Statisticians_group : Message: Re: [Statisticians

yuvaraj,

Thanks for ur nice reply. it may be useful for me if u send t detailed proof which u hav.

--- On Wed, 5/3/08, Yuva Raj <send2yuvraj@...> wrote:

From: Yuva Raj <send2yuvraj@...>
Subject: Re: [Statisticians_group] doubts reg S and s
To: Statisticians_group@...
Cc: thivya_friends@..., "Yuv Raj" <yuvastats@...>
Date: Wednesday, 5 March, 2008, 2:42 PM

Hi

Hope you know about the concept of Population and Sample in Statistics.

When we are doing hypothesis testing, we will be taking sample from the population and perform appropriate tests. After finishing the tests we have to approximate (project) those results to population.

For example, if a company is producing 39,750 units per day. What statisticians will do is, they will take a sample from that population and perform hypothesis testing also calculate the mean and SD (parameters) , called sample mean (x-Bar) and sample standard deviation (s).

You do not want to sit and calculate mean and standard deviation for your population data.

Once it is calculated for sample, it has to be replaced for population. You cannot simply substitute the value of sample to the population. Here comes the property of estimators called unbiased-ness. Says that, once you calculated for sample and while estimating (projecting) to the population, the sample parameter has to satisfy the following condition it is

        Expectation [sample parameter] = Population Parameter

        E[x-Bar] = meu

In this sense, If we use

        $S^2=\frac{1}{n}\sum_{i=1}^n(X_i-\overline{X}\,)^2$

You cannot replace for Population S.D. Sigma

i. e.,

$\operatorname{E}(S^2)=\frac{n-1}{n}\sigma^2\neq\sigma^2.$



That’s why we use

$\begin{displaymath}\var(x) = S^2 = \frac{\sum_{i=1}^n (x_i-\bar{x})^2}{n-1} \end{displaymath}$

Now we can replace our calculated value of sample standard deviation to the population parameter sigma.

I can send you, if you are looking for detailed proof.

Regards

Yuvaraj.

yuvastats@gmail. com

thivya praba <thivya_friends@ yahoo.co. in> wrote:

Hi friends,

     I would be helpful if any one can clear my doubts.

     we use (n-1) d.f in S and n in s. v replace s in place of S, if v r not known abt the pop variance. but my doubt s,

1. if v r known with n, v can find both S and s.

then wat s t necessity of s?

2.why does S has d.f (n-1) and s as n?

Thanks in advance.

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