The Standard Deviation may be a measure of how opened up numbers are.
Its symbol is σ (the greek letter sigma)
The formula is easy: it’s the root of the Variance. So now you ask, “What is that the Variance?”
The Variance is defined as:
The average of the squared differences from the Mean.
To calculate the variance follow these steps:
Work out the Mean (the simple average of the numbers)
Then for every number: subtract the Mean and square the result (the squared difference).
Then compute the typical of these squared differences. (Why Square?)
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 therefore the variance .
Your initiative is to seek out the Mean:
Mean = 600 + 470 + 170 + 430 + 3005
so the mean (average) height is 394 mm. Let’s plot this on the chart:
Now we calculate each dog’s difference from the Mean:
To calculate the Variance, take each difference, square it, then average the result:
σ2 = 2062 + 762 + (−224)2 + 362 + (−94)25
= 42436 + 5776 + 50176 + 1296 + 88365
So the Variance is 21,704
And the variance is simply the root of Variance, so:
σ = √21704
= 147 (to the closest mm)
And the good thing about the quality Deviation is that it’s useful. Now we will show which heights are within one variance (147mm) of the Mean:
So, using the quality Deviation we’ve a “standard” way of knowing what’s normal, and what’s size or extra small.
Rottweilers are tall dogs. And Dachshunds are a touch short, right?
normal distrubution 1 sd = 68%
We can expect about 68% of values to be within plus-or-minus 1 variance .
Read Standard Gaussian distribution to find out more.
Also try the quality Deviation Calculator.
But … there’s alittle change with Sample Data
Our example has been for a Population (the 5 dogs are the sole dogs we have an interest in).
But if the info may be a Sample (a selection taken from a much 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 an equivalent , including how we calculated the mean.
Example: if our 5 dogs are just a sample of a much bigger population of dogs, we divide by 4 rather than 5 like this:
Sample Variance = 108,520 / 4 = 27,130
Sample variance = √27,130 = 165 (to the closest mm)
Think of it as a “correction” when your data is merely a sample.
Here are the 2 formulas, explained at variance Formulas if you would like to understand more:
The “Population Standard Deviation”:
The “Sample Standard Deviation”: Looks complicated, but the important change is to
divide by N-1 (instead of N) when calculating a Sample Variance.