Standard Deviation

The standard deviation is related to the mean (or average) of the set of data. The standard deviation will show a relationship to how the rest of your data stacked up to the mean, was it close to the mean or was it spread out all over the place.

To find the standard deviation, you must find the square root of the variance. The variance is the average of the squared differences from the mean.  

Example: The class had the test scores of 71, 77,77,77,78,79,82,83,84,84,88, and you want to find the standard of deviation of these grades.

First, you must find the mean. So add them up and divide by the total number of items, which will give you 80 as your mean.    
Next, you need to find the variance, which just means how far away from the mean is each number.
     71-80=-9
     77-80=-3
     77-80=-3
     77-80=-3
     78-80=-2
     79-80=-1
     82-80=2
     83-80=3
     84-80=4
     84-80=4
     88-80=8
Now, take this differences and square them and then add them all together. Remember all the negative numbers will become positive when you square them.
-9²+-3²+-3²+-3²+-2²+-1²+2²+3²+4²+4²+8²=222
Your next step is to divide this total by total number of scores which will give you your variance of 20.182.

To get the standard deviation, you take the square root of the variance. In this case it would be 4.5.

Standard deviation is divided into three groups.So in this example you would take your mean of 80 and subtract 4.5 from it and add 4.5 to it to get the range of the first standard deviation, which would be 75.5 to 84.5. This shows that the majority of students passed the test. The second standard of deviation which would include 95% of the students test scores would range from 71 to 89, just adding and subtracting the standard deviation amount to last numbers. The final third standard deviation would include 99.8% of the data and the test scores would range from 66.5 to 93.5.