Mean & Standard Deviation

By Richard Mycroft (P2053) on Friday, February 11, 2000 - 08:15 pm :

Is it possible for two different sets of data to have the same mean and standard deviation? Can you show me why?

Richard

By Dan Goodman (Dfmg2) on Friday, February 11, 2000 - 09:13 pm :

The set of data

(- 2, - 1, 1, 2)

and

(- \sqrt{2.75}, - 1.5, 1.5, \sqrt{2.75})

have the same mean (0) and the same standard deviation (

\sqrt{2.5}

). How did I find this? Well, first of all, it is easy to construct two different sets of data with the same mean. Choose any set of positive numbers, and the negative of each of these numbers and the mean will be 0. To make the standard deviation the same, I reduced the problem a bit. First of all, the variance is the square of the standard deviation, so if this is the same, the SD is the same. One formula for the variance of some data is

Var = (1 / n) \sum_{i = 1}^{n} x_{i}^{2} - (Mean)^{2}

. If the mean is 0, we just need to find two sets of numbers whose sum of squares adds up to the same thing. Let's try for

n = 2

to start with. We need to find

x

y

z

w

such that

x^{2} + y^{2} = z^{2} + w^{2}

. If we choose

x

y

and

z

, we can find a

w

which works,

w = \sqrt{x^{2} + y^{2} - z^{2}}

. I chose

x = 1

y = 2

and

z = 1.5

and got

w = \sqrt{2.75}

. Does this help? Can you try and find two sets of data with mean not equal to zero, and a different number of data points using something like this method?

By Richard Mycroft (P2053) on Friday, February 11, 2000 - 11:01 pm :

Thanks Dan
How about 2 9 11 18 and 3 6 14 17?
(SD = sqrt(32.5) and Mean = 10)
I have to admit that I cheated slightly by using 0 as the mean at first. After reading your solution, I got:
x=1, y=8, z=4, w=7 (chosen because 64+1 = 16+49)
I then added 10 to everything, to get this answer.
I haven't tried a different number of data points yet.

Richard

By Dan Goodman (Dfmg2) on Friday, February 11, 2000 - 11:26 pm :

You've got the idea. Yup, your example is nicer because you found integer points, and I was expecting you to start with something of mean 0 and translate, I wouldn't have bothered doing it any other way, although you could try finding a set that wasn't symmetric about the mean. Well done.