Statistical simulation of the variance of the exercise 1.1

Variance

In the simulation programs use the following variance:

$\begin{displaymath}V(x)=\sum^{k}{\frac{n_{i}\cdot (x_{i}-\overline{x})^2}{n-1}}\end{displaymath}$

The exercise use this variance:

$\begin{displaymath}V(x)=\sum^{k}{\frac{n_{i}\cdot (x_{i}-\overline{x})^2}{n}}\end{displaymath}$

The solution is to multiply the variance of the simulation by:

$\begin{displaymath}V(x)=\sum^{k}{\frac{n_{i}\cdot (x_{i}-\overline{x})^2}{n-1}}\cdot \frac{n-1}{n}\end{displaymath}$
Statistical simulation of the variance with R-Project
> x <- c(rep(47,1),rep(48,3),rep(49,2),rep(50,8),rep(51,3),rep(52,2)
,rep(53,1));

> ni<- table(x);


> n<-sum(ni)

> var(x)*(n-1)/n

[1] 2.1475
Statistical simulation of the variance with Mathematica
x := Join[ConstantArray[47, 1], ConstantArray[48, 3], 
   ConstantArray[49, 2], ConstantArray[50, 8], ConstantArray[51, 3], 
   ConstantArray[52, 2], ConstantArray[53, 1]];
   
ni := Tally [x];

n := Sum[ni[[i, 2]], {i, Length[ni]}];

(Variance[x] (n - 1))/n;

N[%]

2.1475`