Standard Deviation Formula

The defining formual for the standard deviation of a collection of n data values, is given by

Although this is the defining formula, it is a pain to actually use because you need to find the mean, μ, first and then go through the data and find the squared differences between each point and μ, and then get the sum of those values, which you then divide by the number of values, n, and finally take the square root of the answer.

Fortunately, we can use some math to reconfigure the formual into a more convenient version. We start this with a rewrite where we expand the squared difference, much as we learned in Algebra I that (a – b>²=a² – 2ab + b²)

But that sum can be broken apart to be the separate sums of the three terms

However the term

, so we could rewrite the formula as

, Now, with a little slight of hand, we can make this a bit more complex in order to be able to actually simplify it later. We can rewrite

, but that has the factor

which is our definition of μ. Therefore, we can replace

with

and our formula becomes

Of course this reduces to

Then, replacing

with

we get

but that simplifies to

Finally, we can rewrite this as

That final formula may look terrible, but, computationally, it is wonderful. It says that to find the standard deviation we need to know the number of values, n, and the sum of the x_i values which we could call sumx and the sum of the x_i² values which we could call sumsqx. Then the formula becomes

This is the formula that small, and even some larger, calculators use because, as you are entering data values the calculator takes the value that you enter, adds it to the running total sumx, adds the square of the value to the running total sumsqx and increase the count of the data values, n, by one. Then, whenever you ask for the standard deviation the calculator just has to do the last formula above. The system has the extra advantage that if you realize that somewhere in the data entry process you made a mistake in entering a data value then it is easy to delete that value. You hit some button to tell the calculator to remove a value and all it has to do is ask for the value, subtract it from sumx, subtract its square from sumsqx, and decrease the number of data items, n, by one.

Of course, once you have the calculator, or once you have some computer software to do the calculation for you, there is little advantage to knowing these alternative formuli for finding the standard deviation.

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