Central Limit Theorem Calculator
Calculate standard error of the mean and sampling distribution probabilities with step-by-step solutions
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
Identify values
σ = , n =
Calculate √n
√ =
Apply the formula
SEM = σ / √n = / =
Step-by-Step Solution
Identify values
μ = , σ = , n = , x̄ =
Calculate Standard Error
SEM = σ / √n = / √ =
Calculate Z-score
z = (x̄ - μ) / SEM = ( - ) / =
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
- Mean of Sampling Distribution = Population Mean (μ)
- Standard Deviation of Sampling Distribution = σ / √n (Standard Error)
- 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
- Rule of Thumb: n ≥ 30 for CLT to apply to non-normal populations
- For normal populations: CLT applies for any sample size
- Larger samples = Smaller standard error = More precise estimates
- Independence: Samples must be independent for CLT to apply