Central Limit Theorem Calculator
Sampling distribution probability · Standard error · Z-score · Left / Right / Between tails
Probability Mode
Quick Examples
Step-by-Step Solution
Sample Size Sensitivity — Standard Error
Using current σ from Basic tab (or default σ = 12). SE = σ / √n.
| Sample Size (n) | Standard Error (SE) | Relative to n=1 | Interpretation |
|---|
Z-Score Reference Table
Standard normal CDF values Φ(Z) for Z from −3.0 to +3.0.
| Z | Φ(Z) = P(X ≤ Z) | 1 − Φ(Z) | 2·Φ(Z)−1 (two-tail) |
|---|
What is the Central Limit Theorem?
The Central Limit Theorem (CLT) is one of the most fundamental results in all of statistics. It states that the sampling distribution of the sample mean X̄ approaches a normal distribution as the sample size n increases, regardless of the shape of the underlying population distribution. This remarkable result holds as long as the observations are independent and drawn from the same distribution with a finite mean and variance.
In practical terms, for sample size n ≥ 30, the approximation is generally considered good for most populations. If the population is already normally distributed, the CLT applies for any sample size, no matter how small. The CLT is the foundation for many statistical procedures, including hypothesis tests, confidence intervals, quality control charts, and more.
The CLT Formula
When sampling from a population with mean μ and standard deviation σ, the sampling distribution of X̄ satisfies:
- Sampling distribution: X̄ ~ N(μ, σ/√n) for large n
- Standard Error: SE = σ / √n — the standard deviation of the sampling distribution
- Z-statistic: Z = (X̄ − μ) / SE — standardizes the sample mean to a standard normal variable
- Left-tail probability: P(X̄ ≤ x) = Φ(Z)
- Right-tail probability: P(X̄ ≥ x) = 1 − Φ(Z)
- Between probability: P(x₁ ≤ X̄ ≤ x₂) = Φ(Z₂) − Φ(Z₁)
where Φ denotes the standard normal cumulative distribution function (CDF).
When Does the CLT Apply?
- Sample size n ≥ 30: The general rule of thumb for most populations
- Independent observations: Each sampled value must not influence others
- Random sampling: Samples must be drawn randomly from the population
- Already normal population: If the population is normal, CLT applies for any n
- Finite variance: The population must have a finite, well-defined variance
For heavily skewed distributions (e.g., exponential or Pareto), a larger sample size (n ≥ 50 or even n ≥ 100) may be needed before the sampling distribution is approximately normal.
CLT in Practice — Worked Examples
Example 1: Exam Scores
A university records that final exam scores in a statistics course are distributed with population mean μ = 70 and standard deviation σ = 12. A professor randomly selects 36 students. What is the probability that the sample mean score is at most 73?
Step 1: SE = 12 / √36 = 12 / 6 = 2.0
Step 2: Z = (73 − 70) / 2.0 = 1.50
Step 3: P(X̄ ≤ 73) = Φ(1.50) ≈ 0.9332 = 93.32%
There is approximately a 93.32% chance that the sample mean will be 73 or below.
Example 2: Manufacturing Quality Control
A factory produces bolts with a target length of μ = 500 mm and σ = 20 mm. A quality control inspector samples n = 16 bolts. What is the probability that the sample mean length exceeds 505 mm?
Step 1: SE = 20 / √16 = 20 / 4 = 5.0
Step 2: Z = (505 − 500) / 5.0 = 1.00
Step 3: P(X̄ ≥ 505) = 1 − Φ(1.00) ≈ 1 − 0.8413 = 0.1587 = 15.87%
There is about a 15.87% chance that a sample of 16 bolts will have a mean length above 505 mm. Note that n = 16 < 30, so the CLT approximation is less reliable unless bolt lengths are approximately normally distributed.
Example 3: Polling and Surveys
A national survey finds that adult daily screen time has mean μ = 4.5 hours and σ = 1.8 hours. A researcher surveys n = 100 adults. What is the probability the sample mean falls between 4.2 and 4.8 hours?
Step 1: SE = 1.8 / √100 = 1.8 / 10 = 0.18
Step 2: Z₁ = (4.2 − 4.5) / 0.18 ≈ −1.667, Z₂ = (4.8 − 4.5) / 0.18 ≈ 1.667
Step 3: P(4.2 ≤ X̄ ≤ 4.8) = Φ(1.667) − Φ(−1.667) ≈ 0.9044 − 0.0956 = 0.9044 − 0.0956 ≈ 0.9044
There is approximately a 90.44% chance the sample mean falls within this range.
Standard Error Table
This table shows how the standard error decreases as sample size increases for σ = 15 (common IQ-scale example):
| Sample Size (n) | SE = 15/√n | Reduction from n=1 |
|---|---|---|
| 1 | 15.000 | — |
| 5 | 6.708 | 55.3% smaller |
| 10 | 4.743 | 68.4% smaller |
| 25 | 3.000 | 80.0% smaller |
| 30 | 2.739 | 81.7% smaller |
| 50 | 2.121 | 85.9% smaller |
| 100 | 1.500 | 90.0% smaller |
| 200 | 1.061 | 92.9% smaller |
| 500 | 0.671 | 95.5% smaller |
Applications of the CLT
- Hypothesis Testing: The CLT allows the use of Z-tests and t-tests even when the population is not normally distributed, provided the sample is large enough.
- Confidence Intervals: Confidence intervals for population means are constructed using the CLT: X̄ ± z* · (σ/√n).
- Quality Control: Control charts (X̄ charts) in manufacturing use CLT to set control limits, monitoring process stability.
- Polling and Surveys: Pollsters use the CLT to calculate margins of error and determine how many respondents are needed for a given precision.
- Financial Models: Portfolio returns, which are sums of many asset returns, become approximately normally distributed by the CLT, enabling risk calculations.
- Clinical Trials: Sample means of patient outcomes are analyzed using CLT-based statistical tests to determine treatment efficacy.