• A sample of 1,200 reflects national opinion to within ±3%
Guidance for media coverage:
• "Allowing for statistical error, Candidate A is ahead, but the gap with Candidate B could narrow"
• "This poll has a 95% confidence level: in 19 out of 20 polls, the true support would fall within this range"
Conclusion:
A scientifically designed poll grounded in the central limit theorem captured public opinion to within ±3%. A precise understanding of the relationship between sample size and margin of error secured the credibility of the results and made their statistical meaning clear to the public. It also provided a criterion for judging statistical significance even in a close race.
References
Lindeberg, J. W. (1922), "Eine neue Herleitung des Exponentialgesetzes in der Wahrscheinlichkeitsrechnung", Mathematische Zeitschrift 15, 211–225 (DOI 10.1007/BF01494395) — original paper proving the central limit theorem under the Lindeberg condition
Fischer, H. (2011), "A History of the Central Limit Theorem: From Classical to Modern Probability Theory", Springer, New York — the definitive history of the theorem's development from Laplace (1810) to the present
Billingsley, P. (1995), "Probability and Measure", 3rd ed., Wiley, New York — standard probability text with a rigorous measure-theoretic proof of the central limit theorem
Casella, G. & Berger, R. L. (2002), "Statistical Inference", 2nd ed., Duxbury Press, Pacific Grove — standard text deriving the mean and variance of the sample mean (standard error σ/√n) and applying the CLT to statistical inference
