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 = X1 + X2 + + Xn

has expected value

E (S) = n μ

and standard deviation

SD (S) = n σ

Example 1

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

You know that

E (S) = E (X1) + E (X2) + + E (X50) = n μ

E (S) = E (X1) + E (X2) + + E (X50) , = n μ

and that the expected value is

E (S) = 50 1.2 = 60

You also know that

Var (S) = Var (X1) + Var (X2) + + Var (X50) = nσ2

Var (S) = Var (X1) + Var (X2) + + Var (X50) = nσ2

which gives the standard deviation

SD (S) = nσ = 50 0.3 2.12

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