Central Limit Theorem Calculator

In simple words, the central limit theorem says that if you take multiple random samples of a sufficiently large size from a population, the mean of those samples will be close to the mean of the population. Technically it means that the distribution of sample approaches a normal distribution as the sample size increases even if the population distribution is not normal.

Population mean (μ) Population standard deviation (σ) Sample size (n)

Sample Mean (x) = 25

Sample Standard Deviation (s) = 0.8452

Sample Mean (x) = μ = 25

Sample Standard Deviation (s) = σ / √n = 5 / √35 = 0.8452

Note : The sample size should be atleast 30. If the population is normal, then the theorem holds true even for sample size lesser than 30.

Properties of Central Limit Theorem

As per the central limit theorem, sampling distribution must have the following properties -

1. The mean of the sampling distribution is equal to the mean of population.

x = μ

2. The standard deviation of the sampling distribution is equal to the standard deviation of population divided by sample size.

s = σ / √n

Conditions of Central Limit Theorem

To make the central limit theorem valid, it is important to meet the following conditions.

Samples must be selected at random.
Samples must be independent.
Sample size must be at most 10% of a population if sampling is done without replacement.