- 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**.

### 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}= 2.983The population standard deviation is 2.983

### Learn More

Standard Deviation EquationsHow To Calculate Standard Deviation