Standard deviation is a calculation of the dispersion or variation in a set of numbers. If the standard deviation is a small number, it means the data points are close to their average value. If the deviation is large, it means the numbers are spread out, further from the mean or average.

There are two types of standard deviation calculations. Population standard deviation looks at the square root of the variance of the set of numbers. It's used to determine a confidence interval for drawing conclusions (such as accepting or rejecting a hypothesis). A slightly more complex calculation is called sample standard deviation. This is a simple example of how to calculate variance and population standard deviation. First, let's review how to calculate the population standard deviation:

- Calculate the mean (simple average of the numbers).
- For each number: Subtract the mean. Square the result.
- Calculate the mean of those squared differences. This is the
**variance**. - Take the square root of that to obtain the
**population standard deviation**.

## Population Standard Deviation Equation

There are different ways to write out the steps of the population standard deviation calculation into an equation. A common equation is:

σ = ([Σ(x - u)^{2}]/N)^{1/2}

Where:

- σ is the population standard deviation
- Σ represents the sum or total from 1 to N
- x is an individual value
- u is the average of the population
- N is the total number of the population

## Example Problem

You grow 20 crystals from a solution and measure the length of each crystal in millimeters. Here is your data:

9, 2, 5, 4, 12, 7, 8, 11, 9, 3, 7, 4, 12, 5, 4, 10, 9, 6, 9, 4

Calculate the population standard deviation of the length of the crystals.

- Calculate the mean of the data. Add up all the numbers and divide by the total number of data points.(9+2+5+4+12+7+8+11+9+3+7+4+12+5+4+10+9+6+9+4) / 20 = 140/20 = 7
- Subtract the mean from each data point (or the other way around, if you prefer... you will be squaring this number, so it does not matter if it is positive or negative).(9 - 7)
^{2}= (2)^{2}= 4

(2 - 7)^{2}= (-5)^{2}= 25

(5 - 7)^{2}= (-2)^{2}= 4

(4 - 7)^{2}= (-3)^{2}= 9

(12 - 7)^{2}= (5)^{2}= 25

(7 - 7)^{2}= (0)^{2}= 0

(8 - 7)^{2}= (1)^{2}= 1

(11 - 7)^{2}= (4)2^{2}= 16

(9 - 7)^{2}= (2)^{2}= 4

(3 - 7)^{2}= (-4)2^{2}= 16

(7 - 7)^{2}= (0)^{2}= 0

(4 - 7)^{2}= (-3)^{2}= 9

(12 - 7)^{2}= (5)^{2}= 25

(5 - 7)^{2}= (-2)^{2}= 4

(4 - 7)^{2}= (-3)^{2}= 9

(10 - 7)^{2}= (3)^{2}= 9

(9 - 7)^{2}= (2)^{2}= 4

(6 - 7)^{2}= (-1)^{2}= 1

(9 - 7)^{2}= (2)^{2}= 4

(4 - 7)^{2}= (-3)2^{2}= 9 - Calculate the mean of the squared differences.(4+25+4+9+25+0+1+16+4+16+0+9+25+4+9+9+4+1+4+9) / 20 = 178/20 = 8.9

This value is the variance. The variance is 8.9 - The population standard deviation is the square root of the variance. Use a calculator to obtain this number.(8.9)
^{1/2}= 2.983

The population standard deviation is 2.983

## Learn More

From here, you might wish to review the different standard deviation equations and learn more about how to calculate it by hand.

## Sources

- Bland, J.M.; Altman, D.G. (1996). "Statistics notes: measurement error."
*BMJ*. 312 (7047): 1654. doi:10.1136/bmj.312.7047.1654 - Ghahramani, Saeed (2000).
*Fundamentals of Probability*(2nd ed.). New Jersey: Prentice Hall.