Download 12.3 Measures of Dispersion

12.3 MEASURES OF DISPERSION
Range: The difference between the highest and lowest data values in a data set, indicates the total
spread of the data: Range = highest value – lowest value.
Standard deviation:
 (data value  mean)
Compares each data value with the mean:
2
. The
n 1
smaller the standard deviation, the less the variation in the group. A large standard deviation
indicates great diversity within the group.
1. Given test scores of 58, 28, 76, 73, 69, 84, 76, 97, 83, and 76, find:
a. Range
b. Standard Deviation
2. Another class had the following test scores: 67, 92, 90, 67, 85, 47, 67, 72, 66, 67. Find:
a. Range
b. Standard deviation
c. How do they compare and differ from the class in #1?
x
f
38
1
52
2
55
3
67
3
68
5
72
4
81
2
89
3
91
2
99
1
Use the above frequency distribution to find:
a. Mean of frequency distribution:
b. Median:
c. Mode:
d. Standard deviation:
x
38
52
55
67
68
72
81
89
91
99
1
2
3
3
5
4
2
3
2
1
Deviation
Deviation2
f
Total
4. The scores listed below are the final grades of 15 students. Construct a stem and leaf plot. Find
mean, median, mode, midrange, range, and standard deviation.
Average
67
81
67
72
78
62
67
77
82
76
52
79
68
69
76
12.4 THE NORMAL DISTRIBUTION
Normal Distribution: The graph of a normal distribution is a bell-shaped curve and is symmetric
about a vertical line through its center. The mean, median, and mode of a normal distribution are all
equal and occur at the center of the distribution. Empirical Rule: About 68% of all data values of a
normal distribution lie within 1 standard deviation of the mean (in both directions), about 95% are
within 2 standard deviations, and about 99.7% are within 3 standard deviations. Z-Scores describe
how many standard deviations an item in a normal distribution lies above or below the mean. Data
items above the mean have position z-scores; those below the mean are negative; and the mean has
a z-score of 0: z  score  data item  mean . Percentiles: If n% of the items in a distribution are
s tan dard deviation
less than a particular data item, then that data item is in the nth percentile of the distribution.
1. Suppose 300 chemistry students take a midterm exam and that the distribution of their scores
can be treated as normal. Find the number of scores that:
a. Lie within one standard deviation.
b. Lie within two standard deviations.
c. Lie within three standard deviations.
2. If the average grade of the above chemistry students is 72 with a standard deviation of 9, find the
scores that are exactly:
a. One standard deviation from the mean.
b. Two standard deviations from the mean.
c. Three standard deviations from the mean.
3. What percentage of the 300 students score:
a. between 63 and 81?
b. between 72 and 81?
c. Above 90?
4. Given the midterm in problem 1, what is the z-score of a student scoring
a. 75
b. 62
5. Relate the above z-scores to percentiles using the Table 12.10 on page 672.
6. What do the above percentiles mean?