Tchebysheff' s and the Empirical Rule

Needs["Statistics`DescriptiveStatistics`"]

Needs["Statistics`DataManipulation`"]

Seriously useful tools for describing the  distribution of 
        values in the  variable .  It takes mean and the standard 
        deviation and allows you to describe the distribution in 
        non - technical terms .   You only need two values and two 
        functions to go beyond the description of the mean and 
        measure of dispersion .

Tchebysheff' s Theorem

Given a number k greater than or equal to one, and a set of n measurements, at least

1 - (1/(k^2))

of the distribution lies in the interval within  k*(standard deviation) from the mean .

As mentioned in class,  
    there is a relationship between the value of k and the 
      percentage of the distribution around the mean value .  Here is
       the formula that show this .

k  =  (Sqrt[(1/((1 - P)))])

\(where\ P\ is\ the\ percentage\ of\ values\ around\ the\ center\ 
      \((P\ cannot\ be\ equal\ to\ 1)\) . 
		
Here\ is\ a\ table\ of\ 
      percentage\ values\ \ around\ the\ mean\ \((P)\)\ \((column\ one)\)\ and
      \ k\ values\ \(\((column\ two)\) . \)\)

TableForm[
      Table[{P,  Sqrt[(1/(1 - P))]}, {P, 0.05, 0.95, 0.05}]]

In class,  
    we discussed  the starting wages of econ undergraduates .  The 
      following two values (mean and standard deviation) were 
      given .

MeanWages = 32900

SDWages = 3850

\(The\ function\ below\ generates\ the\ values\ in\ the\ 
      table . \ 
Column\ 1\ of\ the\ table\ is\ the\ percentage\ of\ values\ 
      around\ the\ mean\ $32, 900, \ 

    column\ 2\ is\ the\ lower\ bound\ of\ the\ range, \ 

    column\ 3\ has\ the\ upper\ bound\ of\ the\ \(range . \)\)

TableForm[
      Table[{P, (MeanWages - SDWages*Sqrt[(1/(1 - P))]),  
          (MeanWages + SDWages*Sqrt[(1/(1 - P))])}, {P, 0.05, 
          0.95, 0.05}]]

Interpretation using 50  % values,  
    based on the mean and standard deviation values,  
    at least 50  % of the econ undergrads  are expected to obtain 
      starting wages from     $27, 455.30 to  $38, 344.70 .

Empirical Rule

\(\(Tchebysheff' s\ Theorem\ applies\ to\ any\ data\ distribution\ 
      \((symmetric, \ skewed\ to\ the\ left, \ skewed\ to\ the\ right)\) . \ 
        If\ we\ know\ that\ the\ distribution\ is\ symmetric\ and\ unimodal\ 
      \((bell\ shaped)\), \ 
    we\ can\ use\ the\ empirical\ rule . 
		The\ Empirical\ Rule\ is\ 
      limited\ in\ that\ it\ tell\ us\ three\ sets\ of\ \(ranges . \)\ \)\)

For 68  % of the distribution

μ ± σ

For 95  % of the distribution

μ ± 2*σ

For almost 100  % of the distribution

μ ± 3*σ

Here are the 68,  
    95 and 100 percentage ranges for the wage example .

TableForm[
      Table[{MeanWages - i*SDWages,  MeanWages + i*SDWages},  {i, 1, 3}] ]