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Firefighter Math: 9.6 Standard Deviation

Standard deviation, or variance, is a measure of the closeness of each term to the average (mean). If the terms are all close to the mean or average, then the standard deviation will be small and the mean can be considered an accurate approximation of the distribution.

To calculate to the standard deviation:
1. Compute each deviation as discussed on Section 9.4. 
2. Now, square each deviation and add the results all together. 
3. Finally, divide the calculated total by the number of terms minus one, and take the square root.

Example 1 - Find the standard deviation of this set of fire run distances: 9.0 chains, 12 chains, 11.5 chains, 12 chains, 9.5 chains. 

Step 1. Find the average (see Section 9.3).

9.0 + 12.0 + 11.5 + 12.0 + 9.5 = 54, 54 ÷ 5 = 10.8

Step 2. Find the deviation of each term (see Section 9.4).
9.0 - 10.8 = -1.8
12 - 10.8 = 1.2
11.5 - 10.8 = 0.7
12.0 - 10.8 = 1.2
9.5 - 10.8 = -1.3


Step 3. Square each deviation above.
(-1.8)2 = 3.24
(1.2)2 = 1.44
(0.7)2 = 0.49 
(1.2)2 = 1.44 
(-1.3)2 = 1.69 


Step 4. Add the squared deviations together.
3.24 + 1.44 +0.49 + 1.44 + 1.69 = 8.30 

Step 5. Divide the sum of the squared deviations by the number of terms minus 1.
8.30 / (5-1) = 2.075

Step 6. Find the square root of the difference computed in Step 5.
√2.075 = 1.44

The standard deviation is 1.44 chains.

 

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