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Standard Deviation

Standard Deviation

Author: Katherine Williams

Identify how to calculate the standard deviation or variation of a data set.

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This tutorial talks about standard deviation. Standard deviation is the preferred measure of spread for symmetric or approximately symmetric distributions. Other measures of spread include range, variance, interquartile range. But standard deviation is preferred for symmetric distributions.

Now, the calculation is typically done with technology. I will go through an example of how to do it by hand. But you can easily find an online calculator or any kind of spreadsheet software, and that will do it for you as well.

Now, the standard deviation tells us that we can expect a randomly selected value to be about 1 standard deviation from the mean. So the mean minus 1 standard deviation up to the mean plus 1 standard deviation is where we can expect any randomly selected value to fall in. Now, that does not mean that all values fall within that range, just that if we randomly select value, we can expect it will be there.

Now, the variance is something else that we'll talk about. It's another measure of spread, but it's slightly easier to calculate. Let's look at the formula for standard deviation.

Now, with standard deviation, there are two different formulas. It depends on whether you have a population or a sample. Essentially, the formulas are the same with a few minor differences. For the most part, they will give you a very similar result, except for very, very small samples. We are going to show how to use the formula for a sample, because most of the data sets that you'll see and encounter will be samples.

One thing to note is this symbol here. This is the lowercase sigma. It's used to represent standard deviation. You'll be seeing this one a lot. So it's good to know.

Now, let's look through the steps on how to calculate standard deviation of a sample by hand. Here, this example, the heights of the Celtic starting line are 76 inches, 77 inches, 75 inches, 81 inches, and 83 inches. We're going to be using our sample standard deviation formula. But these steps will kind of walk us through how to go through this formula.

First, we need to find the mean. Then we're going to list out all the x values, find the difference between x and the mean, square those differences. Then, we find the variance. We take that sum of all of the x minus the x bar squareds and then divide by 1 less than the number of terms we have. And then, we're going to square root that variance.

So let's look at the numbers. So here, we have that same set of data from before. And I'm going to be calculating the standard deviation. Now, I started by setting up this chart. I like to use it because it helps me to keep my thoughts organized and my math organized.

I just started by writing out all the x values from our data set. The first thing we need to do is find the mean. So the mean of this set of five numbers is 78.4. Then, we're going to take each x value and subtract the mean. So we're going to do 76 minus 78.4, and we get negative 2.4. 77 minus 78.4, we get negative 1.4, and so on.

Then, once we have the difference between the x and the mean calculated, we're going to square those differences. When we do that-- so negative 2.4 squared yields 5.76. Now, at this point, all these numbers should be positive, because when you're squaring them, you have positive results.

Now, once we have this column, our next step is to take the sum of it. So when I add up those five numbers, the 5.76 plus the 1.96 plus 11.56 plus 6.7 plus 21.16, we get 47.2. We're on the way to finding our variance. In order to get the variance, I need to take that sum and divide by 1 less than the number of terms we have. So we have five terms. So I'm going to divide by four. So we have our sum of 47.2. And then we divide by 4 and get 11.8. So this here is our variance.

Now, in order to go from the variance to the standard deviation, I need to take the square root. So when I take the square root of 11.8, I get 3.44. And that is our standard deviation. So the standard deviation is 3.44. And then the units on that are going to be inches. On the variance, our units are inches squared.

So this kind of walks you through how to calculate the standard deviation by hand. This has been your tutorial on the standard deviation.

Terms to Know
Standard Deviation

A typical amount by which we would expect a data point to differ from the mean. Typically, about half to two-thirds of the data points fall within one standard deviation of the mean.


The square of standard deviation. While it has some uses in statistics, it is not a practical unit of measurement. It is calculated the same way as standard deviation, but without the square root.

Formulas to Know
Standard Deviation of a Sample

s equals square root of fraction numerator 1 over denominator n minus 1 end fraction sum from i equals 1 to n of open parentheses X subscript i minus X with bar on top close parentheses squared end root

n equals n u m b e r space o f space d a t a space p o i n t s
X subscript i equals e a c h space o f space t h e space v a l u e s space i n space t h e space d a t a space s e t
X with bar on top equals m e a n space o f space t h e space d a t a space s e t

Variance of a Sample

s squared equals fraction numerator 1 over denominator n minus 1 end fraction sum from i equals 1 to n of left parenthesis X subscript i minus X with bar on top right parenthesis squared