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July 8th, 2012, 10:44 AM   #1
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why is standard deviation squared?

Hello,

I am absolutely baffeled by the fact that standard deviation s = sqrt(1/n Sum((value-mean)^2))
Why is it not just simply:
1/n(sum((abs(value-mean))

Why is it squared and then rooted again?
Why isn't just the absolute value taken of the difference between the values and the mean?
That would make so much more sense.

Also, according to my book, if you don't root it, it's called the sample variance. What is the significance of this sample variance? it seems to me to be a completely meaningless number, because the square seems to be completely arbitrary.

Obviously, it isn't nonsence, because otherwise people wouldn't do it, I just really want to know why they do it this way!

Thank you!
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July 8th, 2012, 11:39 AM   #2
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Re: why is standard deviation squared?

Quote:
Originally Posted by questioner1
Hello,

I am absolutely baffeled by the fact that standard deviation s = sqrt(1/n Sum((value-mean)^2))
Why is it not just simply:
1/n(sum((abs(value-mean))

Why is it squared and then rooted again?
Why isn't just the absolute value taken of the difference between the values and the mean?
That would make so much more sense.

Also, according to my book, if you don't root it, it's called the sample variance. What is the significance of this sample variance? it seems to me to be a completely meaningless number, because the square seems to be completely arbitrary.

Obviously, it isn't nonsence, because otherwise people wouldn't do it, I just really want to know why they do it this way!

Thank you!
The sample variance is the average of the squares of the differences between the sample values and the mean. The standard deviation is the square root of the sample variance. The squaring has the effect of creating positive numbers in all cases so that the sample values don't end up simply canceling each other out (as they otherwise inevitably would). You are correct that you could get the same effect by using absolute values and there is a statistic called mean absolute value. It is not as widely used because it lacks certain mathematical properties that the square root of mean squared differences has. Let me see if I can find a lucid discussion. Wish me luck. My mathematical look ups today have produced even worse dreck than usual, pedagogically speaking.
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July 8th, 2012, 11:47 AM   #3
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Re: why is standard deviation squared?

Here's an article that argues FOR more widespread teaching and use of the mean absolute deviation (as they call it)

http://www.leeds.ac.uk/educol/documents/00003759.htm
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July 8th, 2012, 12:38 PM   #4
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Re: why is standard deviation squared?

It is a lot easier to calculate the variance rather than the mean absolute difference.
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July 8th, 2012, 01:19 PM   #5
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Re: why is standard deviation squared?

The reason why standard deviations will likely always hold sway over mean absolute differences is that standard deviations have strict probabilistic interpretations under the normal distribution. Of course, statisticians can get a little fudgy about when normal distributions truly hold, but even if things are just approximate, that's better than having a statistic from which no probabilities can be calculated.

But the mean absolute difference may make at least a bit of a comeback. It too has some advantages, as discussed in the paper I linked to.
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July 8th, 2012, 01:20 PM   #6
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Re: why is standard deviation squared?

Quote:
Originally Posted by mathman
It is a lot easier to calculate the variance rather than the mean absolute difference.
How so?
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July 9th, 2012, 12:42 PM   #7
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Re: why is standard deviation squared?

Quote:
Originally Posted by johnr
Quote:
Originally Posted by mathman
It is a lot easier to calculate the variance rather than the mean absolute difference.
How so?
To get the first two moments, take the average and the average of the squares. The mean and variance can be calculated from these two numbers. The mean absolute difference requires the calculation of the mean first, and then go back to the original data to get the differences.

For computer programming purposes, the first needs one loop, while the second needs two loops.
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July 9th, 2012, 04:28 PM   #8
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Re: why is standard deviation squared?

Quote:
Originally Posted by mathman
Quote:
Originally Posted by johnr
Quote:
Originally Posted by mathman
It is a lot easier to calculate the variance rather than the mean absolute difference.
How so?
To get the first two moments, take the average and the average of the squares. The mean and variance can be calculated from these two numbers. The mean absolute difference requires the calculation of the mean first, and then go back to the original data to get the differences.

For computer programming purposes, the first needs one loop, while the second needs two loops.
Interesting. I wouldn't have guessed that this was the case.
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