Algorithms

Notation	Description
`n`	Total records (items)
`X_i`	Value of `i`th item of `X`
`Y_i`	Value of `i`th item of `Y`
`w_i`	Weight of `i`th item
`N`	Total of frequency
`R(X)_i`	Corresponding rank for `X_i`
`\barX`	Mean of `X` [?]
`\text{VAR}(X)`	Variance of sample for `X` [?]
`\text{VAR}_p(X)`	Variance of population for `X` [?]
`\text{SD}(X)`	Standard Deviation of sample for `X` [?]
`\text{SD}_p(X)`	Standard Deviation of population for `X` [?]
`\text{COV}(X, Y)`	Covariance of sample for `X` and `Y` [?]
`\text{COV}_p(X, Y)`	Covariance of population for `X` and `Y` [?]

• HISTOGRAM'S NUMBER OF BINS

  Square-root choice:

  `k = \ceil{ \sqrt{N} \ }`

  Sturges' formula:

  `k = \ceil{ \log_2 N } + 1`

  Rice rule:

  `k = \ceil{ 2N^(1/3) }`

• Q-Q PLOT

  Fractional Ranks:

  `f r_\text{dist}(X_i) = (R(X)_i - 3/8) / (n + 1/4)`     (Blom, 1958)

  Scores:

  `a_\text{dist}(X_i) = F_\text{dist}^(-1) (f r_\text{dist}(X_i))`

  where `F_\text{dist}^(-1)` is the inverse cumulative specified distribution function.

  Plot:

  `(X_i, a_\text{dist}(X_i)) \qquad\quad i = 1, 2, ..., n`

• PEARSON CORRELATION

  `r_{xy} = {\text{COV}(X, Y)} / {\text{SD}(X) \ \text{SD}(Y)}`

  Significance level for `r_{xy}` is based on

  `t = r_{xy} \sqrt{(N-2)/(1-r_{ij}^2)}`

  which, under the null hypothesis, is distributed as a `t` with `N-2` degrees of freedom.

• SPEARMAN'S RANK CORRELATION

  Coming soon...

• ONE-SAMPLE T TEST

  Coming soon...

• ONE-WAY ANOVA

  Coming soon...

• F TEST

  Coming soon...

• LINEAR REGRESSION

  Coming soon...