Math Is Fun Forum

over_score

Member

Registered: 2007-03-12

Posts: 7

Standard Deviation

Why is it calculated as the 'root mean square', and not the 'mean square root',
The second one seems more logical to me, I'm guessing the first one is used as it does something along the lines of over compensates?

MathsIsFun

Administrator

Registered: 2005-01-21

Posts: 7,711

Re: Standard Deviation

Coz you square 'em, then you average 'em, then you square root that.

So it is the square root of the average of the squares ... root mean square

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"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Standard Deviation

Let’s take the numbers 1, 2, 3.

The root mean square is √[(1² + 2² + 3²)∕3] = √(14⁄3) ≈ 2.17.

The mean square root is (√1 + √2 + √3)∕3 ≈ 1.38.

The two are different.

over_score

Member

Registered: 2007-03-12

Posts: 7

Re: Standard Deviation

Pardon, change my question to "why is it the 'root mean square' and not the 'mean root square' " - the second one appears more logical to me as it's simply the 'mean distance from the mean - ignoring negative values' the first one obviously comes up with a similar result... but why do we use the first one instead of the second one? Do you understand where I'm coming from?
Cheers

MathsIsFun

Administrator

Registered: 2005-01-21

Posts: 7,711

Re: Standard Deviation

You can use the second one if you want, but it would measure things in a different way and need to be applied differently. Plus you would have trouble with the square root of negative values (which are imaginary).

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"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Standard Deviation

What over_score is saying is that his proposed method of finding the standard deviation is to do it the same way you'd find the mean, but first getting rid of any minus signs.

For example, with the set {-2, -3, 6}:

The mean would be equal to (-2 - 3 + 6)/3 = 1/3
The standard deviation would be equal to ( |2-1/3| + |-3-1/3| + |-6-1/3| )/3 = 34/9.

I have no idea why it became convention to involve squaring in working out the SD rather than just using that though.

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Why did the vector cross the road?
It wanted to be normal.

over_score

Member

Registered: 2007-03-12

Posts: 7

Re: Standard Deviation

Exactly, mathsyperson, what advantage does the standard deviation (root mean square), have over the more logical 'mean distance from the mean'?
A statement like 'the standard deviation is slightly different to the mean distance from the mean in that it does xxx' is what I'm after
I guess I'll sit down and figure out *how* it is different, post that, then pose the question *why* we do it the 'root mean square' (rms) way rather than the (more logcial imo) 'mean distance from the mean' way.

MathsIsFun

Administrator

Registered: 2005-01-21

Posts: 7,711

Re: Standard Deviation

Now we are talking three methods? RMS, "mean square root", and the absolute value one?

One effect of  "mean square root" would be to reduce the impact of extreme values, whereas the RMS increases the impact.

Altering Jane's example (in lieu of a proper analysis):

Let’s take the numbers 1, 2, 10.

The root mean square is √[(1² + 2² + 10²)∕3] = √(105⁄3) ≈ 5.92

The mean square root is (√1 + √2 + √10)∕3 ≈ 1.86

But then, shouldn't we be squaring the result of your method to be comparative?

It is all valid, of course ... with Statistics you process your data in the way you wish, but you need to justify your choices!

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"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

over_score

Member

Registered: 2007-03-12

Posts: 7

Re: Standard Deviation

No no no, all my fault, I made a typo!!
Start again, there are two methods we are talking:
1. The method for finding the standard deviation is the 'root mean squared' (or rms).
2. What I think is more logical is the 'mean distance from the mean' or 'mean root square'.
NB only the order of root & mean have changed between the two methods.

For example, if we have the values 100, 400, 600, 900
The mean is 500,
1. The SD is 336.65
2. The 'mean distance from the mean' is 250

Why is the first figure, the RMS, more useful, or why is it the preferred, used method - and not the (more logical imo) MRS 'mean distance from the mean'.