Standard Deviation - Basics

Standard Deviation

Standard deviation is a measurement used in statistics of the amount a number varies from the average number in a series of numbers. The standard deviation tells those interpreting the data, how reliable the data is or how much difference there is between the pieces of data by showing how close to the average all of the data is.

• A low standard deviation means that the data is very closely related to the average, thus very reliable.
• A high standard deviation means that there is a large variance between the data and the statistical average, thus not as reliable.

Read more at http://examples.yourdictionary.com/examples-of-standard-deviation.html#WVIR7t8FCIs2TfrI.99

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 (the squared difference).
• 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.
Your first step is to find the 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:

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.
Rottweilers are tall dogs. And Dachshunds are a bit short ... but don't tell them!

But ... there is a small change with Sample Data

Our 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

Here are the two formulas:

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.