Central Limit Theorem Calculator

How to Calculate Using the Central Limit Theorem

The Central Limit Theorem (CLT) is a fundamental concept in statistics that describes the distribution of sample means from a population. It states that the distribution of sample means approximates a normal distribution as the sample size becomes larger, regardless of the shape of the population distribution.

Central Limit Theorem Formulas

The key formulas for the Central Limit Theorem are:

1. Sample Mean: \[ \mu_{\bar{x}} = \mu \]
2. Sample Standard Deviation: \[ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \]

Where:

• μ is the population mean
• σ is the population standard deviation
• n is the sample size
• μ_x̄ is the sample mean
• σ_x̄ is the sample standard deviation

Calculation Steps

1. Identify the population mean (μ), population standard deviation (σ), and sample size (n)
2. Calculate the sample mean (μ_x̄) by using the population mean
3. Calculate the sample standard deviation (σ_x̄) by dividing the population standard deviation by the square root of the sample size

Example Calculation

Let's calculate the sample mean and standard deviation for a population with the following parameters:

• Population Mean (μ) = 100
• Population Standard Deviation (σ) = 15
• Sample Size (n) = 30

Steps:

1. Calculate the sample mean: \[ \mu_{\bar{x}} = \mu = 100 \]
2. Calculate the sample standard deviation: \[ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} = \frac{15}{\sqrt{30}} \approx 2.74 \]

Therefore, according to the Central Limit Theorem:

• The sample mean is 100
• The sample standard deviation is approximately 2.74

Visual Representation

A visual representation can help understand the Central Limit Theorem. Here's a diagram showing the distribution of sample means:

This diagram illustrates the normal distribution of sample means. The red dashed line represents the sample mean, which is equal to the population mean. The blue curve shows the probability density of the sample means, which becomes more concentrated around the population mean as the sample size increases.