SOLUTION: Find a set of at least 5 data values that has a mean of 20 and a standard deviation of 6.

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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