9.6 Standard Deviation

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.

Practice

Exercise 1 - Joe wants to determine the slope with a clinometer and records the following numbers: 30, 27, 29, 30, 31, 30, 32 degrees. What are the average, range, and standard deviation of these values?

1. What is the average?

Select the correct answer.

2. What is the range of Joe's measurements?

Select the correct answer.

3. What is the standard deviation of the numbers? First, compute the deviations from the average (see Question 1). Select the correct answer from the options below.

The correct answer is a. Subtract the average from each number to compute the deviation, e.g. 27 - 30 = -3.

Compute the sum of the squares of the deviations and select the appropriate answer below.

The correct answer is b, 15. Squaring each deviation yields 0 + 9 + 1 + 0 + 1 + 4 = 15.

What is the next step?

The correct answer is b, divide the sum by 6. The sum is divided by the number of elements minus 1.

4. What is the standard deviation?

The correct answer is c, 1.6. Dividing by n - 1 yields 15/6 = 2.5. The standard deviation is the square root of 2.5, or 1.6.