Standard Deviation and Variance
Deviation just means how far from the normal
Standard Deviation
The Standard Deviation (σ) is a measure of how spread
out numbers are.
The formula is easy: it is the square root of the Variance.
So now you ask, "What is the Variance?"
Variance
The Variance (which is the square of the standard deviation, ie: σ^{2}) is defined as:
The average of the squared
differences from the Mean.
In other words, follow these steps:
1. Work out the Mean (the simple average
of the numbers)
2. Now, for each number subtract the Mean and then square the result
(the squared difference).
3. Then work out the average of those squared differences. (Why
Square?)
Example
You and your friends have just measured the heights of your dogs
(in millimeters):
The heights (at the shoulders) are: 600mm, 470mm, 170mm, 430mm
and 300mm.
Find out the Mean, the Variance, and the Standard Deviation.
Answer:
Mean = 
600 + 470 + 170 + 430 + 300

= 
1970

= 394 


5

5

so the average height is 394 mm. Let's plot this on the chart:
Now, we calculate each dogs difference from the Mean:
To calculate the Variance, take each difference, square it, and
then average the result:
Variance: σ^{2} = 
206^{2} + 76^{2} + (224)^{2}
+ 36^{2} + (94)^{2}

= 
108,520

= 21,704 


5

5

So, the Variance is 21,704.
And the Standard Deviation is just the square root of Variance,
so:
Standard Deviation: σ = √21,704 = 147
And the good thing about the Standard Deviation is that it is useful.
Now we can show which heights are within one Standard Deviation
(147mm) of the Mean:
So, using the Standard Deviation we have a "standard"
way of knowing what is normal, and what is extra large or extra
small.
Rottweillers are tall dogs. And Dachsunds are a bit
short ... but don't tell them!
*Note: Why square ?
Squaring each difference makes them all positive numbers (to avoid
negatives reducing the Variance)
And it also makes the bigger differences stand out. For example
100^{2}=10,000 is a lot bigger than 50^{2}=2,500.
But squaring them makes the final answer really big, and so unsquaring
the Variance (by taking the square root) makes the Standard Deviation
a much more useful number.
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