Descriptive Statistics Formulas

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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)


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