Question 639183: Find a set of at least 5 data values that has a mean of 20 and a standard deviation of 6.
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! Start with the set
20,20,20,20,20
That has a mean of 20, but a standard deviation of 0. Then increase one value to get 21, but decrease another value (to balance things out) to get 19.
So we might have
19, 20, 20, 20, 21
Now find the standard deviation. It will be roughly 0.70710678 (use a calculator here).
Repeat the last 3 steps to find another standard deviation.
We can keep going and do this by trial and error, but there's a much easier way
This way involves using the definition of sample standard deviation.
sigma = sqrt(sum((x-mu)^2)/(n-1))
6 = sqrt(sum((x-mu)^2)/(n-1))
6^2 = sum((x-mu)^2)/(n-1)
36 = sum((x-mu)^2)/(n-1)
36 = sum((x-mu)^2)/(5-1)
36 = sum((x-mu)^2)/4
36*4 = sum((x-mu)^2)
144 = sum((x-mu)^2)
sum((x-mu)^2) = 144
(x1-mu)^2+(x2-mu)^2+(x3-mu)^2+(x4-mu)^2+(x5-mu)^2 = 144
(x1-20)^2+(x2-20)^2+(x3-20)^2+(x4-20)^2+(x5-20)^2 = 144
(20-x-20)^2+(20-20)^2+(20-20)^2+(20-20)^2+(20+x-20)^2 = 144
(20-x-20)^2+(0)^2+(0)^2+(0)^2+(20+x-20)^2 = 144
(-x)^2+0+0+0+(x)^2 = 144
x^2 + x^2 = 144
2x^2 = 144
x^2 = 144/2
x^2 = 72
x = sqrt(72)
x = 8.48528137423857
So the data set of
20,20,20,20,20
becomes
20-8.48528137423857,20,20,20,20+8.48528137423857
11.5147186257614,20,20,20,28.48528137423857
So the data set
11.5147186257614,20,20,20,28.48528137423857
has a mean of 20 and a standard deviation of (roughly) 6
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If you want to be exact, then the data set
20-sqrt(72), 20, 20, 20, 20+sqrt(72)
has a mean of 20 and a standard deviation of exactly 6
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