# How to Calculate Standard Deviation

by RJBradley

## Step-by-step tutorial for computing the standard deviation of a set of numbers with an example.

The standard deviation is a *statistic*, a number that is calculated from another set of numbers. Other statistics include the mean and median, which tell you where the "center" is in a set of data. The standard deviation tells you how spread out the data is, how much it deviates from the center. People who crunch numbers for a living use statistical calculators and software to compute the standard deviation, but if you know the formula you can also compute it with a simple hand calculator.

### The Formula

The formula for the standard deviation of a set of numerical data is sqrt(∑(m-x_{i})^{2}/n), where * m* is the mean of the set, the

*are the elements of the set, and*

**x**_{i}*is the number of elements in the set. To be more accurate, this is the formula for*

**n***population standard deviation*. You use this formula when your data represents the entire sample space.

If your set of data is a merely a sample from a larger set, the formula for standard deviation is slightly different: sqrt(∑(m-x_{i})^{2}/(n-1)). This is called, conveniently, the *sample standard deviation*. Notice that instead of dividing by n, you divide by n-1.This has the effect of slightly increasing the standard deviation. The larger the value of n, the less of a difference this makes.

### Sample Exercise in Computing the Standard Deviation

Alice and Bob are thinking about selling their home. To come up with a reasonable selling price, they take a sample of other homes in their city that are for sale and are of a similar size and age. Here are the numbers they come up with:

- $100,000
- $140,000
- $150,000
- $150,000
- $175,000
- $190,000
- $230,000

The mean selling price of the 7 homes is $162,143. Since this is a sample of a larger set, they should use the second formula, sqrt(∑(162143-x_{i})^{2}/6). The final answer is $41,217.

### Another Example

An anthropologist surveys all of the adult residents in a small village to find out how many different languages people can speak. Out of the 1000 adults, 681 speak only one language, 205 speak two languages, 109 speak three languages, 4 speak four languages, and 1 person can speak six languages.

The mean number of languages spoken by a person is [681*1 + 205*2 + 109*3 + 4*4 + 1*6]/1000 = 1440/1000 = 1.44.

The population standard deviation is sqrt(∑(1.44-x_{i})^{2}/1000), or

sqrt[(681*(1.44-1)^{2} + 205*(1.44-2)^{2} + 109*(1.44-3)^{2} + 4*(1.44-4)^{2} + 1*(1.44-6)^{2})/1000]

= sqrt[(131.84 + 64.29 + 255.53 + 26.21 + 20.79)/1000]

= sqrt[498.66/1000]

= 0.7062.

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## Comments

Rose on 01/20/2012This takes me back to my university days!

EKlein on 01/19/2012Good explanation of the difference between population s.d. and sample s.d.

RJBradley on 01/13/2012Yes, they can do that, or they can consider +/- 1 standard deviation from the mean to be an "average" range. Realistically, though, their sample size is pretty small and they'd be better off contacting a real estate agent!

TerriRexson on 01/13/2012So how do Alice and Bob use the standard deviation to come up with a selling price for their house? Do they add the standard deviation to the mean and then go a bit under that to be at the higher end of normal?