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any rules applied to find assumed mean

how to find assumed mean while solving problems like standard deviation or finding mean by direct method?

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2 Answers

Dear Shaik

Let Mr is the real mean, then Mr = ( ∑(i = 1 to n) xi ) / n ----------(1)

and then n(Mr) = ∑(i = 1 to n) xi  ------------(1)'

Let Ma is the assumed mean, then (x1 - Ma) + (x2-Ma) + (x3-Ma) + ... + (xn -Ma) = R ----------(2)

 

If the assumed mean is the same as real mean then R should be zero.

If R is not zero, then assumed mean is not the same as real mean.

The equation (2) can be modified as below.

(x1 + x2 + x3+ .... + xn) - n(Ma) = R   ---------------(2)'

From equation (1)' (x1+x2+...+xn) = ∑(i = 1 to n) xi =n(Mr)  plug this to equation (2)'

Then n(Mr) - n(Ma) = R -----------------(3)

Divide the equation by n gives Mr = Ma + R/n -------------(4)

This is the relation between the real mean and the assumed mean.

The standard deviation is σ = √( ∑(i = 1 to n) ( xi - Mr)2 / n )-------------(5)

If you want to use assumed mean for standard deviation, you can use equation (4) to equation (5)

then σ =  √( ∑(i = 1 to n)( xi - Ma - R/n )2  / n )

 

I hope you understand this and this will help you.

 

 

 

Ok, I'm a little fuzzy about what exactly you're looking for, so I'll give an example that Wikipedia did on their site (I'm good at math, not creative math- I can't just whip up numbers). But I understand assumed mean to be just like this. If this isn't what you're needing, we'll talk it though a little more, and I'll try to find a better way to explain it than wikipedia.

The mean of the following numbers is sought:
219, 223, 226, 228, 231, 234, 235, 236, 240, 241, 244, 247, 249, 255, 262
Suppose we start with a plausible initial guess that the mean is about 240. Then the deviations from this "assumed" mean are the following:
−21, −17, −14, −12, −9, −6, −5, −4, 0, 1, 4, 7, 9, 15, 22
In adding these up, one finds that:
22 and −21 almost cancel, leaving +1,
15 and −17 almost cancel, levaing −2,
9 and −9 cancel,
7 + 4 cancels −6 − 5,
and so on. We are left with a sum of −30. The average of these 15 deviations from the assumed mean is therefore −30/15 = −2. Therefore that is what we need to add to the assumed mean to get the correct mean:
correct mean = 240 − 2 = 238.