Statistics mathematics formulas, Cheat Sheet of Statistics

Download Statistics mathematics formulas and more Cheat Sheet Statistics in PDF only on Docsity! Stat 101 Formulas Brian Powers, Summer 2014 Sample Statistics Sample mean ?̅? = 1 𝑛 ∑ 𝑥𝑖 𝑛 𝑖=1 Sample variance 𝑠2 = 1 𝑛 − 1 ∑(𝑥𝑖 − ?̅?)2 𝑛 𝑖=1 = ∑ 𝑥𝑖 2 − 𝑛?̅?2 𝑛 − 1 Sample standard deviation 𝑠 = √𝑠2 = √ 1 𝑛 − 1 ∑(𝑥𝑖 − ?̅?)2 𝑛 𝑖=1 5-Number summary 𝑄0 = 𝑚𝑖𝑛𝑖𝑚𝑢𝑚 𝑄1 = 1𝑠𝑡 𝑞𝑢𝑎𝑟𝑡𝑖𝑙𝑒 𝑄2 = 𝑚𝑒𝑑𝑖𝑎𝑛 𝑄3 = 3𝑟𝑑 𝑞𝑢𝑎𝑟𝑡𝑖𝑙𝑒 𝑄4 = 𝑚𝑎𝑥𝑖𝑚𝑢𝑚 Range 𝑅𝑎𝑛𝑔𝑒 = 𝑚𝑎𝑥𝑖𝑚𝑢𝑚 − 𝑚𝑖𝑛𝑖𝑚𝑢𝑚 = 𝑄4 − 𝑄0 Inter-Quartile Range 𝐼𝑄𝑅 = 𝑄3 − 𝑄1 Fences for Outliers 𝑄1 − 1.5 ∗ 𝐼𝑄𝑅, 𝑄3 + 1.5 ∗ 𝐼𝑄𝑅 Simple Linear Regression Sample Covariance 𝐶𝑜𝑣(𝑥, 𝑦) = ∑ 𝑥𝑖𝑦𝑖 − 𝑛?̅??̅? (𝑛 − 1) Sample Correlation 𝑟 = 𝐶𝑜𝑣(𝑥, 𝑦) 𝑠𝑥𝑠𝑦 = ∑ 𝑥𝑖𝑦𝑖 − 𝑛?̅??̅? (𝑛 − 1)𝑠𝑥𝑠𝑦 Regression Model ?̂? = 𝑎 + 𝑏𝑥 Slope 𝑏 = 𝑟 𝑠𝑦 𝑠𝑥 Intercept 𝑎 = ?̅? − 𝑏?̅? Residual 𝑟𝑒𝑠𝑖𝑑𝑖 = 𝑦𝑖 − ?̂?𝑖 = 𝑦𝑖 − (𝑎 + 𝑏𝑥𝑖) Normal Distribution Standardize 𝑧 = 𝑥 − 𝜇 𝜎 Un-Standardize 𝑥 = 𝜇 + 𝑧𝜎 68/95/99.7 Rule 𝑃(−1 < 𝑍 < 1) ≈ .68 𝑃(−2 < 𝑍 < 2) ≈ .95 𝑃(−3 < 𝑍 < 3) ≈ .997 kth Percentile 𝑥 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝑃(𝑋 < 𝑥) = 𝑘% Stat 101 Formulas Brian Powers, Summer 2014 Probability Complement Rule 𝑃(𝐴𝐶) = 1 − 𝑃(𝐴) General Addition Rule 𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 ∩ 𝐵) Multiplication Rule for Independent Events 𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴) ∗ 𝑃(𝐵) General Multiplication Rule 𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴) ∗ 𝑃(𝐵|𝐴) = 𝑃(𝐵) ∗ (𝑃(𝐴|𝐵) Conditional Probability 𝑃(𝐴|𝐵) = 𝑃(𝐴 ∩ 𝐵) 𝑃(𝐵) A and B are Independent if: 1) 𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴) ∗ 𝑃(𝐵) 2) 𝑃(𝐴) = 𝑃(𝐴|𝐵) 3) 𝑃(𝐵) = 𝑃(𝐵|𝐴) Random Variables Expected Value 𝜇 = 𝐸(𝑋) = ∑ 𝑥𝑖 ∗ 𝑃(𝑋 = 𝑥𝑖) 𝑘 𝑖=1 = ∑ 𝑥𝑖 ∗ 𝑝𝑖 𝑘 𝑖=1 Variance 𝜎2 = 𝑉𝑎𝑟(𝑋) = 𝐸((𝑋 − 𝜇)2) = 𝐸(𝑋2) − 𝜇2 = ∑(𝑥 − 𝜇)2 ∗ 𝑝𝑖 𝑘 𝑖=1 Linearity of Expected Value 𝐸(𝑎𝑋) = 𝑎𝐸(𝑋) 𝐸(𝑋 + 𝑏) = 𝐸(𝑋) + 𝑏 𝐸(𝑋 + 𝑌) = 𝐸(𝑋) + 𝐸(𝑌) Variance of a Linear Combination 𝑉𝑎𝑟(𝑎𝑋) = 𝑎2𝑉𝑎𝑟(𝑋) 𝑉𝑎𝑟(𝑋 + 𝑏) = 𝑉𝑎𝑟(𝑋) 𝑉𝑎𝑟(𝑎𝑋 + 𝑏𝑌) = 𝑎2𝑉𝑎𝑟(𝑋) + 𝑏2𝑉𝑎𝑟(𝑌) + 2𝑎𝑏𝐶𝑜𝑣(𝑋, 𝑌) Variance of Linear Combination of Independent X,Y 𝑉𝑎𝑟(𝑎𝑋 + 𝑏𝑌) = 𝑎2𝑉𝑎𝑟(𝑋) + 𝑏2𝑉𝑎𝑟(𝑌) 𝑆𝐷(𝑋1 + 𝑋2 + ⋯ + 𝑋𝑛) = √𝑛𝑆𝐷(𝑋)