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?”

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?)

dogs on graph shoulder heights

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 therefore the variance .

Your initiative is to seek out the Mean:

dogs on graph: mean

Answer:

Mean = 600 + 470 + 170 + 430 + 3005

= 19705

= 394

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:

dogs on graph: deviation

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

Variance

σ2 = 2062 + 762 + (−224)2 + 362 + (−94)25

= 42436 + 5776 + 50176 + 1296 + 88365

= 1085205

= 21704

So the Variance is 21,704

And the variance is simply the root of Variance, so:

Standard Deviation

σ = √21704

= 147.32…

= 147 (to the closest mm)

 

dogs on graph: standard deviation

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?

Using

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.

Formulas

Here are the 2 formulas, explained at variance Formulas if you would like to understand more:

 

The “Population Standard Deviation”:

square root of [ (1/N) times Sigma i=1 to N of (xi - mu)^2 ]
square root of [ (1/(N-1)) times Sigma i=1 to N of (xi - xbar)^2 ]

The “Sample Standard Deviation”: Looks complicated, but the important change is to

divide by N-1 (instead of N) when calculating a Sample Variance.