Formulas: Statistical Measures

Module: M03

Arithmetic Mean

$\bar{X}=\frac{{\sum }_{i=1}^n{X}_i}{n}$

Weighted Mean

${\bar{X}}_w={\sum }_{i=1}^n{w}_i{X}_i$

Geometric Mean

${\bar{X}}_G=\sqrt[n]{X_1\cdot X_2\cdots X_n}$

Harmonic Mean

${\bar{X}}_H=\frac{n}{{\sum }_{i=1}^n\frac{1}{X_i}}$

Range

$\text{Range}=X_{\max }-X_{\min }$

Mean Absolute Deviation (MAD)

$MAD=\frac{{\sum }_{i=1}^n|X_i-\bar{X}|}{n}$

Population Variance

${\sigma }^2=\frac{{\sum }_{i=1}^N(X_i-\mu {)}^2}{N}$

Sample Variance

$s^2=\frac{{\sum }_{i=1}^n(X_i-\bar{X}{)}^2}{n-1}$

Standard Deviation

$\sigma =\sqrt{{\sigma }^2}s=\sqrt{s^2}$

Coefficient of Variation

$CV=\frac{s}{\bar{X}}$

Target Downside Deviation

$s_{\text{target}}=\sqrt{\frac{{\sum }_{\text{for all }{X}_i<B}(X_i-B{)}^2}{n-1}}$

where $B$ = target return (benchmark)

Skewness (Sample)

$S_k=\frac{n}{(n-1)(n-2)}\sum {\left(\frac{X_i-\bar{X}}{s}\right)}^3$

Excess Kurtosis (Sample)

$K_E=\frac{n(n+1)}{(n-1)(n-2)(n-3)}\sum {\left(\frac{X_i-\bar{X}}{s}\right)}^4-\frac{3(n-1{)}^2}{(n-2)(n-3)}$

Covariance (Sample)

$Cov(X,Y)=\frac{{\sum }_{i=1}^n(X_i-\bar{X})(Y_i-\bar{Y})}{n-1}$

Correlation

$\rho (X,Y)=\frac{Cov(X,Y)}{{\sigma }_X\cdot {\sigma }_Y}$