Maureen P Sullivan/Getty Images Updated on October 05, 2019 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: 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 Population
Standard Deviation Equation
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)22 = 16
(9 - 7)2 = (2)2 = 4
(3 - 7)2 = (-4)22 = 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)22 = 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.
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Let’s calculate the variance of the follow data set: 2, 7, 3, 12, 9.
The first step is to calculate the mean. The sum is 33 and there are 5 data points. Therefore, the mean is 33 ÷ 5 = 6.6. Then you take each value in data set, subtract the mean and square the difference. For instance, for the first value:
(2 - 6.6)2 = 21.16
The squared differences for all values are added:
21.16 + 0.16 + 12.96 + 29.16 + 5.76 = 69.20
The sum is then divided by the number of data points:
69.20 ÷5 = 13.84
The variance is 13.84. To get the standard deviation, you calculate the square root of the variance, which is 3.72.
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