Standard Deviation and Variance

Standard Deviation and Variance
Standard DeviationThe Standard Deviation is a measure of how spread out numbers are.
Its symbol is σ (the greek letter sigma)
The formula is easy: it is the square root of the Variance. So now you ask, "What is the Variance?"
Variance
The average of the squared differences from the Mean.

 for each number: subtract the Mean and square the result (the squared difference).

Then work out the average of those squared differences.

Now we calculate each dog's difference from the Mean:

To calculate the Variance, take each difference, square it, and then average the result:

So, the Variance is 21,704.

And the Standard Deviation is just the square root of Variance, so:

Standard Deviation: σ = √21,704 = 147.32... = 147 (to the nearest mm)

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.

But if the data is a Sample (a selection taken from a bigger Population), then the calculation changes!

When you have "N" data values that are:

• The Population: divide by N when calculating Variance (like we did)
• A Sample: divide by N-1 when calculating Variance

All other calculations stay the same, including how we calculated the mean.

Example: if our 5 dogs were just a sample of a bigger population of dogs, we would divide by 4 instead of 5 like this:

Sample Variance = 108,520 / 4 = 27,130

Sample Standard Deviation = √27,130 = 164 (to the nearest mm)

Think of it as a "correction" when your data is only a sample.

That is nice! The Standard Deviation is bigger when the differences are more spread out ... just what we want!
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