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} $