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.

MEAN, MEDIAN OR MODE
WHAT DO THESE CRAZY WORDS MEAN?

The mean is the average of a set of numbers. You use this a lot in sports such as batting averages, bowling scores, etc. Teachers will use this a lot to find the class average of a test score. To find the mean, you simply add up all the numbers in the set and divide the total by the number of items.

The median is the middle number when you line up the set from least to greatest. Remembering that the median is in the middle of the road may help you remember what to do for this number. To find the median, simply put the numbers in the set in order from least to greatest. Then cross off the first and last number, repeating this until you get to the middle number, which is your median. If you have an even number, you will have two numbers left. In this case, you add the two numbers together and divide by two to get your median.

Mode is just the number that occurs the most times in the set. To find the mode, put the numbers in order from least to greatest, and find the number that occurs the most. You can have two modes which is called bimodal.