\[\bar{x} = \frac{\sum x_i}{n} = \frac{x_1 + x_2 + \cdots + x_n}{n}\]

\[\bar{x} = \frac{\sum f_ix_i}{\sum f_i}\]

• Middle value when data is ordered
• If \(n\) is odd: median is the \(\frac{n+1}{2}\)th value
• If \(n\) is even: median is average of \(\frac{n}{2}\)th and \(\frac{n}{2}+1\)th values

The value that occurs most frequently in the data set

\[\text{Range} = \text{Maximum} - \text{Minimum}\]

\[\text{IQR} = Q_3 - Q_1\]

\[\sigma^2 = \frac{\sum(x_i - \mu)^2}{n}\]

\[\sigma^2 = \frac{\sum x_i^2}{n} - \mu^2\]

\[\sigma = \sqrt{\sigma^2} = \sqrt{\frac{\sum(x_i - \mu)^2}{n}}\]

• New mean: \(\bar{x}_{\text{new}} = \bar{x} + k\)
• Standard deviation unchanged: \(\sigma_{\text{new}} = \sigma\)
• Variance unchanged: \(\sigma^2_{\text{new}} = \sigma^2\)

• New mean: \(\bar{x}_{\text{new}} = k\bar{x}\)
• New standard deviation: \(\sigma_{\text{new}} = |k|\sigma\)
• New variance: \(\sigma^2_{\text{new}} = k^2\sigma^2\)

\[0 \leq P(A) \leq 1\]

\[P(A') = 1 - P(A)\]

\[P(A \cup B) = P(A) + P(B) - P(A \cap B)\]

\[P(A \cup B) = P(A) + P(B)\]

\[P(A \cap B) = P(A) \times P(B|A)\]

\[P(A|B) = \frac{P(A \cap B)}{P(B)}\]

\[P(A \cap B) = P(A) \times P(B)\]

Also: \(P(A|B) = P(A)\) and \(P(B|A) = P(B)\)

\[E(X) = \sum x_i P(X = x_i)\]

\[\text{Var}(X) = E(X^2) - [E(X)]^2\]

\[\text{Var}(X) = \sum(x_i - \mu)^2 P(X = x_i)\]

\[\sigma = \sqrt{\text{Var}(X)}\]

For \(Y = aX + b\):
• \(E(Y) = aE(X) + b\)
• \(\text{Var}(Y) = a^2\text{Var}(X)\)

\[X \sim B(n, p)\]

\[P(X = r) = \binom{n}{r}p^r(1-p)^{n-r}\]

\[E(X) = np\]

\[\text{Var}(X) = np(1-p)\]

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

\[Z \sim N(0, 1)\]

\[Z = \frac{X - \mu}{\sigma}\]

\[X = \mu + Z\sigma\]

\[r = \frac{\sum(x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum(x_i - \bar{x})^2 \sum(y_i - \bar{y})^2}}\]

\[y = ax + b\]

\[r^2\]

\[^nP_r = \frac{n!}{(n-r)!}\]

\[\binom{n}{r} = \frac{n!}{r!(n-r)!}\]

\[n! = n \times (n-1) \times (n-2) \times \cdots \times 2 \times 1\]