Download 5.6 MEAN, MEDIAN, MODE Textbook Reference Section 12.1

SECTION 5.6
Mean, Median, Mode 221
Textbook Reference Section 12.1
5.6 MEAN, MEDIAN, MODE
CLAST OBJECTIVES
" Determine the mean, median, and mode
" Recognize properties and interrelations among the mean, median, and mode
" Infer relations and make accurate predictions from studying statistical data
" Interpret real world data involving frequency and cumulative frequency
tables
Mean, median, and mode are called measures of central tendency because each describes the
center of the data sample.
Mean (Average)
The mean or average is a calculation and is found by dividing the sum of the data values by
the number of data values.
Sum of data values
Mean =
Number of data values
Median
The median is the middle position when data values are ranked from low to high. To find the
median rank the data in ascending order. If there is an odd number of data values, then the
median is the value that lies in the middle position. If there is an even number of data value,
then the median is the average of the two middle values.
Mode
The mode is the most frequently occurring (popular) value.
Example
a) Find the mean, median, and mode
of the following set:
{5, 20, 35, 20, 50, 45, 20, 10, 25, 45}
Solution
There are 10 elements in the set.
Mean:
5 + 20 + 35 + 20 + 50 + 45 + 20 + 10 + 25 + 45
10
=
275
= 27.5
10
Median: 5, 10, 20, 20, 20, 25, 35, 45, 45, 50
Since we have 10 data values, the middle positions
are the 5 th and 6 th positions. Note there are four
data values below the 5 th position and 4 data
values above 6 th positions. 20 and 25 are in the
5 th and 6 th positions. When we average 20 and
20 + 25 45
25, we get
=
= 22.5
5
2
Thus, the median is 22.5.
Mode:
The mode is 20 as it appears most frequently.
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222 CHAPTER 5
Probability and Statistics
Check Your Progress 5.6
1. Consider the scores of nine students: {84, 88, 52, 87, 70, 66, 95, 84, 71}. Find the mean,
median, and mode.
2. Consider the following set of babies’ birth-weight (in pounds): {6.5, 7, 10, 8, 7, 7.5, 8, 7}.
Find the mean, median, and mode.
Comparing Mean, Median, and Mode
Mean
Median
Mode
Consider three cases.
Case 1: Symmetric
1
2
3
Case 2: Right - Skewed
1
2
4
5
6
7
8
Median
Mode
3
4
9
Mean
5
6
Mode
Case 3: Left - Skewed
Median
Mean
1
2
3
4
5
6
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SECTION 5.6
Examples
b) The graph below shows the number of hours
that college students spend in the game room
per week with the years spent in college.
Compare the mean, median, and mode.
Mean, Median, Mode 223
Solutions
Hours
The distribution is symmetric. Thus the
mean, median, and mode are equal.
1
2
3
Years in College
4
5
c) The graph below shows the distribution of
scores on an algebra quiz. Compare the mean,
median, and mode.
The distribution is right-skewed. This
indicates that the mode is less than the
median, which is less than the mean.
Frequency
25
20
15
10
5
0
65
70
75
80
85
90
95
Algebra Quiz Scores
d) Half of the students scored 90 on an English
test. Most of the remaining students scored 70
and the last few scored 30. Compare the mean,
median, and mode.
The distribution is skewed left. This
indicates that the mean is less than the
median, which is less than the mode.
30
70
90
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224 CHAPTER 5
Probability and Statistics
Number of Babies
Check Your Progress 5.6
3. The ounces of milk consumed per day by 6-month old babies is shown in the graph
below. Compare the mean, median, and mode.
6
5
4
3
2
1
0
22
24
26
28
30'
32
34
Ounces of Milk
4. The distribution of placement exam scores are shown below. Discuss the mean,
median, and mode.
15
10
5
0
95
90
85
80
75
70
65
60
Placement Test Scores
5. More than half of the shoes in a store are priced at $34.00. Most of the others are
priced at $45.00, and a few are priced at $61.00. Discuss the mean, median, and mode.
6. Half of the scores on a History test were 95. Most of the remaining scores were 75. A
few were 40. Discuss the mean, median, and mode.
Applications of Mean, Median, and Mode
Mode: The category with the highest frequency, percent, or proportion.
Mean: Multiply each category by its proportion and add.
Median: Organize categories in ascending order in one column. Write the corresponding
frequencies or proportions in an adjacent column. Starting at the top, add the frequencies or
proportions up to the fifty-percent point. The corresponding category will be the median.
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SECTION 5.6
Examples
e) The table below shows the proportion of
students taking a number of college-level
classes this semester. Find the mean,
median, and mode.
No. of classes
0
1
2
3
4
5
Proportion
0.02
0.02
0.11
0.23
0.54
0.08
Mean, Median, Mode 225
Solutions
Mean:
= 0(0.02) + 1(0.02) + 2(0.11)
+ 3(0.23) + 4(0.54) + 5(0.08)
= 0 + 0.02 + 0.22 + 0.69 + 2.16 + 0.4
= 3.49
Median: (Find the 50 % mark)
0.02 + 0.02 + 0.11 + 0.23 = 0.38 Å 38 %
0.38 + 0.54 = 0.92 Å 92 %
This indicates that the 50 % - mark falls in
the “ 4 Classes ” category. Thus the median
is 4 classes.
Mode:
The mode is 4 classes because the “ 4
Classes” category has the largest proportion.
f) In the table below, the scores on a
Spanish test are listed with the
corresponding percentile rank. What
percent of students scored between 60 and
80? What percent scored above 80? What
percent scored below 60?
Score
90
80
70
60
40
50
Percentile Rank
Between 60 and 80: 93 % – 34 % = 59 %
99
93
72
34
16
5
Above 80: 99 % – 93 % = 6 %
Below 60: 34 %
Check Your Progress 5.6
For Questions 7 and 8, use the information found in the table below. The table shows the
results of a survey in which people were asked the number of times per week that they
shop at a grocery store.
No. of Trips to
Grocery Store
0
1
2
3
4
Percent Men
18
45
12
17
8
Percent Women
7
12
53
13
15
7. Find the median number of trips to the grocery store for men.
8. Find the mean number of trips to the grocery store for women.
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226 CHAPTER 5
Probability and Statistics
For Questions 9 and 10, use the information in the table below, which shows the
distribution of the total family income for a certain county for the year 2000.
Income Level ($)
0 – 9,999
10,000 – 19,999
20,000 – 29,999
30,000 – 39,999
40,000 – 49,999
50,000 – 59,999
60,000 – 69,999
70,000 and over
Percent of Families
12
14
16
13
13
9
9
14
9. What percentage of families has incomes of at least $30,000?
10. What percentage of families has incomes between $20,000 and $49,999?
See If You
Remember
SECTIONS 5.1 – 5.5
1. How many 5-digit numbers can be formed from the digits 0 – 9, if digits cannot be
repeated?
2. Seven candidates are running for three city council seats. How many different groups
can be formed?
3. What are the odds in favor of randomly drawing a green ball from a box that contains 2
yellow, 3 red, 5 green, and 6 white balls?
4. A fair die is rolled three times. What is the probability of getting a three, then a two,
and then a one?
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