Central Limit Theorem Calculator - Calculate Sampling Distribution
Calculate sampling distribution mean, standard error, and confidence intervals using Central Limit Theorem with population parameters
Population & Sample Parameters
Sampling Distribution
What is a Central Limit Theorem Calculator?
A Central Limit Theorem Calculator computes the sampling distribution of the sample mean based on population parameters, showing how sample means are distributed around the population mean.
This calculator is used for:
- Sampling Distribution - Understand sample mean variability
- Confidence Intervals - Calculate population parameter estimates
- Statistical Inference - Make predictions about populations
- Research Design - Determine required sample sizes
To calculate precise confidence intervals for your estimates, explore our Confidence Interval Calculator to determine statistical ranges for population means and proportions.
To understand the empirical rule for normal distributions, check out our Empirical Rule Calculator to see how data distributes within standard deviations of the mean.
To work with discrete probability distributions in sampling, visit our Binomial Distribution Calculator to calculate probabilities for binary outcomes over multiple trials.
To test hypotheses about your data, use our Chi-Square Calculator to determine if observed data differs significantly from expected values.
How Central Limit Theorem Works
The CLT uses these formulas:
Where μ is population mean, σ is population standard deviation, n is sample size, and Z is the critical value for the confidence level.
Key Concepts Explained
Standard Error
SE measures the variability of sample means. As sample size increases, SE decreases, meaning sample means become more precise estimates of the population mean.
Sample Size Requirement
The CLT generally requires n ≥ 30. With smaller samples, the population should be approximately normal for CLT to apply.
Confidence Interval
A range of values likely to contain the true population mean. Higher confidence levels produce wider intervals.
How to Use This Calculator
Enter Population Parameters
Input population mean and standard deviation
Set Sample Size
Choose your sample size (n ≥ 30 recommended)
Select Confidence Level
Choose 90%, 95%, or 99% confidence level
Benefits of This Calculator
- Instant Calculations - Get sampling distribution instantly
- Confidence Intervals - Automatic CI computation
- CLT Validation - Checks if sample size is adequate
- Educational Tool - Learn about sampling distributions
- Research Planning - Estimate required sample sizes
- Professional Analysis - Suitable for statistical inference
Factors Affecting Results
- Sample Size - Larger samples reduce standard error
- Population Variance - Higher variance increases SE
- Confidence Level - Higher confidence widens intervals
- Population Distribution - Non-normal requires larger n
- Independence - Samples must be independent
- Sampling Method - Random sampling is essential
Frequently Asked Questions
What is the Central Limit Theorem?
The Central Limit Theorem (CLT) states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the population's distribution, provided the samples are independent and identically distributed.
What is the formula for the Central Limit Theorem?
For the sampling distribution of the mean: Mean = μ, Standard Error = σ/√n, where μ is the population mean, σ is the population standard deviation, and n is the sample size.
What sample size is needed for the Central Limit Theorem?
Generally, a sample size of n ≥ 30 is considered sufficient for the CLT to apply. For populations that are already approximately normal, smaller sample sizes may suffice.
What is standard error in the Central Limit Theorem?
Standard error (SE) is the standard deviation of the sampling distribution of the mean, calculated as SE = σ/√n. It measures how much sample means vary from the population mean.
Why is the Central Limit Theorem important?
The CLT is fundamental to statistical inference, enabling hypothesis testing and confidence intervals even when the population distribution is unknown. It justifies using normal distribution methods for large samples.
When does the Central Limit Theorem not apply?
The CLT may not apply with very small samples (n < 30), when observations are not independent, when sampling from populations with extreme skewness or heavy tails, or when the population variance is infinite.