This is a simple example of how to calculate sample variance and sample standard deviation. First, let's review the steps for calculating the sample standard deviation:

- Calculate the mean (simple average of the numbers).
- For each number: subtract the mean. Square the result.
- Add up all of the squared results.
- Divide this sum by one less than the number of data points (N - 1). This gives you the sample variance.
- Take the square root of this value to obtain the sample 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 sample 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 preferâ€¦ 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) / 19 = 178/19 = 9.368

This value is the**sample variance**. The sample variance is 9.368 - The population standard deviation is the square root of the variance. Use a calculator to obtain this number.(9.368)
^{1/2}= 3.061

The population standard deviation is 3.061

Compare this with the variance and population standard deviation for the same data.