Standard deviation formula
The standard deviation formula measures the spread of the data about the mean value. It's useful in comparing sets of data which might have the same mean but a different range. For example, the mean of the following 2 is the same: 15, 15, 15, 14, 16 and 2, 7, 14, 22, 30. But the 2nd is clearly more spread out. If a set has a low standard deviation, the values aren't spread out too much.
Just like when working out the mean, the method is different if the data is given to you in groups.
Non-Grouped Data
Non-grouped data is just a list of values. The standard deviation formula is given by the formula:
s means 'standard deviation'.
S means 'the sum of'.
-x means 'the mean'
Example
Find the standard deviation of 4, 9, 11, 12, 17, 5, 8, 12, 14
First work out the mean: 10.222
Now, subtract the mean individually from each of the numbers given and square the result. This is equivalent to the (x - )² step. x refers to the values given in the question.
ow add up these results (this is the 'sigma' in the formula): 139.55
Divide by n. n is the number of values, so in this case is 9. This gives us: 15.51
And finally, square root this: 3.94
The standard deviation could often be calculated easier with a calculator and this might be acceptable in some exams. Check your calculator's manual to see how to calculate it on yours.
NB: If you have a set of numbers (e.g. 1, 5, 2, 7, 3, 5 and 3), if each number is increased by the same amount (e.g. to 3, 7, 4, 9, 5, 7 and 5), the standard deviation will be the same and the mean will have increased by the amount each of the numbers were increased by (2 in this case). This is because the standard deviation measures the spread of the data. Increasing each of the numbers by 2 doesn't make the numbers any more spread out, it just shifts them all along.
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