Page 162 - Probability Demystified

CHAPTER 9 The Normal Distribution 151

SOLUTION:

Since the largest data value is 40 and the smallest data value is 16, the range is
40 16 ¼ 24.

Another measure that is also used as a measure of variability for individual
data values is called the standard deviation. This measure was also used in
Chapter 7.
The steps for computing the standard deviation for individual data
values are

Step 1: Find the mean.
Step 2: Subtract the mean from each value and square the differences.
Step 3: Find the sum of the squares.
Step 4: Divide the sum by the number of data values minus one.
Step 5: Take the square root of the answer.

EXAMPLE: Find the standard deviation for 32, 18, 15, 24, and 11.

SOLUTION:

32 þ 18 þ 15 þ 24 þ 11 100
Step 1: Find the mean: ¼ ¼ 20
5 5
Step 2: Subtract the mean from each value and square the differences:

2
32 20 ¼ 12 12 ¼ 144
2
18 20 ¼ 2 ð 2Þ ¼ 4
2
15 20 ¼ 5 ð 5Þ ¼ 25
2
24 20 ¼ 4 ð4Þ ¼ 16
2
11 20 ¼ 9 ð 9Þ ¼ 81
Step 3: Find the sum of the squares:
144 þ 4 þ 25 þ 16 þ 81 ¼ 270
Step 4: Divide 270 by 5 1 or 4: 270 4 ¼ 67.5
p ffiffiffiffiffiffiffiffiffi
Step 5: Take the square root of the answer 67:5 ¼ 8:22 (rounded)
The standard deviation is 8.22.

Recall from Chapter 7 that most data values fall within 2 standard
deviations of the mean. In this case, 20 2 (8.22) is 3.56 < most