Calculate Standard Deviation

Every day, we drown in a sea of data which is generated and bombarded at us every minute. Especially, if you work in a field like finance, that is always the case. There is no way one can make a sense of the numerical data without statistical tools. Statistical techniques can give you a condensed view of the trend shown by a given data set. They can give you the gist of the nature of certain data sets. In this article, I demonstrate how to calculate standard deviation in simple steps.

What is Standard Deviation?
Before we go in to standard techniques of calculating standard deviation, let us see what do we mean by standard deviation and get some basic statistical ideas cleared, that you need to know before you can calculate standard deviation.

Standard deviation can be defined as the 'Square Root of a Quantity called the Variance'! Roughly put, variance is the mean value of square of deviations of data points from the mean value. To understand standard deviation, one must understand what is variance, which inevitably necessitates understanding what is mean value of the set of numbers. So, let us see what is the mean of certain set of numbers or data points. The mean is the average value of a set of numbers. To calculate the mean value of a set of numbers is very simple. Take the set of numbers and add them together. Divide the sum of all these numbers by the total quantity of numbers that you added. That will give you the mean value of numbers. In mathematics, understanding things is simpler through worked out examples. We learn math better, by doing the math. So, let us see how to stepwise, calculate standard deviation.

Steps to Calculate Standard Deviation
Let us begin our example, which will demonstrate calculating standard deviation. Keep a calculator in hand. I assume you know how to handle a calculator and know how to execute the standard mathematical operations of adding, subtracting, multiplying and dividing, as well as squaring (multiplying the number by itself) and taking a square root, using a calculator.

Step 1: Calculate the Mean Value
First let us take a set of data points and get their mean value.

Step	Result
Get the Data Set of Numbers or a 'population' of numbers as statisticians call it.	Consider the data points to be 1, 5, 8, 7, 13, 11, 4
How many numbers are there? Count the number of data points	The number of data points is 7.
Add the data points and note the sum	1 + 5 + 8 + 7 + 13 + 11 + 4 = 49
Divide the Sum by the total number of data points to get the mean value.	49 / 7 = 7

So, the mean value of our set of data points is '7'.

Step 2: Calculate the Variance
Next step is to calculate the variance of this set of data points. Here is how we go about it.

Step	Result
Calculate the deviation of each number from the mean value, that is the difference between a data point and its mean. That is subtract the mean (7 here ) from all the data points (1,5,8,7,13,11, 4). The deviation may be negative or positive.	(1-7) = -6, (5-7) = -2, (8-7) = 1, (7-7) = 0, (13-7) = 6, (11-7) = 4, (4-7) = -3
Square all the deviations or differences that you got for the set of points which are -6, -2, 1, 0, 6, 4, -3	(-6)2 = 36, (-2)2 = 4, (1)2 = 1, (0)2 = 0, (6)2 = 36, (4)2 = 16, (-3)2 = 9
Add all the square of deviating values which in this case are (36, 4, 1, 0, 36, 16, 9).	36 + 4 + 1 + 0 + 36 + 16 + 9 = 102
Divide the sum of the square of deviations by the total number of data points, to get the variance. That is in this case, divide 102 by 7. This assumes that the data points make a complete population and we calculate the 'population standard deviation'. If the data points come from random sampling from a parent set of data points, then the sum of square of deviations, should be divided by the 'One minus the total number of data points' that is (7-1) = 6. In that case, we would be calculating 'sample standard deviation'. Here we will divide by 7 as we are not random sampling.	102 / 7 = 14.57143

Thus, the variance of this data set is 14.57143.

Step 3: Take the Square Root of Variance
Let us see what is the last step which will directly give you the value of standard deviation.

Step	Result
Calculate the square root of variance (√14.57143 in this case) by using a calculator.	√14.57143 = 3.8173

So, finally we have calculated the standard deviation of this data set, which is '3.8173'.

You now know how to calculate standard deviations. What do you do with it? How is it useful? Standard deviation is a measure of how spread out is your data from the mean value. A high value of standard deviation means you have a data set which is highly spread out from the mean. While a low value of standard deviation means that your data points are tightly bound around the mean value. Standard deviation is an important quantity, when comparing two or more data sets.