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 $S$ ?

You know that

\begin{matrix} E (S) = E (X_{1}) + E (X_{2}) + \dots + E (X_{50}) \\ = n \cdot μ \end{matrix}

E (S) = E (X_{1}) + E (X_{2}) + \dots + E (X_{50}), = n \cdot μ

and that the expected value is

E (S) = 50 \cdot 1.2 = 60

You also know that

\begin{array}{l} Var (S) & = Var (X_{1}) + Var (X_{2}) + \\ \dots + Var (X_{50}) \\ = n σ^{2} \end{array}

\begin{array}{l} Var (S) = Var (X_{1}) + Var (X_{2}) + \dots + Var (X_{50}) = n σ^{2} \end{array}

which gives the standard deviation

SD (S) = \sqrt{n} σ = \sqrt{50} \cdot 0.3 \approx 2.12

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What Is a Normal Distribution in Statistics?

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