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Encyclopedia>Statistics and Probability>Probability and Combinatorics>Probability Distributions>What Is the Central Limit Theorem in Statistics?

If you make $n$ independent trials with the same random variable $X$ , which has expected value $μ$ and standard deviation $σ$ , you can then sum the random variables. That sum $S$ will be approximately normally distributed when $n$ is sufficiently large.

Rule

The Central Limit Theorem

S = X_{1} + X_{2} + \dots + X_{n}

has expected value

E (S) = n \cdot μ

and standard deviation

SD (S) = \sqrt{n} \cdot σ

Example 1

You let $X_{i}$ be a randomly selected cheese at the cheese counter. The expected value of the weight of such a cheese is $μ = 1.2 kg$ , with a standard deviation of $σ = 0.3 kg$ . If you let $S$ denote the sum of 50 such cheeses, what is the expectation value and the standard deviation for $S$ ?