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Formula


Big Dog 04

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Hey BD 04:

 

Say that x1, x2, x3, x4, x5, ...., xn are n draws from a (random) sample. Then:

 

Step 1: Compute the mean, i.e. m =[ Sum xi (i=1,..., n) ] / n

Step 2: Compute the squared deviation of each observation from its mean, i.e.

For x1 --------> (x1-m)^2

For x2---------> (x2-m)^2

.....

For xn---------> (xn-m)^2

Step 3: The variance is V= [(x1-m)^2 + (x2-m)^2 + .... + (xn-m)^2 ] / n

Step 4: The s.d. is s.d. = V^(1/2)

 

Example: Let x1=10, x2= 20 and x3=30

Then:

(1) m=20

(3) V = [ (10-20)^2 + 0 + (30-20)^2] / 3 = 200/3

(4) s.d. = (20/3) ^ (1/2)

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In that case, you calculate it as follows:

Short Method:

sqrt [Σfidi^2 / Σfi - (Σfidi/Σfi) ^2)]

 

Where--> di --> Deviations from the arbitrary/assumed/working mean (Xw)--> X - Xw

fi --> frequency of xi

 

In this case:

 

Let us assume that the working mean (Xw) is : 2

So di = X - Xw = -2, -1, 0, 1, 2

Σfidi = -2*2 + -1*6 + 0*3 + 1*2 + 2*4 = -4 + -6 + 0 + 2 + 8 = 0

Therefore second fraction within the root is ZERO.

 

First fraction:

Σfidi^2

8 + 6 + 0 + 2 + 16 = 32

 

Σfi = 17

 

Therfore Standard Deviation = sqrt 32/17.

 

So approximating:

S.D. is slightly less than sqrt 2

 

or S.D.

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I did the "finding SD for freq distribution sum" using this formula and I found this one quite easy:

SD=sqr root of(E [f(x-mean)^2]/N)

 

create 3 additional columns for the sum like this: First calculate the mean.

mean=0(2)+1(6)+2(3)+3(2)+4(4)/17= 34/17 = 2

 

x f (x-mean) (x-mean)^2 f(x-mean)^2

0 2 -2 4 8

1 6 -1 1 6

2 3 0 0 0

3 2 1 1 2

4 4 2 4 16

----------

E[f(x-mean)^2]= 32

-----------

 

a)SD= sqr root of(E [f(x-mean)^2]/N)= sqr root of(32/17)=1.4 approximately

 

b)Mode=x with highest frequency=1

 

c)median=2

 

d)range=biggest x - smallest x=4-0=4

 

This entire sum took me 2 minutes to do.

I hope the above expln. will make u understand better.

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separate the 5 column numbers under their heading. I typed it in a tabular form but i guess the browser ignores spaces.

Column 1 heading=x

Column 2 heading=f

Column 3 heading=(x-mean)

column 4 heading=(x-mean)^2

column 5 heading=f(x-mean)^2

Add the numbers of the 5th column to get the number 32.

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