8-2 measures of central tendency and...
TRANSCRIPT
MAIN IDEADescribe a set of data using mean, median, mode, and range.
New Vocabularymeasures of central
tendencymeanmedianmode
Math Online
glencoe.com• Extra Examples• Personal Tutor• Self-Check Quiz
Key ConceptMean
Words The mean of a set of data is the sum of the data divided by the number of items in the data set. The mean is also referred to as average.
Examples data set: 1 cm, 1 cm, 5 cm, 2 cm, 2 cm, 4 cm, 2 cm, 5 cm
mean: 1 + 1 + 5 + 2 + 2 + 4 + 2 + 5 ___
8 or 2.75 cm
8-2
The number of pennies in each cup represents Jack’s scores for
five math quizzes.
8 7 9 6 10
Move the pennies among the cups so that each cup has the same
number of pennies.
1. What was Jack’s average score for the five quizzes? 8 points
2. If Jack scores 14 points on the next quiz, how many pennies would
be in each cup? 9
A number used to describe the center of a set of data is a measure of
central tendency. The most common of these measures is the mean.
Find the Mean
1 QUIZ SCORES The table shows the quiz scores Quiz Scores
47 40 43 45
49 41 49 44
43 41 44 49
44 50 41 44
for 16 students. Find the mean.
mean = 47 + 40 + ... + 44 __
16
sum of data number of data items
= 714 _
16 or 44.625
a. MONEY Adam earned $14, $10, $12, $15, and $13 by doing chores
around the house. What is the mean amount Adam earned doing
these chores? $12.80
Measures of Central Tendency and Range
402 Chapter 8 Statistics: Analyzing Data
402_0402_0408_CH08_L2_874046 402402_0402_0408_CH08_L2_874046 402 9/22/07 12:51:06 PM9/22/07 12:51:06 PM
Key ConceptMedian
Words In a data set that has been ordered from least to greatest, themedian is the middle number if there is an odd number of data items. If there is an even number of data items, the median is the mean of the two numbers closest to the middle.
Example data set: 7 yd, 11 yd, 15 yd, 17 yd, 20 yd, 20 yd
median: 15 + 17 _
2 or 16 yd The median divides
the data in half.
ModeWords The mode of a set of data is the number that occurs most
often. If there are two or more numbers that occur most often, all of them are modes.
Example data set: 50 mi, 45 mi, 45 mi, 52 mi, 49 mi, 56 mi, 56 mi
modes: 45 mi and 56 mi
Vocabulary LinkMedian
Everyday Use the middle paved or planted section of a highway, as in median strip.
Math Use the middle number of the ordered data.
Two other common measures of central tendency are median and mode.
Find the Mean, Median, and Mode
2 MOVIE RENTALS The number
of DVDs rented during one
week at Star Struck Movie
Rental is shown in the table.
What are the mean, median,
and mode of the data?
mean: 55 + 34 + 35 + 34 + 57 + 78 + 106
___ 7 = 399
_ 7 or 57
median: 34, 34, 35, 55, 57, 78, 106 First, write the data in order.
median
mode: 34 It is the only value that occurs more than once.
The mean is 57 DVDs, the median is 55 DVDs, and the mode is
34 DVDs.
b. BICYCLES The sizes of the bicycles
owned by the students in Ms. Garcia’s
class are listed in the table. What are the
mean, median, and mode of the data?
c. FOOTBALL The points scored in each game by Darby Middle
School’s football team for 9 games are 21, 35, 14, 17, 28, 14, 7, 21,
and 14. Find the mean, median, and mode. 19, 17, 14
24.25 in., 24 in., 24 in.24.25 in., 24 in., 24 in.
Star Struck Movie RentalDaily DVD Rentals
S M T W TH F S
55 34 35 34 57 78 106
Students’ Bicycle Sizes (in.)
20 24 20 26
24 24 24 26
24 29 26 24
Lesson 8-2 Measures of Central Tendency and Range 403
403_0402_0408_CH08_L2_874046 403403_0402_0408_CH08_L2_874046 403 9/22/07 12:51:18 PM9/22/07 12:51:18 PM
Comparing Measures Another way to solve Example 3 is to find the measures before 98 is added to the data set. Then find the measures after 98 is added to the data set. Then compare.
3 The maximum length in feet of several whales is listed below.
46, 53, 33, 53, 79
If the maximum length of the Blue Whale, 98 feet, is added to this
list, which of the following statements would be true?
A The mode would decrease. C The mean would increase.
B The median would decrease. D The mean would decrease.
Read the ItemYou are asked to identify which statement would be true if the data
value 98 was added to the data set.
Solve the ItemUse number sense to eliminate possibilities.
The mode, 53, will remain unchanged since the new data value
occurs only once. So, eliminate answer choice A.
Since the new data value is greater than each value in the data set, the
median will not decrease. So, eliminate answer choice B.
The remaining two answer choices refer to the mean. Since 98 is
greater than each value in the data set, the mean will increase, not
decrease. So, the answer is C.
d. If the maximum length of the Orca Whale, 30 feet, is added to the
list in Example 3, which of the following statements would be true?
F The mode would decrease. H The mean would increase.
G The median would increase. J The mean would decrease.
In addition to the mean, median, and mode, you can also use the range to
describe a set of data. Below are some guidelines for using these measures.
Concept SummaryMean, Median, Mode, and Range
Measure Most Useful When...
Mean · data set has no outliers
Median · data set has outliers· there are no big gaps in the middle of the data
Mode · data set has many identical numbers
Range · describing the spread of the data
JJ
404 Chapter 8 Statistics: Analyzing Data
404_0402_0408_CH08_L2_874046 404404_0402_0408_CH08_L2_874046 404 9/22/07 12:51:21 PM9/22/07 12:51:21 PM
Choose Mean, Median, Mode, or Range
4 PLANTS The line plot
shows the height of
desert cacti. Would the
mean, median, mode,
or range best represent
the heights?
mean: 1 + 2 + 2 + ... + 30
__ 14
or 8.8
median: 7th term + 8th term
__ 2 =
5 + 5 _
2 or 5
mode: 5
range: 30 - 1 or 29
The mean of 8.8 misrepresents the score. The median or mode
represents the height of the cacti well.
e. GAMES The table shows the cost of
various board games. Would the
mean, median, mode, or range best
represent the costs? Explain.
Examples 1, 2(pp. 402–403)
Find the mean, median, and mode for each set of data. Round to the nearest
tenth if necessary.
1. Miles traveled on the weekend: 29, 14, 80, 59, 78, 30, 59, 69, 55, 50
2. Team Number of Wins
Eagles 10
Hawks 8
Zipps 9
Falcons 11
3.
Example 3(p. 404)
4. MULTIPLE CHOICE During the week, the daily low temperatures were 52°F,
45°F, 51°F, 45°F, and 48°F. If Saturday’s low temperature of 51°F is added,
which statement about the data set would be true? C
A The mean would decrease. C The mode would increase.
B The median would decrease. D The mode would decrease.
Example 4(p. 405)
5. SHOES The line plot shows
80655045 10060 95908570 7555
× ×
×
×
×
×
×
×
×
×
×
×
×
×
× ××
×
×
×
×
×
×
×
×the price of athletic shoes.
Which measure best describes
the data: mean, median, mode,
or range? Explain. See margin.
e. The mean is $21.50,
the median is $18.50,
the mode is $40, and
the range is $28. The
mean or median could
be used to represent
the data.
e. The mean is $21.50,
the median is $18.50,
the mode is $40, and
the range is $28. The
mean or median could
be used to represent
the data.
★ indicates multi-step problem★ indicates multi-step problem
1. 52.3; 57; 591. 52.3; 57; 59
2. 9.5; 9.5; none
3. 26.5, 25.5, 23
and 25
2. 9.5; 9.5; none
3. 26.5, 25.5, 23
and 25
★★
Median Median When there is an odd number of data, the median is the middle number of the ordered set. When there is an even number of data, the median is the mean of the two middle numbers. 14820 18 22 26 306 281610 124 20 24
× ×
×
×
×
××
×
×
×
×
×
× ×
29262322 3225 313027 2824
×
×
× ×
×
×
×
×
×
×
× ×
×
×
×
×
Board Game Costs ($)
12 15 40 22
14 40 15 17
20 18 40 19
16 21 19 16
Lesson 8-2 Measures of Central Tendency and Range 405
0402_0408_CH08_L2_874046.indd 4050402_0408_CH08_L2_874046.indd 405 9/24/07 5:13:20 PM9/24/07 5:13:20 PM
Real-World LinkThe International Space Station measures 356 feet by 290 feet, and contains almost an acre of solar panels.Source: The World Almanac
Find the mean, median, and mode for each set of data. Round to the nearest
tenth if necessary.
6. Number of dogs groomed each week: 65, 56, 57, 75, 76, 66, 64 65.6; 65; none
7. Daily number of boats in a harbor: 93, 84, 80, 91, 94, 90, 78, 93, 80
8. Scores earned on a math test: 95, 90, 92, 94, 91, 90, 98, 88, 89, 100
For Exercises
6–1130–3212–13
See Examples
1, 234
HOMEWORK HELPHELP
9. Prices of books: $10, $18, $11, $6, $6, $5, $10, $11, $46, $7, $6, $8 $12; $9; $6
10. Cost Number of Coats
$75 8
$80 3
$85 6
11.
12. MUSIC The line plot shows the number of weeks that songs have been on
the Top 20 Country Songs list. Would the mean, median, mode, or range
best represent the data? Explain. See margin.
6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44
Country Songs Number of Weeks in Top 20
× ×× × ××× × × × ××
× × ×
× × × × ×
13. SPACE Twenty-seven countries have sent people into space. The table
shows the number of individuals from each country. Which measure best
describes the data: mean, median, mode, or range? Explain. See margin.
People in Space
267 1 9 8 1 1 1 1 1
97 1 1 1 3 1 1 2 1
11 2 1 1 5 1 1 1 1
Source: The World Almanac
Find the mean, median, and mode for each set of data. Round to the nearest
tenth if necessary.
14. Weight in ounces of various insects: 6.1, 5.2, 7.2, 7.2, 3.6, 9.0, 6.5, 7.4, 5.4
15. Prices of magazines: $3.50, $3.75, $3.50, $4.00, $3.00, $3.50, $3.25
16. Daily low temperatures: -2°F, -8°F, -2°F, 0°F, -1°F, 1°F, -2°F, -1°F
REASONING Determine whether each statement is always, sometimes, or never
true about the data set {8, 12, 15, 23}. Explain your reasoning.
17. If a value greater than 23 is added, the mean will increase.
18. If a value less than or equal to 8 is added, the mean will decrease.
19. If a value between 8 and 23 is added, the mean will remain unchanged.
Exercise Levels
A: 6–13
B: 14–24
C: 25–29
Exercise Levels
A: 6–13
B: 14–24
C: 25–29
11. 46, 45, 4411. 46, 45, 44
★★
★★
14. 6.4, 6.5, 7.2 15. $3.50, $3.50, $3.5014. 6.4, 6.5, 7.2 15. $3.50, $3.50, $3.50
-1.9°F, -1.5°F, -2°F-1.9°F, -1.5°F, -2°F
17–19.
See margin.
17–19.
See margin.
50443836 5642 545246 4840
× × ×
××
×
× × × × × ×× ××
×
7. 87; 90; 80 and 93 8. 92.7; 91.5; 90
$79.40; $80; $75
406 Chapter 8 Statistics: Analyzing Data
406_0402_0408_CH08_L2_874046 406406_0402_0408_CH08_L2_874046 406 9/22/07 12:51:27 PM9/22/07 12:51:27 PM
See pages 687, 711.EXTRA PRACTICEPRACTICE
DINOSAURS For Exercises 20–22, use the lengths of the dinosaurs
shown below.
Tyrannosaurus RexLength: 480 inches
CorythosaurusLength: 396 inches
ParasaurolophusLength: 480 inches
20. What is the mean length of the dinosaurs? 452 inches
21. One of the largest dinosaurs ever is the Brachiosaurus. Its length was
960 inches. If this data value is added to the lengths of the dinosaurs
above, how will it affect the mean? Explain your reasoning.
22. Which measure best describes the data if the length of the Brachiosaurus is
included: mean, median, mode, or range? If the length of the Brachiosaurus
is not included? Explain any similarities or differences.
23. SPORTS The table shows the points scored Hawks Lacrosse Team Points Scored
11 15 12 10 10 10 13
14 13 13 10 15 12
by a lacrosse team so far this season. The
team will play 14 games this season. How
many points need to be scored during the
last game so that the average number of
points scored this season is 12? Explain. 10 points; See margin for explanation.
24. FIND THE DATA Refer to the Data File on pages 16–19. Choose some data and
then describe it using the mean, median, mode, and range. See students’ work.
H.O.T. Problems 25. OPEN ENDED Give an example of a set of data in which the mean is not the
best representation of the data set. Explain why not.
26. Which One Doesn’t Belong? Identify the term that does not have the same
characteristic as the other three. Explain your reasoning.
mean median range mode
27. REASONING Determine whether the median is sometimes, always, or never
part of the data set. Explain your reasoning. See margin.
28. CHALLENGE Without calculating, would the mean, median, or mode be most
affected by eliminating 1,000 from the data shown? Which would be the
least affected? Explain your reasoning. See margin.
50, 100, 75, 60, 75, 1,000, 90, 100
29. MATHWRITING IN According to the U.S. Census Bureau, the typical
number of family members per household is 2.59. State whether this
measure is a mean or mode. Explain how you know.
21. Sample answer:
A length of 960
inches is much
greater than the
other pieces of
data. So, if this
piece of data is
added, the mean
will increase.
21. Sample answer:
A length of 960
inches is much
greater than the
other pieces of
data. So, if this
piece of data is
added, the mean
will increase.
22. Median or
mode; Mean,
median, or mode;
Sample answer:
The length of the
Brachiosaurus
is an outlier, so
the mean
misrepresents the
data if this value
is included.
22. Median or
mode; Mean,
median, or mode;
Sample answer:
The length of the
Brachiosaurus
is an outlier, so
the mean
misrepresents the
data if this value
is included.
25. Sample answer:
4, 5, 2, 2, 3, 3, 1, 0,
1, 2, 4, 68, 5; the
mean, 7.7, is
not the best
representation
since it is greater
than all the data
items except one.
25. Sample answer:
4, 5, 2, 2, 3, 3, 1, 0,
1, 2, 4, 68, 5; the
mean, 7.7, is
not the best
representation
since it is greater
than all the data
items except one.
26. range; mean,
median, and mode
are used to
describe the center
of data. Range
describes the
spread of the data.
26. range; mean,
median, and mode
are used to
describe the center
of data. Range
describes the
spread of the data.
See margin. See margin.
Lesson 8-2 Measures of Central Tendency and Range 407
407_0402_0408_CH08_L2_874046 407407_0402_0408_CH08_L2_874046 407 9/22/07 12:51:33 PM9/22/07 12:51:33 PM
30. The table below shows the number of
soup labels collected in one week by
each homeroom in grade 7.
Classroom Number of Soup Labels
Mr. Martin 138
Ms. Davis 125
Mr. Cardona 89
Mrs. Turner 110
Mr. Wilhelm 130
Mrs. LaBash ?
Which number could be added to the
set of data in order for the mode and
median of the set to be equal? C
A 89 C 125
B 110 D 130
31. An antique dealer purchased
5 antiques for a total of $850.00.
He later bought another antique
for $758.00. What is the mean cost
of all the antiques? H
F $151.60 H $268.00
G $170.00 J $321.60
32. Gina found the mean and median of
the following list of numbers.
5, 7, 7
If the number 11 was added to this list,
which of the following statements
would be true? A
A The mean would increase.
B The mean would decrease.
C The median would increase.
D The median would decrease.
33. TEMPERATURE The table shows record high temperatures for July Temperatures
99 98 96 98 98
97 100 103 103 103
100 95 100 103 105
Kentucky in July. Make a line plot of the data. (Lesson 8-1)
Find the simple interest earned to the nearest cent for each
principal, interest rate, and time. (Lesson 7-8)
34. $1,250, 3.5%, 2 years $87.50
35. $569, 5.5%, 4 months $10.43
36. FOOD The United States produced almost 11 Uses of Apples in the United States
FreshFruit59%
Juice andCider23%
CannedFruit11%
Other7%
Source: U.S. Apple Association
billion pounds of apples in a recent year. Use
the information in the graph to find how many
pounds of apples were used to make juice and
cider. (Lesson 7-1) about 2.53 billion
37. Name the property shown by the statement
4 × 6 = 6 × 4. (Lesson 1-8) Commutative (×)
PREREQUISITE SKILL Name the place value of the underlined digit. (p. 669)
38. 581 hundreds 39. 6,295 ones 40. 4,369 thousands 41. 2.84 tenths
See margin.See margin.
408 Chapter 8 Statistics: Analyzing Data
408_0402_0408_CH08_L2_874046 408408_0402_0408_CH08_L2_874046 408 9/22/07 12:51:38 PM9/22/07 12:51:38 PM