how do i determine a required future variation given a to date mean &
variation and a known number of future parts?

for example 
.... say I have 20 parts with mean 22.4 units and standard deviation 7.33 units

.... I need to achieve a total goal of better than a mean of 25 AND std dev of 5

.... I have the opportunity to achieve this with 28 more parts (ie
overall 48 parts) ... I know their mean needs to be < 27.5 but what
does their variation need to be & how is this calcualted?

tia

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Clarification of Question by baz229-ga on 19 May 2006 11:51 PDT

OK to assume my data is normally distributed to an acceptable
confidence level ... the 20 parts ... and the 28 to come

Let X(i) (i=1,2,...,20) be the numbers you already have and Y(i)
(i=1,2,...,28) be the new numbers. So the mean of X is 22.4 and the
standard deviation of X is 7.33.

The sum of X and Y must be 48*25 = 1200 to make the overall mean 25.
Since the sum of X is 20*22.4 = 448, the sum of Y must be 1200 - 448 =
752, which makes the mean of Y equal to 752/28 = 26.86.

Also the standard deviation of X is given to be 7.33. This means 53.73
= 7.33^2 = the [sum of X^2 - (mean of X)^2/20]/20 = [sum of X^2 -
22.4^2/20]/20 . (This is a common formula for the standard deviation.
Here I'm assuming this is a "population mean". If it's a "sample mean"
divide by 19 instead of 20.) So 1074.6 = 53.73*20 = sum of X^2 - 
25.09 . So the sum of X^2 = 1099.7.

We want the standard deviation of X and Y to be 5. So 25 = 5^2 = [sum
of X^2 +sum of Y^2 - 27.5^2/48 ]/48 by the same formula. So 1200 =
25*48 =  1099.7 +sum of Y^2 - 15.76 . So the sum of Y^2 = 116.06 .

Once more using the same standard deviation formula, we get the
standard deviation of Y is sqrt[(sum of Y^2 - (mean of Y)^2/28)/28 ]=
sqrt [ (116.06 - 26.86^2/28 )/28 ] =  2.03 .

I made a common error in the standard deviation formula. Sorry about
that. The corrected formula for a standard deviation is sd^2 = (sum of
squares - n*average^2)/n . The "n" in the formula was in the wrong
place. Making these corrections, I get (check me): sum of X^2 =
11109.8 . This forces sum of Y^2 to be 20090.2 if the overall sd is to
be 5. But then the sd of Y is imaginary since sum of Y^2 is less than
20200.87 which is 28*(26.86)^2.