Statistics Formula Sheet

A quick-reference sheet of essential statistics and probability formulas. Covers descriptive statistics, probability rules, distributions, confidence intervals, hypothesis tests, and regression. Each formula links to its full definition page.

Descriptive Statistics

Mean

\bar{x} = \frac{\sum x_i}{n}

Population Std Dev

\sigma = \sqrt{\frac{\sum(x_i-\mu)^2}{N}}

Sample Std Dev

s = \sqrt{\frac{\sum(x_i-\bar{x})^2}{n-1}}

Variance

s^2 = \frac{\sum(x_i-\bar{x})^2}{n-1}

IQR

\text{IQR} = Q_3 - Q_1

Z-Score

z = \frac{x - \mu}{\sigma}

Probability Rules

Complement

P(A') = 1 - P(A)

Addition Rule

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

Conditional

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

Multiplication Rule

P(A \cap B) = P(A) \cdot P(B|A)

Bayes' Theorem

P(A|B) = \frac{P(B|A)\,P(A)}{P(B)}

Counting & Combinatorics

Permutations

P(n,r) = \frac{n!}{(n-r)!}

Combinations

C(n,r) = \binom{n}{r} = \frac{n!}{r!(n-r)!}

Distributions

Binomial

P(X=k) = \binom{n}{k}p^k(1-p)^{n-k}

Binomial Mean

\mu = np

Binomial Std Dev

\sigma = \sqrt{np(1-p)}

Normal (Empirical Rule)

68\%-95\%-99.7\%

Confidence Intervals

CI for Mean (σ known)

\bar{x} \pm z^*\frac{\sigma}{\sqrt{n}}

CI for Mean (σ unknown)

\bar{x} \pm t^*\frac{s}{\sqrt{n}}

CI for Proportion

\hat{p} \pm z^*\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}

Margin of Error

E = z^*\frac{\sigma}{\sqrt{n}}

Linear Regression

Regression Line

\hat{y} = a + bx

Slope

b = r\frac{s_y}{s_x}

Correlation

r = \frac{1}{n-1}\sum\left(\frac{x_i-\bar{x}}{s_x}\right)\left(\frac{y_i-\bar{y}}{s_y}\right)

R-Squared

R^2 = r^2