Standard Deviation Calculation

Statistics is all about number crunching and data analysis. With the sharp scalpel of statistical tools, you can analyze tons of pure numerical data and actually derive some important conclusions from it. Be it the social sciences or pure sciences, most research projects, require you to analyze data sets of varied kinds. One of the most important statistical parameters, which can help you analyze the degree of spread in a data set, is standard deviation. It is a statistical tool, that is frequently used in applied mathematics, where a large number of data points need to be analyzed. The aim of writing this Buzzle article is to illustrate standard deviation calculation.

Standard Deviation: Definition

Let me clearly define standard deviation right at the start and provide you with the standard deviation formula. Standard deviation is the square root of a quantity called the 'Variance'. Variance of a data set is the average value of square of deviation of data points from a mean value. In simpler words, it gives you an idea about the deviation of a set of data points from its mean value. The definition seems complicated, because the standard deviation (σ) formula presented below, is itself complicated.

σ = √[{Σ_n=1^N (x_n - X_Mean)²}/{N - 1}]

Here X_Mean is the mean value of all the data points, N is the total number of points and x_n are all the data points. Calculating standard deviation can be substantially simplified if you split the whole calculation into small steps. Let us see how to calculate standard deviation in the next section, through an actual example.

Standard Deviation Calculation: Example

The best way of knowing how to find standard deviation is to work through an actual example, all by yourself. Larger is the data set you are analyzing, more careful you need to be while calculating. Here is a step-wise algorithm for standard deviation calculation, with a solved example.

Standard Deviation Calculation Step 1: Calculate the Mean Value
The first step is quite simple and it is purely calculating the mean value for the whole data set.

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

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

Standard Deviation Calculation Step 2: Calculate the Variance
Variance calculation is the next important step in calculating standard deviation. You need to be very careful in this step, as it's the most complex.

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.

Standard Deviation Calculation Step 3: Take the Square Root of Variance
Last step is the simplest of all. With this step, you will conclude the entire calculation.

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

Finally we get what we were looking for, which is the standard deviation of the data set, that we studied and it's '3.8173'. Hope the example of standard deviation calculation has left no doubt in your mind, about how to go about it. You can do this calculation using any spread sheet software like Microsoft Excel or Openoffice.org Calc.