Standard Deviation (biostat)

[Arosa Hya]

Standard Deviation
Simple Example of Calculating Standard DeviationLet's say we wanted to calculate the standard deviation for the amounts of gold coins pirates on a pirate ship have.
There are 100 pirates on the ship. In statistical terms this means we have a population of 100. If we know the amount of gold coins each of the 100 pirates have, we use the standard deviation equation for an entire population:

http://standard-deviation.appspot.com/images/standard-deviation-1.png

http://standard-deviation.appspot.com/images/where.png

http://standard-deviation.appspot.com/images/standard-deviation-def-1.png

http://standard-deviation.appspot.com/images/value-1.png

http://standard-deviation.appspot.com/images/mean.png

http://standard-deviation.appspot.com/images/population.png
What if we don't know the amount of gold coins each of the 100 pirates have? For example, we only had enough time to ask 5 pirates how many gold coins they have. In statistical terms this means we have a sample size of 5 and in this case we use the standard deviation equation for a sample of a population:
http://standard-deviation.appspot.com/images/standard-deviation-2.png
http://standard-deviation.appspot.com/images/where.png

http://standard-deviation.appspot.com/images/value-2.png
http://standard-deviation.appspot.com/images/standard-deviation-def-2.png
http://standard-deviation.appspot.com/images/value-2.png
http://standard-deviation.appspot.com/images/mean.png
http://standard-deviation.appspot.com/images/sample-size.png


The rest of this example will be done in the case where we have a sample size of 5 pirates, therefore we will be using the standard deviation equation for a sample of a population.
Here are the amounts of gold coins the 5 pirates have:
4, 2, 5, 8, 6.
Now, let's calculate the standard deviation:


1. Calculate the mean:

http://standard-deviation.appspot.com/images/mean-def-1.png
http://standard-deviation.appspot.com/images/mean-def-2.png
http://standard-deviation.appspot.com/images/mean-def-3.png
http://standard-deviation.appspot.com/images/mean-def-4.png


2. Calculate http://standard-deviation.appspot.com/images/x-x.png for each value in the sample:


http://standard-deviation.appspot.com/images/x1-x1.png
http://standard-deviation.appspot.com/images/x2-x1.png
http://standard-deviation.appspot.com/images/x3-x1.png
http://standard-deviation.appspot.com/images/x4-x1.png
http://standard-deviation.appspot.com/images/x5-x1.png


3. Calculate http://standard-deviation.appspot.com/images/sumx-x.png:
http://standard-deviation.appspot.com/images/sumx-x1.png
http://standard-deviation.appspot.com/images/sumx-x2.png
http://standard-deviation.appspot.com/images/sumx-x3.png


4. Calculate the standard deviation:
http://standard-deviation.appspot.com/images/standard-deviation-2.png
http://standard-deviation.appspot.com/images/sdans1.png
http://standard-deviation.appspot.com/images/sdans2.png
The standard deviation for the amounts of gold coins the pirrates have is 2.24 gold coins.