Central Limit Theorem Calculator
Apply the Central Limit Theorem to analyze sampling distributions, calculate standard error, and determine confidence intervals. Our statistics calculator provides comprehensive CLT analysis with step-by-step explanations.
Last updated: December 15, 2024
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CLT typically applies well when n ≥ 30
CLT Analysis
Sampling Distribution Mean:
μₓ̄ = 50
Standard Error:
σₓ̄ = 1.8257
Distribution:
Approximately Normal
Analysis:
According to the Central Limit Theorem, the sampling distribution of sample means has mean μ = 50 and standard error σ/√n = 1.8257. Since n ≥ 30, the distribution is approximately normal regardless of the population distribution.
Calculation Steps:
- Population parameters: μ = 50, σ = 10
- Sample size: n = 30
- Sampling distribution mean: μₓ̄ = μ = 50
- Standard error: σₓ̄ = σ/√n = 10/√30 = 1.8257
- Distribution: Approximately Normal (CLT applies)
Central Limit Theorem:
- • Sample means approach normal distribution as n increases
- • μₓ̄ = μ (sampling distribution mean equals population mean)
- • σₓ̄ = σ/√n (standard error decreases with larger samples)
- • Works regardless of population distribution shape (when n ≥ 30)
Quick Example Result
For μ = 50, σ = 10, n = 30:
μₓ̄ = 50, σₓ̄ = 1.8257
Standard error = 10/√30 ≈ 1.8257, Distribution: Approximately Normal
How This Calculator Works
Our Central Limit Theorem calculator applies fundamental principles of statistical inference to analyze sampling distributions. The calculator uses the CLT frameworkto determine how sample means behave and provides tools for confidence intervals and probability calculations.
The Central Limit Theorem Formula
μₓ̄ = μσₓ̄ = σ / √nX̄ ~ N(μ, σ²/n)These formulas show that sample means cluster around the population mean with decreasing variability as sample size increases. The CLT guarantees normality for large samples regardless of the original population distribution.
Shows how sampling distributions become normal as sample size increases
Statistical Foundation
The Central Limit Theorem is one of the most important results in statistics, providing the foundation for confidence intervals, hypothesis testing, and statistical inference. It explains why the normal distribution appears so frequently in statistical applications and why we can make probabilistic statements about sample means even when we don't know the population distribution.
- Sample means are unbiased estimators of population means (μₓ̄ = μ)
- Standard error decreases proportionally to 1/√n, not 1/n
- CLT applies regardless of population distribution shape when n is large
- Forms the basis for constructing confidence intervals and conducting hypothesis tests
Sources & References
- Introduction to Mathematical Statistics - Robert V. Hogg, Joseph McKean, Allen T. Craig (8th Edition)Comprehensive treatment of CLT and sampling distributions
- American Statistical Association - Statistical Education GuidelinesProfessional standards for teaching CLT concepts
- NIST Engineering Statistics Handbook - Sampling Distributions and CLTPractical applications and computational methods
Need help with other statistical calculations? Check out our normal distribution calculator and confidence interval calculator.
Get Custom Calculator for Your PlatformExample Analysis
Given Parameters:
- Population mean: μ = 100mm
- Population std dev: σ = 5mm
- Sample size: n = 25
CLT Application:
- Sampling mean: μₓ̄ = 100mm
- Standard error: σₓ̄ = 5/√25 = 1mm
- Distribution: X̄ ~ N(100, 1²)
- 95% CI: X̄ ± 1.96(1) = X̄ ± 1.96mm
Result: Sample means follow N(100, 1²) distribution
Even though n = 25 < 30, the CLT applies if the population is approximately normal. Sample means will cluster tightly around 100mm with standard error of only 1mm, making quality control monitoring very precise.
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