Descriptive Statistics Formulas
Combinations
(𝑛) = 𝑛! / 𝑥!(𝑛−𝑥)! x
Probability of Ranges in Normal Distribution
-Below some value 𝑃(−∞ < 𝑧 ≤ 𝑧𝑈) = 𝐹(𝑧𝑈) -Above some value 𝑃(𝑧𝐿 ≤𝑧<+∞)=1−𝐹(𝑧𝐿) -Between two values 𝑃(𝑧𝐿 ≤𝑧≤𝑧𝑈)=𝐹(𝑧𝑈)−𝐹(𝑧𝐿) where 𝐹(𝑧) is the CDF of the standard normal distribution
Standard Normal Distribution
-Convert x to z-score: 𝑧=𝑥−𝜇 / 𝜎 -Convert z-score to x: 𝑥 = 𝜇 + 𝜎𝑧
Median
Get the corresponding value or midpoint of two values -If 𝑖 is not an integer, round up and use 𝑥𝑖 -If 𝑖 is an integer, use the midpoint 𝑥𝑖+𝑥𝑖+1 / 2
Independent Events
P(A|B) = P(A) P(B|A) = P(B)
Relative frequency
frequency / total # obs.
Range
largest value - smallest value
Class Width
min value of the next class - min value of this class
Mean and Variance of Binomial
np and np(1 - p)
Frequency
number of observations in each category
Percent frequency
rel. freq. x 100
Mean of Sampling Distribution
𝐸(𝑥̅) = 𝜇
Multiplication Law - Independent events
𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴)𝑃(𝐵)
Multiplication Law
𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴)𝑃(𝐵|𝐴)
Addition Law - Mutually Exclusive events
𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵)
Addition Law
𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 ∩ 𝐵)
Addition Law - Complements
𝑃(𝐴) = 1 − 𝑃(𝐴𝑐)
Conditional Probability
𝑃(𝐴|𝐵) = 𝑃(𝐴 ∩ 𝐵) / 𝑃(𝐵)
Bayes' Theorem
𝑃(𝐴|𝐵) = 𝑃(𝐴)𝑃(𝐵|𝐴) / 𝑃(𝐵)
Probability of Ranges in Continuous Uniform
𝑃(𝑎≤𝑥≤𝑏) = (𝑏−𝑎)×𝑓(𝑥) (width) x (height)
Standard Deviation
𝑆𝐷 or 𝑠 = √𝑠^2
Standard Error of the Sample Mean
𝑆𝐸(𝑥̅) = (√𝜎^2 / n) = 𝜎 / 𝑛
Variance of Sampling Distribution
𝑉(𝑥̅) = 𝜎^2 / 𝑛
Binomial Probability Function
𝑓 ( 𝑥 ) = ( 𝑛 ) 𝑝^𝑥 ( 1 − 𝑝 )^( 𝑛 − 𝑥 ) x
Discrete Uniform Probability Function
𝑓(𝑥) = 1 / N (for x = 1...N)
Continuous Uniform Probability Density Function
𝑓(𝑥)= 1 / 𝐵−𝐴, if𝑥∈[𝐴,𝐵]
Geometric Mean
𝑟̅ = √(𝑟1)(𝑟2)(𝑟3)...(𝑟𝑛) or 𝑟̅ = √(𝑟1)(𝑟2)(𝑟3)...(𝑟𝑛) or 𝑟̅ = [(𝑟1)(𝑟2)(𝑟3)...(𝑟𝑛)]^1/𝑛
Coefficient of Variation
𝑠 / 𝑥̅ or (𝑠/𝑥̅ × 100)%
Covariance
𝑠 𝑥 𝑦 = ∑ ( 𝑥 𝑖 − 𝑥 ̅ ) ( 𝑦 𝑖 − 𝑦̅ ) / 𝑛−1
Variance
𝑠^2 = ∑(𝑥𝑖 − 𝑥̅)^2 / 𝑛−1
Arithmetic Mean
𝑥 ̅ = ∑ 𝑥𝑖 / 𝑛
Sampling Distribution of Sample Mean
𝑥̅ ~ Normal (𝜇, 𝜎^2 / 𝑛 )
Weighted Arithmetic Mean
𝑥̅𝑊 = ∑𝑥𝑖 × 𝑤𝑖 / ∑𝑤𝑖
z-Score
𝑧𝑖 = 𝑥𝑖 − 𝑥̅ / 𝑠
Mean of Probability Distribution
𝜇 = ∑𝑥 𝑓(𝑥)
Mean and Variance of Poisson
𝜇 and 𝜇
Variance of Probability Distribution
𝜎2 = ∑(𝑥 − 𝜇)2𝑓(𝑥)