Chapter 8. Graphs and Statistics
8.1
8.2
8.3
8.4
Reading Graphs and Tables
Mean, Median and Mode
Equations in Two Variables; The Rectangular
Coordinate System
Graphing Linear Equations
1
8.1 Reading Graphs and Tables
1. Reading tables
2. Read bar graphs
3. Read pictographs
4. Read circle graphs
5. Read line graphs
6. Read histograms and frequency polygon
2
8.1 Reading Graphs and Tables
1. Reading tables
Weight
Not over
(pounds)
1
2
3
4
5
6
7
8
9
10
11
12
Postage Rate for Priority Mail 2009
Zones
Local,
1&2
$4.95
4.95
5.50
6.10
6.85
7.55
8.30
8.80
9.25
9.90
10.55
11.20
3
$4.95
5.20
6.25
7.10
8.15
9.25
10.30
10.70
11.45
12.35
13.30
14.20
4
$4.95
5.75
7.10
8.15
9.45
10.75
12.05
13.10
13.95
15.15
16.40
17.60
5
$4.95
7.10
9.05
10.80
12.70
14.65
46.55
17.95
19.15
20.75
22.40
24.00
6
$4.95
7.60
9.90
11.95
13.75
15.50
17.30
18.80
20.30
22.50
24.75
26.95
7
$4.95
8.10
10.60
12.95
15.20
17.50
19.75
21.70
23.60
25.90
28.20
30.50
8
$4.95
8.70
11.95
14.70
17.15
19.60
22.05
24.75
27.55
29.95
32.40
34.80
e.g.1 POSTAL RATES Refer to the table of priority mail
postal rates. Find the cost of mailing a 5 43 -pound
package by priority mail to post zone 6. $15.50
3
8.1 Reading Graphs and Tables
2. Read bar graphs – the graph shows maximum speed of animals
zebra
reindeer
lion
giraffe
elephant
coyote
chicken
cheetah
cat (domestic)
0
10
20
30
40
50
60
70
80
e.g.2 SPEED OF ANIMALS Refer to the bar graph.
a. What is the maximum speed of chicken? 9 mph
b. How much greater is the maximum speed of a zebra compared to
that of an elephant?
15 mph
c. What animals have the same maximum speed? Giraffe and reindeer
4
8.1 Reading Graphs and Tables
2. Read bar graphs – the graph shows national income by industry
4000
3500
3000
2500
2000
1500
1000
500
0
wholesale
retail
service
yr 1990
yr 2000
yr 2007
e.g.3 THE US ECONOMY Refer to the bar graph (y-axis in billions)
a. What income was generated by whole sales in 2000? $500 Bi
b. What income was generated by service sector in 1990? $1000 Bi
c. By what amount did the income from the service sector increase
from 1990 through 2007? $2,500 Bi
5
8.1 Reading Graphs and Tables
3. Read pictographs
Pizzas ordered during final exam week
= 12 pizzas
Men’s
Residence
Hall
Women’s
Residence
Hall
Co-ed
Residence
Hall
e.g.4 PIZZA DELIVERIES In the pictograph, what information does
the middle row of pizzas give?
54 pizzas were ordered during the final exam week in women’s residence hall.
6
8.1 Reading Graphs and Tables
4. Read circle graphs
S. Africa 10%
China 12%
2008 Word Gold Production
78 millions troy ounces
Russia 7%
Australia 10%
Canada 4%
U.S. 10%
Other 47%
e.g.5 GOLD PRODUCTION Refer to the circle graph.
a. What percent of the total was combined productions of
24%
Australia, the U.S., and Canada in 2008
b. How many ounces of gold did South Africa produce in
2008?
7.8 millions ounces
7
8.1 Reading Graphs and Tables
5. Read line graphs
450
ATM in the U.S.
(y-axis in thousands)
400
350
300
250
2000
2001
2002
2003
2004
2005
2006
2007
e.g.6 Refer to the line graph.
a. How many ATMs were there in the U.S. in 2005? 390k
b. Find the increase in the number of ATMs between 2000
About 50k
and 2001?
c. In which year did the number of ATMs first exceed
2002
350,000?
8
8.1 Reading Graphs and Tables
6. Read histograms and frequency polygon
Weight of Carry-on Luggages
970
Frequency
1100
900
700
500
300
100
-100
540
430
200
120
3.5
7.5
11.5
15.5
Weight (lb)
19.5
23.5
e.g.8 CARRY-ON LUGGAGE Refer to the histogram.
a. How many passengers carried luggage in the
120 passengers
4-to-7-pound range?
b. How many passengers carried luggage in the
1940 passengers
8-to-19-pound range?
9
8.2 Mean, Median and Mode
1. Find the mean (average) of a set of values
2. Find the weighted mean of a set of values
3. Find the median of a set of values
4. Find the mode of a set of values
5. Use the mean, median, mode, and range to describe a set
of values
10
8.2 Mean, Median and Mode
1. Find the mean (average) of a set of values (review)
Add all values then divide the sum by number of values.
e.g. Given a set of 4 values {1, 2, 3, 4} the average is
1+ 2 + 3+ 4
= 2 .5
4
e.g.2 DELIVERING PIZZA In the month of March,
delivery persons at a Pizza Hut store drove 2,945
miles in their pizza delivery car. On average, how
many miles were driven per day in the delivery car?
(comment: sum is already done)
95 mi per day
11
8.2 Mean, Median and Mode
2. Find the weighted mean of a set of values
e.g.4 FINDING GPA Find the semester grade point
average for a student that received the following
grades. Round to the nearest hundredth.
course
grade credit
COMP 101
D
3
total points:
MATH 150
B
4
1·3 + 3·4 + 4·2 + 2·1 + 3·3 = 34
ART HIST 175 A
2
total credits: 3 + 4 + 2 + 1 + 3 = 13
HEALTH 090
C
1
34
GPA =
= 2 .62
13
SPAN 130
B
3
12
8.2 Mean, Median and Mode
3. Find the median of a set of values
e.g. Find the median of the following set of values:
3, 6, 7, 4, 5, 8, 6
Step 1. Arrange the values in increasing order:
3, 4, 5, 6, 6, 7, 8
Step 2. The median is the value in the middle, 6.
Median = 6
13
8.2 Mean, Median and Mode
3. Find the median of a set of values
What if there are two values in the middle? Let throw
away one value, 8, from the above set. The rearranged
values are:
3, 4, 5, 6, 6, 7
There are two numbers in the middle, 5 and 6. The median
is the average of these two values:
median =
5+6
= 5 .5
2
14
8.2 Mean, Median and Mode
4. Find the mode of a set of values
The mode of a set of values is the value that occurs most
often.
Using the previous example, consider a set with values:
3, 6, 7, 4, 5, 8, 6
The mode of this set = 6. Because 6 occurs twice, all
other values occur just once.
A data set can have no mode, or bimodal.
15
8.2 Mean, Median and Mode
5. Use the mean, median, mode, and range to describe a set
of values
The range of a set of values is the difference between the
largest value and the smallest value.
e.g.8 MICROSCOPES The weights of six different glass
lenses used in microscopes are:
5.28g, 5.44g, 5.36g, 5.32g, 5.44g, 5.50g.
Find the mean, median, mode, and range.
Mean = 5.39g, Median = 5.40g, mode = 5.44g, range = 0.22g
16
8.3
Equations in Two Variables; The Rectangular
Coordinate System
1. Determine whether an ordered pair is a solutions of an
equation
2. Complete ordered-pair solutions of equations
3. Construct a rectangular coordinate system
4. Plot ordered pairs and determine the coordinates of a
point
17
8.3
Equations in Two Variables; The Rectangular
Coordinate System
1. Determine whether an ordered pair is a solutions of an
equation
We want use table and graph to describe equations.
Suppose a hike is walking at a speed of 4 mph. Then the
distance he walked and time spent is related by equation:
d = 4t (equation with two variables)
We can describe the distance-time relation by:
d
28
d = 4t
d = 4t
t
1
2
3
4
d
4
8
12
16
24
20
16
12
8
4
0
1
2
3
4
5
6
7
t
18
8.3
Equations in Two Variables; The Rectangular
Coordinate System
1. Determine whether an ordered pair is a solutions of an
equation
Consider equations:
2x + y = 12
x = 3, y = 6 is a solution of this equation. We say (x, y) =
(3, 6) is a solutions of this equation. Here (x, y) = (3, 6) is
an ordered pair.
e.g.1 is (1, –5) a solution of
3x + y = –2 ?
yes
e.g.2 is (–5, –2) a solution of
y=x+7 ?
no
19
8.3
Equations in Two Variables; The Rectangular
Coordinate System
2. Complete ordered-pair solutions of equations
e.g.3 Complete the following ordered pair so that each one
is a solution of the equation:
5x + 3y = 15.
a. (0, 5 )
b. (4, − 53 )
e.g.4 Complete the table of solutions for equation:
5x + 2y = –9
x
3
–1
y
–12
–2
(x, y)
( 3 , –12)
(–1, –2 )
20
8.3
Equations in Two Variables; The Rectangular
Coordinate System
2. Complete ordered-pair solutions of equations
Rectangular coordinate system
y-axis
3
Quadrant II
Quadrant I
2
1
Origin
−3
−2
x-axis
−1
1
2
3
−1
Quadrant III
−2
Quadrant IV
−3
21
8.3
Equations in Two Variables; The Rectangular
Coordinate System
4. Plot ordered pairs and determine the coordinates of a
point
For an ordered pair, 1st number is the x-coordinate, 2nd
number is the y-coordinate. For example,
(3, 2)
x-coordinate y-coordinate
–
–
–
–
x-coordinate positive, move to right
x-coordinate negative, move to left
y-coordinate positive, move up
y-coordinate negative, move down
22
8.3
Equations in Two Variables; The Rectangular
Coordinate System
4. Plot ordered pairs and determine the coordinates of a
point
e.g.5 Graph (plot) the points (ordered pairs) (5, 4), (6, –3),
(0, 4.5), (0, –1), (–6, 0), and (–2, − 25 ).
y
6
(0, 4.5)
(5, 4)
4
2
(–6, 0)
−6
−4
−2
(0, –1)
5⎞
⎛
⎜ − 2,− ⎟−2
2⎠
⎝
2
4
6
x
(6, –3)
−4
−6
23
8.3
Equations in Two Variables; The Rectangular
Coordinate System
4. Plot ordered pairs and determine the coordinates of a
point
e.g.6 Find the coordinates of points A, B, C and D.
y
6
4
2
B
−6
−4
−2
(–3.5, –2.5)
C
(0, 1)
2
(4, 0)
A
4
6
x
−2
−4
(2, –4)
D
−6
24
8.4 Graphing Linear Equations
1. Construct a table of solutions
2. Graph linear equations that are solved for y
3. Graph linear equations by finding intercepts
4. Graph linear equations of the form y = b and x = a
25
8.4 Graphing Linear Equations
1. Construct a table of solutions
2. Graph linear equations that are solved for y
Graphing Linear Equations solved for y by plotting points
1). Find three ordered pairs that solution of the equation by selecting
three values for x and calculating the corresponding values of y.
2). Plot the three ordered pairs on a rectangular coordinate system.
3). Draw a straight line passing through the points. If the there is any
point(s) not on the line, check your calculations.
26
8.4 Graphing Linear Equations
1. Construct a table of solutions
2. Graph linear equations that are solved for y
e.g.1 Graph: y = 3x – 1
y
6
x
–1
0
1
y (x, y)
–4 (–1,–4)
–1 (0,–1)
2 (1,2)
4
y = 3x – 1
2
−6
−4
−2
2
4
6
x
−2
−4
−6
27
8.4 Graphing Linear Equations
1. Construct a table of solutions
2. Graph linear equations that are solved for y
e.g.2 Graph: y = − 14 x + 3
y
6
x
–4
0
4
y (x, y)
4 (–4,4)
3 (0,3)
2 (4,2)
4
1
y = − x+3
4
2
−6
−4
−2
2
4
6
x
−2
−4
−6
28
8.4 Graphing Linear Equations
1. Construct a table of solutions
2. Graph linear equations that are solved for y
e.g.3 Graph: y = 30x
y
60
x
–1
0
1
y (x, y)
–30 (–1,–30)
0 (0,0)
30 (1,30)
40
y = 30x
20
−6
−4
−2
2
4
6
x
−20
−40
−60
29
8.4 Graphing Linear Equations
3. Graph linear equations by finding intercepts
The point where a line intersects the x- or y-axis is called
an intercept.
Finding Intercepts
To find the y-intercept, substitute 0 for x in the given
equation and solve for y.
To find the x-intercept, substitute 0 for y in the given
equation and solve for x.
30
8.4 Graphing Linear Equations
3. Graph linear equations by finding intercepts
e.g.4 Graph 2x – 3y = 6 by finding the x- and y-intercepts.
y
6
x-intercept = (3, 0)
y-intercept = (0, –2)
4
2
−6
−4
−2
2x – 3y = 6
2
4
6
x
−2
−4
−6
31
8.4 Graphing Linear Equations
3. Graph linear equations by finding intercepts
e.g.5 Graph 40x + 7y = –280 by finding the x- and
y
y-intercepts.
60
x-intercept = (–7, 0)
y-intercept = (0, –40)
40
20
−6
−4
−2
2
4
6
x
−20
−40
40x + 7y = –280
−60
32
8.4 Graphing Linear Equations
4. Graph linear equations of the form y = b and x = a
e.g.6 Graph: y = –3
y
6
x
–2
0
3
y
–3
–3
–3
(x, y)
(–2,–3)
(0,–3)
(3,–3)
4
2
−6
−4
−2
2
−2
4
y = –3
6
x
Horizontal line
−4
−6
33
8.4 Graphing Linear Equations
4. Graph linear equations of the form y = b and x = a
e.g.7 Graph: x = 2
y
6
x y (x, y)
2 –2 (2,–2)
2 0 (2, 0)
2 3 (2, 3)
4
x=2
Vertical line
2
−6
−4
−2
2
4
6
x
−2
−4
−6
34