Section 6 Sample Mean & Variance

We measure a random variable X on an independent random sample of size n collected from a population in which the variable is normally distributed

Population

\[\Huge X \sim Normal(\mu, \sigma^2)\]

Sample

\[ \Huge x = (x_1, x_2, ..., x_n) \]

Sample Mean

\[ \Huge \bar{x} = \frac{1}{n}\sum\limits_{i=1}^{n} x_{i} \]

\[ \Huge Mean(\bar{x}) = \mu \]

Sample Variance

\[ \Huge Var(x) = s_x^2 = \frac{1}{n-1}\sum\limits_{i=1}^{n} (x_i-\bar{x})^2 \]

\[ \Huge Mean(s_x^2) = \sigma^2 \]