Central Limit Theorem Calculator
Calculate sampling distribution properties and demonstrate the Central Limit Theorem
Calculate Central Limit Theorem Statistics
The average of the entire population
The spread of the population data (must be positive)
Number of observations in each sample (recommend ≥ 30 for CLT)
Central Limit Theorem Results
Sampling Distribution
Population Parameters
Confidence Intervals for Sample Mean
Central Limit Theorem Formulas
Sample Mean: x̄ = μ = 60
Standard Error: SE = σ/√n = 35/√49 = 5.0000
Sampling Distribution: X̄ ~ N(μ, σ²/n)
Central Limit Theorem Analysis
Example: Age Distribution in a City
Population Parameters
Population: All residents of a city
Population mean age (μ): 60 years
Population standard deviation (σ): 35 years
Sample size (n): 49 randomly selected individuals
CLT Application
Sample mean (x̄): 60 years (same as population mean)
Standard error: SE = 35/√49 = 35/7 = 5 years
Distribution: X̄ ~ N(60, 5²)
95% Confidence Interval: [50.2, 69.8] years
CLT Conditions
Random Sampling
Data must be sampled randomly
Independence
Sample values must be independent
Sample Size
Minimum of 30 observations
Population Size
Sample ≤ 10% of population
Key Insights
Sample mean equals population mean
Standard error = σ/√n
Larger samples → smaller standard error
Distribution approaches normal regardless of population shape
Understanding the Central Limit Theorem
What is the Central Limit Theorem?
The Central Limit Theorem (CLT) is one of the most important concepts in statistics. It states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the shape of the population distribution.
Key Properties
- •Sample mean equals population mean (μ)
- •Standard error decreases as sample size increases
- •Distribution becomes normal with sufficient sample size
- •Works for any population distribution shape
Mathematical Formulation
If X̄ is the sample mean, then:
X̄ ~ N(μ, σ²/n)
Where:
• μ = population mean
• σ² = population variance
• n = sample size
Applications
- •Confidence interval estimation
- •Hypothesis testing
- •Quality control processes
- •Survey sampling
Why is CLT Important?
The Central Limit Theorem allows us to make inferences about population parameters using sample statistics, even when we don't know the population distribution. This makes it fundamental to statistical inference, hypothesis testing, and confidence interval construction.