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Statistics mathematics formulas, Cheat Sheet of Statistics

University of Illinois - Chicago Statistics

Statistics formulas are sample statistics, simple linear regression, normal distributions, probability, random variables and central limit theorem.

Typology: Cheat Sheet

2021/2022

Uploaded on 02/07/2022

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Stat 101 Formulas

Brian Powers, Summer 2014

Sample Statistics

Sample mean 𝑥=1

𝑛∑𝑥𝑖

𝑛

𝑖=1

Sample variance

𝑠2=1

𝑛−1∑(𝑥𝑖−𝑥)2

𝑛

𝑖=1 =∑𝑥𝑖2−𝑛𝑥2

𝑛−1

Sample standard deviation

𝑠=√𝑠2=√1

𝑛−1∑(𝑥𝑖−𝑥)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−𝑛𝑥𝑦

(𝑛−1)

Sample Correlation

𝑟=𝐶𝑜𝑣(𝑥,𝑦)

𝑠𝑥𝑠𝑦=∑𝑥𝑖𝑦𝑖−𝑛𝑥𝑦

(𝑛−1)𝑠𝑥𝑠𝑦

Regression Model

𝑦=𝑎+𝑏𝑥

Slope 𝑏=𝑟𝑠𝑦

𝑠𝑥

Intercept 𝑎=𝑦−𝑏𝑥

Residual

𝑟𝑒𝑠𝑖𝑑𝑖=𝑦𝑖−𝑦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%

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Sample Statistics

Sample mean

𝑥̅ =

𝑛

𝑖= Sample variance

𝑠^2 =

𝑛 − 1 ∑(𝑥𝑖^ − 𝑥̅)

2

𝑛

𝑖=

∑ 𝑥𝑖^2 − 𝑛𝑥̅^2

Sample standard deviation

𝑠 = √𝑠^2 = √^

𝑛 − 1 ∑(𝑥𝑖^ − 𝑥̅)

2

𝑛

𝑖=

5-Number summary 𝑄 0 = 𝑚𝑖𝑛𝑖𝑚𝑢𝑚 𝑄 1 = 1𝑠𝑡 𝑞𝑢𝑎𝑟𝑡𝑖𝑙𝑒 𝑄 2 = 𝑚𝑒𝑑𝑖𝑎𝑛 𝑄 3 = 3𝑟𝑑 𝑞𝑢𝑎𝑟𝑡𝑖𝑙𝑒 𝑄 4 = 𝑚𝑎𝑥𝑖𝑚𝑢𝑚 Range 𝑅𝑎𝑛𝑔𝑒 = 𝑚𝑎𝑥𝑖𝑚𝑢𝑚 − 𝑚𝑖𝑛𝑖𝑚𝑢𝑚 = 𝑄 4 − 𝑄 0

Inter-Quartile Range 𝐼𝑄𝑅 = 𝑄 3 − 𝑄 1 Fences for Outliers 𝑄 1 − 1.5 ∗ 𝐼𝑄𝑅, 𝑄 3 + 1.5 ∗ 𝐼𝑄𝑅

Simple Linear Regression

Sample Covariance

𝐶𝑜𝑣(𝑥, 𝑦) =

Sample Correlation

𝑟 =

𝑠𝑥𝑠𝑦^ =

Regression Model 𝑦̂ = 𝑎 + 𝑏𝑥 Slope 𝑏 = 𝑟

Intercept 𝑎 = 𝑦̅ − 𝑏𝑥̅ Residual 𝑟𝑒𝑠𝑖𝑑𝑖 = 𝑦𝑖 − 𝑦̂𝑖 = 𝑦𝑖 − (𝑎 + 𝑏𝑥𝑖)

Normal Distribution

Standardize

𝑧 =

Un-Standardize 𝑥 = 𝜇 + 𝑧𝜎 68/95/99.7 Rule 𝑃(−1 < 𝑍 < 1) ≈. 𝑃(−2 < 𝑍 < 2) ≈. 𝑃(−3 < 𝑍 < 3) ≈. kth Percentile 𝑥 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝑃(𝑋 < 𝑥) = 𝑘%

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) =

A and B are Independent if:

Random Variables

Expected Value

𝜇 = 𝐸(𝑋) = ∑ 𝑥𝑖 ∗ 𝑃(𝑋 = 𝑥𝑖)

𝑘

𝑖=

𝑘

𝑖= Variance

𝜎^2 = 𝑉𝑎𝑟(𝑋) = 𝐸((𝑋 − 𝜇)^2 ) = 𝐸(𝑋^2 ) − 𝜇^2 = ∑(𝑥 − 𝜇)^2 ∗ 𝑝𝑖

𝑘

𝑖= 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) = 𝑎^2 𝑉𝑎𝑟(𝑋) 𝑉𝑎𝑟(𝑋 + 𝑏) = 𝑉𝑎𝑟(𝑋) 𝑉𝑎𝑟(𝑎𝑋 + 𝑏𝑌) = 𝑎^2 𝑉𝑎𝑟(𝑋) + 𝑏^2 𝑉𝑎𝑟(𝑌) + 2𝑎𝑏𝐶𝑜𝑣(𝑋, 𝑌) Variance of Linear Combination of Independent X,Y 𝑉𝑎𝑟(𝑎𝑋 + 𝑏𝑌) = 𝑎^2 𝑉𝑎𝑟(𝑋) + 𝑏^2 𝑉𝑎𝑟(𝑌) 𝑆𝐷(𝑋 1 + 𝑋 2 + ⋯ + 𝑋𝑛) = √𝑛𝑆𝐷(𝑋)

Population mean μ (n<30, σ known) (^) 𝑥̅ ± 𝑡𝛼/ 2 ∗ 𝜎 √𝑛^

Population mean μ (σ unknown) (^) 𝑥̅ ± 𝑡𝛼/ 2 ∗ 𝑠 √𝑛^

Hypothesis Testing

Calculate the Test Statistic Population proportion p (n large) 𝑧 =

Population difference p 1 - p 2 ( n 1 , n 2 large) H 0 : p 1 - p 2 =0 𝑧^ =^

√𝑝̂𝑝𝑜𝑜𝑙𝑒𝑑𝑞̂𝑝𝑜𝑜𝑙𝑒𝑑 ( 𝑛^1

1 +^

And 𝑝̂𝑝𝑜𝑜𝑙𝑒𝑑 =

𝑛 1 + 𝑛 2 =^

Population mean μ (n≥30, σ known) (^) 𝑧 = 𝑥̅^ −^ 𝜇 0 𝜎/√𝑛 Population mean μ (n<30, σ known) (^) 𝑡 = 𝑥̅^ −^ 𝜇 0 𝜎/√𝑛 Population mean μ (σ unknown) (^) 𝑡 = 𝑥̅^ −^ 𝜇 0 𝑠/√𝑛

Conclusion Test Type p-value formula calculator Upper-Tail 𝑃(𝑍 > 𝑧) (^) normalcdf(z,10) Lower-Tail 𝑃(𝑍 < 𝑧) (^) normalcdf(-10,z) Two-Tailed 2 ∗ 𝑃(𝑍 > |𝑧|) (^2) *normalcdf(|z|,10) Reject H 0 if p-value<α

Statistics mathematics formulas, Cheat Sheet of Statistics

Related documents

Partial preview of the text

Download Statistics mathematics formulas and more Cheat Sheet Statistics in PDF only on Docsity!

Sample Statistics

𝑛 − 1 ∑(𝑥𝑖^ − 𝑥̅)

∑ 𝑥𝑖^2 − 𝑛𝑥̅^2

𝑠 = √𝑠^2 = √^

𝑛 − 1 ∑(𝑥𝑖^ − 𝑥̅)

Simple Linear Regression

𝑠𝑥𝑠𝑦^ =

Normal Distribution

Probability

Random Variables

Hypothesis Testing

√𝑝̂𝑝𝑜𝑜𝑙𝑒𝑑𝑞̂𝑝𝑜𝑜𝑙𝑒𝑑 ( 𝑛^1

1 +^

𝑛 1 + 𝑛 2 =^