Statistics - Formulas

Following is the list of statistics formulas used in the Howcodex statistics tutorials. Each formula is linked to a web page that describe how to use the formula.

A

• Adjusted R-Squared - ${R_{adj}^2 = 1 - [\frac{(1-R^2)(n-1)}{n-k-1}]}$

• Arithmetic Mean - $\bar{x} = \frac{_{\sum {x}}}{N}$

• Arithmetic Median - Median = Value of $\frac{N+1}{2})^{th}\ item$

• Arithmetic Range - ${Coefficient\ of\ Range = \frac{L-S}{L+S}}$

B

• Best Point Estimation - ${MLE = \frac{S}{T}}$

• Binomial Distribution - ${P(X-x)} = ^{n}{C_x}{Q^{n-x}}.{p^x}$

C

• Chebyshev's Theorem - ${1-\frac{1}{k^2}}$

• Circular Permutation - ${P_n = (n-1)!}$

• Cohen's kappa coefficient - ${k = \frac{p_0 - p_e}{1-p_e} = 1 - \frac{1-p_o}{1-p_e}}$

• Combination - ${C(n,r) = \frac{n!}{r!(n-r)!}}$

• Combination with replacement - ${^nC_r = \frac{(n+r-1)!}{r!(n-1)!} }$

• Continuous Uniform Distribution - f(x) = $\begin{cases} 1/(b-a), & \text{when $ a \le x \le b $} \\ 0, & \text{when $x \lt a$ or $x \gt b$} \end{cases}$

• Coefficient of Variation - ${CV = \frac{\sigma}{X} \times 100 }$

• Correlation Co-efficient - ${r = \frac{N \sum xy - (\sum x)(\sum y)}{\sqrt{[N\sum x^2 - (\sum x)^2][N\sum y^2 - (\sum y)^2]}} }$

• Cumulative Poisson Distribution - ${F(x,\lambda) = \sum_{k=0}^x \frac{e^{- \lambda} \lambda ^x}{k!}}$

D

• Deciles Statistics - ${D_i = l + \frac{h}{f}(\frac{iN}{10} - c); i = 1,2,3...,9}$

• Deciles Statistics - ${D_i = l + \frac{h}{f}(\frac{iN}{10} - c); i = 1,2,3...,9}$

F

• Factorial - ${n! = 1 \times 2 \times 3 ... \times n}$

G

• Geometric Mean - $G.M. = \sqrt[n]{x_1x_2x_3...x_n}$

• Geometric Probability Distribution - ${P(X=x) = p \times q^{x-1} }$

• Grand Mean - ${X_{GM} = \frac{\sum x}{N}}$

H

• Harmonic Mean - $H.M. = \frac{W}{\sum (\frac{W}{X})}$

• Harmonic Mean - $H.M. = \frac{W}{\sum (\frac{W}{X})}$

• Hypergeometric Distribution - ${h(x;N,n,K) = \frac{[C(k,x)][C(N-k,n-x)]}{C(N,n)}}$

I

• Interval Estimation - ${\mu = \bar x \pm Z_{\frac{\alpha}{2}}\frac{\sigma}{\sqrt n}}$

L

• Logistic Regression - ${\pi(x) = \frac{e^{\alpha + \beta x}}{1 + e^{\alpha + \beta x}}}$

M

• Mean Deviation - ${MD} =\frac{1}{N} \sum{|X-A|} = \frac{\sum{|D|}}{N}$

• Mean Difference - ${Mean\ Difference= \frac{\sum x_1}{n} - \frac{\sum x_2}{n}}$

• Multinomial Distribution - ${P_r = \frac{n!}{(n_1!)(n_2!)...(n_x!)} {P_1}^{n_1}{P_2}^{n_2}...{P_x}^{n_x}}$

N

• Negative Binomial Distribution - ${f(x) = P(X=x) = (x-1r-1)(1-p)x-rpr}$

• Normal Distribution - ${y = \frac{1}{\sqrt {2 \pi}}e^{\frac{-(x - \mu)^2}{2 \sigma}} }$

O

• One Proportion Z Test - ${ z = \frac {\hat p -p_o}{\sqrt{\frac{p_o(1-p_o)}{n}}} }$

P

• Permutation - ${ {^nP_r = \frac{n!}{(n-r)!} }$

• Permutation with Replacement - ${^nP_r = n^r }$

• Poisson Distribution - ${P(X-x)} = {e^{-m}}.\frac{m^x}{x!}$

• probability - ${P(A) = \frac{Number\ of\ favourable\ cases}{Total\ number\ of\ equally\ likely\ cases} = \frac{m}{n}}$

• Probability Additive Theorem - ${P(A\ or\ B) = P(A) + P(B) \\[7pt] P (A \cup B) = P(A) + P(B)}$

• Probability Multiplicative Theorem - ${P(A\ and\ B) = P(A) \times P(B) \\[7pt] P (AB) = P(A) \times P(B)}$

• Probability Bayes Theorem - ${P(A_i/B) = \frac{P(A_i) \times P (B/A_i)}{\sum_{i=1}^k P(A_i) \times P (B/A_i)}}$

• Probability Density Function - ${P(a \le X \le b) = \int_a^b f(x) d_x}$

R

• Reliability Coefficient - ${Reliability\ Coefficient,\ RC = (\frac{N}{(N-1)}) \times (\frac{(Total\ Variance\ - Sum\ of\ Variance)}{Total Variance})}$

• Residual Sum of Squares - ${RSS = \sum_{i=0}^n(\epsilon_i)^2 = \sum_{i=0}^n(y_i - (\alpha + \beta x_i))^2}$

S

• Shannon Wiener Diversity Index - ${ H = \sum[(p_i) \times ln(p_i)] }$

• Standard Deviation - $\sigma = \sqrt{\frac{\sum_{i=1}^n{(x-\bar x)^2}}{N-1}}$

• Standard Error ( SE ) - $SE_\bar{x} = \frac{s}{\sqrt{n}}$

• Sum of Square - ${Sum\ of\ Squares\ = \sum(x_i - \bar x)^2 }$

T

• Trimmed Mean - $\mu = \frac{\sum {X_i}}{n}$