Central Limit Theorem Calculator

Calculate standard error of the mean and sampling distribution probabilities with step-by-step solutions

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Standard Error: σx̄ = σ / √n

The standard deviation of the sampling distribution of the mean

Must be positive

Positive integer (n ≥ 30 recommended)

Must be less than σ

Probability Type

Quick Examples

Invalid Input

Standard Error of the Mean (σ / √)

SEM is % smaller than σ

P(x̄ < ) P(x̄ > ) P(within ± of μ)

Z-score:

Required Sample Size for SEM ≤

Actual SEM:

Standard Error (SEM)

Sample Size (n)

√n

95% Confidence Interval for Sample Means

Lower Bound

Population Mean (μ)

Upper Bound

Margin of Error: ± (using z = 1.96)

Step-by-Step Solution

1

Identify values

σ = , n =

2

Calculate √n

√ =

3

Apply the formula

SEM = σ / √n = / =

Step-by-Step Solution

1

Identify values

μ = , σ = , n = , x̄ =

2

Calculate Standard Error

SEM = σ / √n = / √ =

3

Calculate Z-score

z = (x̄ - μ) / SEM = ( - ) / =

4

Find Probability

Using standard normal table: P =

Sample Size & Standard Error Relationship

Sample Size (n) √n SEM = σ/√n % Reduction
10 3.16 σ × 0.316 68.4%
30 5.48 σ × 0.183 81.7%
100 10.00 σ × 0.100 90.0%
400 20.00 σ × 0.050 95.0%
1000 31.62 σ × 0.032 96.8%

Note: To halve the SEM, you need to quadruple the sample size.

Small Sample Size

Your sample size (n = ) is less than 30. The Central Limit Theorem may not apply unless the population is normally distributed.

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About Central Limit Theorem Calculator

What is the Central Limit Theorem?

The Central Limit Theorem (CLT) is one of the most important concepts in statistics. It states that when you take sufficiently large random samples from a population, the distribution of sample means will be approximately normal (bell-shaped), regardless of the original population's distribution.

Key Concepts

Sampling Distribution of the Mean

When you repeatedly take samples of size n from a population and calculate the mean of each sample, these sample means form their own distribution called the sampling distribution of the mean.

Standard Error of the Mean (SEM)

Formula: σ_x̄ = σ / √n

Where:

  • σ_x̄ = Standard error of the mean
  • σ = Population standard deviation
  • n = Sample size

The SEM measures how much sample means typically vary from the true population mean.

CLT Properties

  1. Mean of Sampling Distribution = Population Mean (μ)
  2. Standard Deviation of Sampling Distribution = σ / √n (Standard Error)
  3. Shape → Approximately normal when n ≥ 30 (or smaller if population is already normal)

Z-Score for Sample Means

z = (x̄ - μ) / (σ / √n)

This allows us to calculate probabilities about sample means using the standard normal distribution.

Practical Applications

1. Quality Control

  • Monitoring production processes
  • Control charts and process capability

2. Survey Sampling

  • Estimating population parameters
  • Margin of error calculations

3. Scientific Research

  • Hypothesis testing
  • Confidence interval construction

4. Business Analytics

  • Customer satisfaction surveys
  • Sales forecasting

Sample Size Considerations

Sample Size (n) SEM Reduction CLT Applicability
10 σ / 3.16 May not apply
30 σ / 5.48 Generally applies
100 σ / 10 Strongly applies
400 σ / 20 Very precise

Important Notes

  1. Rule of Thumb: n ≥ 30 for CLT to apply to non-normal populations
  2. For normal populations: CLT applies for any sample size
  3. Larger samples = Smaller standard error = More precise estimates
  4. Independence: Samples must be independent for CLT to apply