Standard Deviation and Variance

Standard Deviation

The 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 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 each number: subtract the Mean and square the result
• Then work out the average of those squared differences.

Example: You and your friends have just measured the heights of your dogs (in millimeters)

Measurement

The heights (at the shoulders) are: 600mm, 470mm, 170mm, 430mm and 300mm.

Find out the Mean, the Variance, and the Standard Deviation.

Your first step is to find the Mean:

Answer:

                  Mean= (600 + 470 + 170 + 430 + 300) / 5= 394

so the mean (average) height is 394 mm. Let's plot this on the chart:

Mean

Now, we calculate each dogs difference from the Mean:

Variance

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

Variance

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:

Standard Deviation

So, using the Standard Deviation we have a "standard" way of knowing what is normal, and what is extra large or extra small.

There is a small change with Sample Data. The example was for a Population (the 5 dogs were the only dogs we were interested in).

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.

Formulas:

The "Population Standard Deviation"

The "Sample Standard Deviation"