Transcript MS 134 - jsu.edu

Statistics
Chapter 9
Day 1
Unusual Episode
MS133 Final Exam Scores
79
91
71
62
69
78
63
86
78
53
79
58
91
77
79
94
95
67
74
89
86
65
88
96
64
69
49
84
78
75
79
77
78
68
77
Line Plot or Dot Plot
Stem and Leaf
Stem and Leaf
9 1 1 4 5 6
8 6 8 9 6 4
7 9 1 8 8 9 7 9 8 5 9 7 4 8 7
6 2 9 3 5 7 4 9 8
5 3 8
4 9
Ordered Stem and Leaf
9 1 1 4 5 6
8 4 6 6 8 9
7 1 4 5 7 7 7 8 8 8 8 9 9 9 9
6 2 3 4 5 7 8 9 9
5 3 8
4 9
Frequency Table
Grade
Score
Tally
Frequency
Frequency Table
Grade
Score
Tally
Frequency
A
90-100
IIII
5
B
80-89
IIII
5
C
70-79
IIII IIII IIII
14
D
60-69
IIII III
8
F
0-59
III
3
Bar Graph
MS133 Final Exam Grades
F
R
E
Q
U
E
N
C
Y
14
12
10
8
6
4
2
A's
B's
C's
GRADES
D's
F's
Make a Pie Chart
• 5 A’s out of how many grades total?
• 5 A’s out of how many total grades? 35
• What percent of the class made an A?
• 5 A’s out of how many total grades? 35
• What percent of the class made an A?
5/35 ≈ 0.14 ≈ 14%
• What percent of the pie should represent
the A’s?
• 5 A’s out of how many total grades? 35
• What percent of the class made an A?
5/35 ≈ 0.14 ≈ 14%
• What percent of the pie should represent
the A’s? 14%
• How many degrees in the whole pie?
• 5 A’s out of how many total grades? 35
• What percent of the class made an A?
5/35 ≈ 0.14 ≈ 14%
• What percent of the pie should represent
the A’s? 14%
• How many degrees in the whole pie? 360°
• 5 A’s out of how many total grades? 35
• What percent of the class made an A?
5/35 ≈ 0.14 ≈ 14%
• What percent of the pie should represent
the A’s? 14%
• How many degrees in the whole pie? 360°
• 14% of 360° is how many degrees?
• 5 A’s out of how many total grades? 35
• What percent of the class made an A?
5/35 ≈ 0.14 ≈ 14%
• What percent of the pie should represent
the A’s? 14%
• How many degrees in the whole pie? 360°
• 14% of 360° is how many degrees?
.14 x 360° ≈ 51°
A's
14%
• 5 B’s out of 35 grades total ≈ 14% ≈ 51°
A's
14%
B's
14%
A's
14%
• 14 C’s out of 35 grades
• 14 C’s out of 35 grades
• 14/35 = .4 = 40%
• .4 x 360° = 144°
B's
14%
A's
14%
B's
14%
C's
40%
A's
14%
• 8 D’s out of 35 grades total
• 8 D’s out of 35 grades
• 8/35 ≈ .23 ≈ 23% (to the nearest percent)
(keep the entire quotient in the calculator)
• x 360° ≈ 82°
B's
14%
C's
40%
A's
14%
B's
14%
A's
14%
C's
40%
D's
23%
• 3 F’s out of 35 total
• 3 F’s out of 35 grades total
• 3/35 ≈ .09 ≈ 9% (to the nearest percent)
(keep the entire quotient in the calculator)
• x 360° ≈ 31°
• Check the remaining angle to make sure it
is 31°
B's
14%
A's
14%
C's
40%
D's
23%
MS133 Final Exam Grades
B's
14%
A's
14%
C's
40%
F's 9%
D's
23%
Make a Pie Chart
• Gross income: $10,895,000
•
•
•
•
•
Labor: $5,120,650
Materials: $4,031,150
New Equipment: $326,850
Plant Maintenance: $544,750
Profit: $871,600
• Labor: $5,120,650 =
47%
10,895,000
• Materials: $4,031,150 =
37%
10,895,000
• New Equipment: $326,850 =
3%
10,895,000
• Plant Maintenance: $544,750 = 5%
10,895,000
• Profit : $871,600 =
8%
10,895,000
169°
133°
11°
18°
29°
Labor
47%
M ate rials
37%
Profit
8%
5%
3% M ainte nance
Equipm e nt
Histogram
• Table 9.2 Page 527
Eisenhower High School
Boys Heights
F
R
E
Q
U
E
N
C
Y
18
14
10
6
2
64 65 66 67 68 69
70 71 72 73 74
HEIGHTS (inches)
EHS Boys’ Heights
Height Frequency
Relative
Frequency
64
1
65
1
70
14
66
3
71
10
67
7
72
6
68
15
73
2
69
19
74
2
EHS Boys’ Heights
Height Frequency
64
1
Relative
Frequency
.0125
65
1
.0125
70
14
.175
66
3
.0375
71
10
.125
67
7
.0875
72
6
.075
68
15
.1875
73
2
.025
69
19
.2375
74
2
.025
Eisenhower High School
Boys Heights
.25
R
E
L
A
T
I
V
E
F
R
E
Q
U
E
N
C
Y
.20
.15
.10
.05
64 65 66 67 68 69
70 71 72 73 74
HEIGHTS (inches)
EHS Boys’ Heights
18
F
R
E
Q
U
E
N
C
Y
14
10
6
2
64
65
66
67
68
69
70
71
HEIGHTS (inches)
72
73
74
Day 2
Measures of Central Tendency Lab
Print your first name below.
Getting Mean with Tiles
• Use your colored tiles to build a column 9
tiles high and another column 15 tiles high.
Use a different color for each column.
Getting Mean with Tiles
• Use your colored tiles to build a column 9
tiles high and another column 15 tiles high.
Use a different color for each column.
• Move the tiles one at a time from one
column to another “evening out” to create
2 columns the same height.
• What is the new (average) height?
Getting Mean with Tiles
• Move the tiles back so that you have a
column 9 tiles high and another 15 tiles
high.
• Find another method to “even off” the
columns?
Getting Mean with Tiles
• Use your colored tiles to build a column 19
tiles high and another column 11 tiles high.
Use a different color for each column.
• “Even-off” the two columns using the most
efficient method.
• What is the new (average) height?
Getting Mean with Tiles
• If we start with a column x tiles high and
another y tiles high, describe how you
could find the new (average) height?
•
Let’s assume x is the larger number
•
x – y (extra)
•
x – y (extra)
x–y
2
•
x – y (extra)
• y+x–y
2
x–y
2
•
x – y (extra)
• y+x–y
2
2y + x – y
2
2
x–y
2
•
x – y (extra)
• y+x–y
2
2y + x – y
2
2
2y + x - y
2
x–y
2
•
x – y (extra)
• y+x–y
2
2y + x – y
2
2
2y + x - y
2
x+y
2
x–y
2
Homework Questions
Page 538
Measures of Central Tendency
• Mean – “Evening-off”
• Median – “Middle”
• Most – “Most”
Class R
71
77
92
46
67
63
71
76
74
79
77
77
76
70
86
72
72
61
79
72
79
81
77
76
46,61,63,67,70,71,71,72,72,72,74,76,76,76,77,77,77,77,79,79,79,81,86,92
Mean = Sum of all grades
Number of grades
46,61,63,67,70,71,71,72,72,72,74,76,76,76,77,77,77,77,79,79,79,81,86,92
Mean = Sum of all grades
Number of grades
Mean = 1771
24
46,61,63,67,70,71,71,72,72,72,74,76,76,76,77,77,77,77,79,79,79,81,86,92
Mean = Sum of all grades
Number of grades
Mean = 1771
24
x
1771
 73 .8
24
Class S
72
67
71
82
68
72
73
77
76
68
73
69
79
75
69
77
69
71
74
75
76
79
76
78
73
67,68,68,69,69,69,71,71,72,72,73,73,73,74,75,75,76,76,76,77,77,78,79,79,82
Mean =
67,68,68,69,69,69,71,71,72,72,73,73,73,74,75,75,76,76,76,77,77,78,79,79,82
Mean =
1839
25
67,68,68,69,69,69,71,71,72,72,73,73,73,74,75,75,76,76,76,77,77,78,79,79,82
Mean = 1839
25
x
1839
 73 .6
25
Class T
74
40
40
49
74
86
79
82
49
40
96
75
86
40
70
45
81
89
84
61
85
91
85
85
40,40,40,40,45,49,49,61,70,74,74,75,79,81,82,84,85,85,85,86,86,89,91,96
Mean =
40,40,40,40,45,49,49,61,70,74,74,75,79,81,82,84,85,85,85,86,86,89,91,96
Mean = 1686
24
40,40,40,40,45,49,49,61,70,74,74,75,79,81,82,84,85,85,85,86,86,89,91,96
Mean =
1686
24
1686
x
 70 .3
24
Median –”Middle”
Class R:
46,61,63,67,70,71,71,72,72,72,74,76,76,76,77,77,77,77,79,79,79,81,86,92
Class S:
67,68,68,69,69,69,71,71,72,72,73,73,73,74,75,75,76,76,76,77,77,78,79,79,82
Class T:
40,40,40,40,45,49,49,61,70,74,74,75,79,81,82,84,85,85,85,86,86,89,91,96
Median
• Class R:
76
• Class S:
73
• Class T: 77
Mode – “Most”
Class R:
46,61,63,67,70,71,71,72,72,72,74,76,76,76,77,77,77,77,79,79,79,81,86,92
Class S:
67,68,68,69,69,69,71,71,72,72,73,73,73,74,75,75,76,76,76,77,77,78,79,79,82
Class T:
40,40,40,40,45,49,49,61,70,74,74,75,79,81,82,84,85,85,85,86,86,89,91,96
Mode
• Class R:
77
• Class S:
69, 73, 76
• Class T: 40
Range - A measure of dispersion
Greatest - Least
Class R:
46,61,63,67,70,71,71,72,72,72,74,76,76,76,77,77,77,77,79,79,79,81,86,92
Class S:
67,68,68,69,69,69,71,71,72,72,73,73,73,74,75,75,76,76,76,77,77,78,79,79,82
Class T:
40,40,40,40,45,49,49,61,70,74,74,75,79,81,82,84,85,85,85,86,86,89,91,96
Range
• Class R:
92 - 46 = 46
• Class S:
82 – 67 = 15
• Class T: 96 – 40 = 56
Class R
Class S
Class T
73.6
70.3
Median = 76
73
77
Mode =
69,73,76
40
15
56
Mean =
73.8
77
Range = 46
Weighted Mean Example 9.7
Owner/Manager earned $850,000
Assistant Manager earned $48,000
16 employees $27,000 each
3 secretaries $18,000 each
Find the MEAN, MEDIAN, MODE
MEAN
Salary
$18,000
$27,000
$48,000
$850,000
MEAN
Salary
Frequency
$18,000
3
$27,000
16
$48,000
1
$850,000
1
MEAN
Mean =
3(18,000)+16(27,000)+48,000+850,000
21
= 1384000
21
≈ $65,905
MEDIAN
Salary
Frequency
$18,000
3
$27,000
16
$48,000
1
$850,000
1
MEDIAN
Salary
Frequency
Cumulative Frequency
$18,000
3
1–3
$27,000
16
4 - 19
$48,000
1
20
$850,000
1
21
MODE
Salary
Frequency
Cumulative Frequency
$18,000
3
1–3
$27,000
16
4 - 19
$48,000
1
20
$850,000
1
21
RANGE
Salary
Frequency
Cumulative Frequency
$18,000
3
1–3
$27,000
16
4 - 19
$48,000
1
20
$850,000
1
21
• Mean = $65,905
• Median = $27,000
• Mode = $27,000
• Range = $832,000
Grade Point Average
A weighted mean
quality points earned
hours attempted
Quality Points
Every A gets 4 quality points per hour. For
example, an A in a 3 hour class gets 4
quality points for each of the 3 hours,
4x3=12. An A in a 4 hour class gets 4
quality points for each of the 4 hours,
4X4=16 quality points.
Every B gets 3 quality points per hour.
Every C gets 2 quality points per hour.
Every D gets 1 quality points per hour.
No quality points for an F.
Sally Ann’s First Semester Grades
Hours
Grade
3
D
4
F
2
B
3
C
2
C
1
A
Sally Ann’s First semester GPA
to the nearest hundredth
23
 1.53
15
Sally Ann’s Second Semester
Hours
Grade
3
C
3
C
3
B
3
B
Sally Ann’s Second Semester GPA
30
 2 .5
12
Sally Ann’s Cumulative GPA
Total quality points earned
Total hours attempted
Sally Ann’s New GPA
to the nearest hundredth
53
 1.96
27
Day 3
Class X
60, 60, 72, 72, 72, 78, 78, 82, 82, 82, 82, 85, 85, 90, 90
Find the mean, median, mode, and range.
Mean
60, 60, 72, 72, 72, 78, 78, 82, 82, 82, 82, 85, 85, 90, 90
2(60)  3(72)  2(78)  4(82)  2(85)  2(90)
15
Mean
60, 60, 72, 72, 72, 78, 78, 82, 82, 82, 82, 85, 85, 90, 90
2(60)  3(72)  2(78)  4(82)  2(85)  2(90)
15
1170
 78
15
Median – Mode – Range
60, 60, 72, 72, 72, 78, 78, 82, 82, 82, 82, 85, 85, 90, 90
•
•
•
•
Mean = 78
Median = 82
Mode = 82
Range = 30
Standard Deviation
The standard deviation is a measure of
dispersion. You can think of the standard
deviation as the “average” amount each
data is away from the mean. Some data
are close, some are farther. The standard
deviation gives you an average.
Find the standard deviation of class x.
Standard Deviation
60, 60, 72, 72, 72, 78, 78, 82, 82, 82, 82, 85, 85, 90, 90
Mean = 78
Standard Deviation of Class X
2(78  60) 2  3(78  72) 2  2(78  78) 2  4(78  82) 2  2(78  85) 2  2(78  90) 2
15
2(18) 2  3(6) 2  2(0) 2  4(4) 2  2(7) 2  2(12) 2
15
2(18) 2  3(6) 2  2(0) 2  4(4) 2  2(7) 2  2(12) 2
15
2(324)  3(36)  2(0)  4(16)  2(49)  2(144)
15
2(18) 2  3(6) 2  2(0) 2  4(4) 2  2(7) 2  2(12) 2
15
2(324)  3(36)  2(0)  4(16)  2(49)  2(144)
15
648 108 0  64  98  288
15
2(18) 2  3(6) 2  2(0) 2  4(4) 2  2(7) 2  2(12) 2
15
2(324)  3(36)  2(0)  4(16)  2(49)  2(144)
15
648 108 0  64  98  288
15
1206

15
80.4  8.97
Page 558
Example 9.11
Find the mean (to the nearest tenth):
35, 42, 61, 29, 39
Page 558
Example 9.11
Find the mean (to the nearest tenth): ≈ 41.2
Standard deviation (to the nearest tenth):
35, 42, 61, 29, 39
Page 558
Example 9.11
Find the mean (to the nearest tenth): ≈ 41.2
Standard deviation (to the nearest tenth): ≈ 10.8
Box and Whisker Graph
•
•
•
•
Graph of dispersion
Data is divided into fourths
The middle half of the data is in the box
Outliers are not connected to the rest of
the data but are indicted by an asterisk.
Box and Whisker Graph
• Class R:
46,61,63,67,70,71,71,72,72,72,74,76,76,76,77,77,77,77,79,79,79,81,86,92
Median =
Upper Quartile =
Lower Quartile =
40
50
60
70
80
90
100
Outliers
• Any data more than 1 ½ boxes away from
the box (middle half) is considered an
outlier and will not be connected to the
rest of the data.
• The size of the box is called the Inner
Quartile Range (IQR) and is determined
by finding the range of the middle half of
the data.
Box and Whisker Graph
• Class R:
46,61,63,67,70,71,71,72,72,72,74,76,76,76,77,77,77,77,79,79,79,81,86,92
Median =76
Upper Quartile = 78
Inner Quartile Range =
Lower Quartile = 71
Box and Whisker Graph
• Class R:
46,61,63,67,70,71,71,72,72,72,74,76,76,76,77,77,77,77,79,79,79,81,86,92
Median =76
Upper Quartile = 78
Inner Quartile Range = 7
Lower Quartile = 71
IQR x 1.5 =
Box and Whisker Graph
• Class R:
46,61,63,67,70,71,71,72,72,72,74,76,76,76,77,77,77,77,79,79,79,81,86,92
Median =76
Upper Quartile = 78
Inner Quartile Range = 7
Checkpoints for Outliers:
Lower Quartile = 71
IQR x 1.5 = 10.5
Box and Whisker Graph
• Class R:
46,61,63,67,70,71,71,72,72,72,74,76,76,76,77,77,77,77,79,79,79,81,86,92
Median =76
Upper Quartile = 78
Lower Quartile = 71
Inner Quartile Range = 7
IQR x 1.5 = 10.5
Checkpoints for Outliers: 60.5, 88.5
Outliers =
*
40
*
50
60
70
80
90
100
Box and Whisker Graph
• Class R:
46,61,63,67,70,71,71,72,72,72,74,76,76,76,77,77,77,77,79,79,79,81,86,92
Median =76
Upper Quartile = 78
Lower Quartile = 71
Inner Quartile Range = 7
IQR x 1.5 = 10.5
Checkpoints for Outliers: 60.5, 88.5
Outliers = 46, 92
Whisker Ends =
Box and Whisker Graph
• Class R:
46,61,63,67,70,71,71,72,72,72,74,76,76,76,77,77,77,77,79,79,79,81,86,92
Median =76
Upper Quartile = 78
Lower Quartile = 71
Inner Quartile Range = 7
IQR x 1.5 = 10.5
Checkpoints for Outliers: 60.5, 88.5
Outliers = 46, 92
Whisker Ends = 61, 86
*
40
*
50
60
70
80
90
100
Box and Whisker Graph
• Class S:
67,68,68,69,69,69,71,71,72,72,73,73,73,74,75,75,76,76,76,77,77,78,79,79,82
Median =
UQ =
LQ =
IQR =
IQR x 1.5 =
Checkpoints for outliers:
Outliers =
Whisker Ends =
Box and Whisker Graph
• Class S:
67,68,68,69,69,69,71,71,72,72,73,73,73,74,75,75,76,76,76,77,77,78,79,79,82
Median = 73
UQ = 76.5
LQ = 70
IQR = 6.5
IQR x 1.5 = 9.75
Checkpoints for outliers: 60.25, 86.25
Outliers = none
Whisker Ends = 67, 82
*
40
*
50
60
70
80
90
100
Box and Whisker Graph
• Class T:
40,40,40,40,45,49,49,61,70,74,74,75,79,81,82,84,85,85,85,86,86,89,91,96
Median =
UQ =
LQ =
IQR x 1.5 =
Checkpoints for Outliers:
Outliers=
IQR =
Whisker Ends=
Box and Whisker Graph
• Class T:
40,40,40,40,45,49,49,61,70,74,74,75,79,81,82,84,85,85,85,86,86,89,91,96
Median = 77
UQ = 85
LQ = 49
IQR = 36
IQR x 1.5 = 54
Checkpoints for Outliers: -5, 139
Outliers = none
Whisker Ends = 40, 96
*
40
*
50
60
70
80
90
100
Day 4
Homework Questions
Page 561
Statistical Inference
•
•
•
•
Population
Sampling
Random Sampling
Page 576 #2, 4, 5, 17, 18, 19, 21, 22
Example 9.15, Page 569
Getting a random sampling
5,5,2,9,1,0,4,5,3,1,2,4,1,9,4,6,6,9,1,7
5,5,2,9,1,0,4,5,3,1,2,4,1,9,4,6,6,9,1,7
55
24
29
19
10
46
45
69
31
17
5,5,2,9,1,0,4,5,3,1,2,4,1,9,4,6,6,9,1,7
55
24
29
19
10
46
45
69
31
17
65
63
Sample
64 68 65
64 62 64
63
67
Find the mean of the sample
65
63
64
64
68
62
65
64
63
67
Mean = 62 + 2(63) + 3(64) + 2(65) + 67 + 68
10
Sample Mean
Mean = 62 + 2(63) + 3(64) + 2(65) + 67 + 68
10
Mean = 645
10
Mean = 64.5
Standard Deviation of the Sample
62
63
63
64
64
64
65
65
67 68
Standard Deviation of the Sample
62
63
63
64
64
64
65
65
67 68
(64.5  62) 2  2(64.5  63) 2  3(64.5  64) 2  2(64.5  65) 2  (64.5  67) 2  (64.5  68)2
10
Standard Deviation
(64.5  62) 2  2(64.5  63) 2  3(64.5  64) 2  2(64.5  65) 2  (64.5  67) 2  (64.5  68) 2
10
(2.5) 2  2(1.5) 2  3(.5) 2  2(.5) 2  (2.5) 2  (3.5) 2
10
Standard Deviation
(64.5  62) 2  2(64.5  63) 2  3(64.5  64) 2  2(64.5  65) 2  (64.5  67) 2  (64.5  68) 2
10
(2.5) 2  2(1.5) 2  3(.5) 2  2(.5) 2  (2.5) 2  (3.5) 2
10
6.25  2(2.25)  3(.25)  2(.25)  6.25  12.25
10
Standard Deviation
(64.5  62) 2  2(64.5  63) 2  3(64.5  64) 2  2(64.5  65) 2  (64.5  67) 2  (64.5  68) 2
10
(2.5) 2  2(1.5) 2  3(.5) 2  2(.5) 2  (2.5) 2  (3.5) 2
10
6.25  2(2.25)  3(.25)  2(.25)  6.25  12.25
10
30.5
 3.05  1.75
10
Random Sample
• Mean = 64.5
• Standard deviation = 1.75
• Compare the sample to the mean and
standard deviation of the entire population.
(example 9.14)
• Compare our sample to the author’s
sample. (example 9.14)
Beans or Fish
Normal Distribution
• The distribution of many populations form
the shape of a “bell-shaped” curve and are
said to be normally distributed.
• If a population is normally distributed,
approximately 68% of the population lies
within 1 standard deviation of the mean.
About 95% within 2 standard deviations.
About 99.7% within 3 standard deviations.
Normal Curve
x - 3s
x - 2s
x-s
x
x +s
x + 2s
x + 3s
68% of the data is within
1 standard deviation of the mean
< 68% >
x-s
x
x+s
95% of the data is within
2 standard deviations of the mean
<
x - 2s
95%
x
>
x + 2s
99.7% of the data is within
3 standard deviations of the mean
<
99.7%
x - 3s
x
>
x + 3s
Normal Distribution
99.7%
95%
68%
x - 3s
x - 2s
x-s
x
x +s
x + 2s
x + 3s
Normal Distribution Example
• Suppose the 200 grades of a certain
professor are normally distributed. The
mean score is 74. The standard deviation
is 4.3.
• What whole number grade constitutes an
A, B, C, D and F?
• Approximately how many students will
make each grade?
x  74
s.d .  4.3
200 students
61.1
65.4
69.7
74
78.3
82.6
86.9
61.1
•
•
•
•
•
65.4
69.7
A: 83 and above
B: 79 – 82
C: 70 – 78
D: 66 – 69
F: 65 and below
74
78.3
82.6
86.9
200 students
•
•
•
•
•
A: 83 and above
B: 79 – 82
C: 70 – 78
D: 66 – 69
F: 65 and below
5 people
27 people
136 people
27 people
5 people
Normal Distribution
• The graph of a normal distribution is
symmetric about a vertical line drawn
through the mean. So the mean is also the
median.
• The highest point of the graph is the
mean, so the mean is also the mode.
• The area under the entire curve is one.
Normal Distribution
x - 3s
x - 2s
x-s
x
x +s
x + 2s
x + 3s
Standardized form of the normal
distribution (z curve)
-3
-2
-1
0
1
2
3
Z Curve
• The scale on the horizontal axis now
shows a z – Score.
Any normal distribution in standard form will
have mean 0 and standard deviation1.
• 68% of the data will lie between -1 and 1.
• 95% of the data will lie between -2 and 2.
• 99.7% of the data will lie between -3 and
3.
Z- Scores
• By using a z-Score, it is possible to tell if
an observation is only fair, quite good, or
rather poor.
• EXAMPLE: A z-Score of 2 on a national
test would be considered quite good, since
it is 2 standard deviations above the
mean.
• This information is more useful than the
raw score on the test.
Z- Scores
• z – Score of a data is determined by
subtracting the mean from the data and
dividing the result by the standard
deviation.
• z=x-µ
σ
62,62,63,64,64,64,64,66,66,66
• Mean = 64.1
• Standard deviation ≈ 1.45
• Convert these data to a set of z-scores.
62,62,63,64,64,64,64,66,66,66
z-scores:
62, 63, 64, 66
-1.45, -0.76, -0.07, 1.31
Percentiles
• The percentile tells us the percent of the
data that is less than or equal to that data.
Percentile in a sample:
62,62,63,64,64,64,64,66,66,66
• The percentile corresponding to 63 is the
percent of the data less than or equal to
63.
• 3 data out of 10 data = .3 = 30% of the
data is less than or equal to 63.
• For this sample, 63 is in the 30th
percentile.
Percentile in a Population
• Remember that the area under the normal curve
is one.
• The area above any interval under the curve is
less than one which can be written as a decimal.
• Any decimal can be written as a percent by
multiplying by 100 (which moves the decimal to
the right 2 places).
• That number would tell us the percent of the
population in that particular region.
Percentiles
• Working through this process, we can find the
percent of the data less than or equal to a
particular data – the percentile.
• The z-score tells us where we are on the
horizontal scale.
• Table 9.4 on pages 585 and 586 convert the zscore to a decimal representation of the area to
the left of that data.
• By converting that number to a percent, we will
have the percentile of that data.
• If the z-score of a data in a normal distribution is
-0.76,what is it’s percentile in the population?
•
•
•
•
•
Table 9.4 page 585
Row marked -0.7
Column headed .06
Entry .2236
22.36% of the population lies to the left of -0.76
Note the difference in finding the
percentile in a sample and the
entire population.
Interval Example
• Show that 34% of a normally distributed
population lies between the z scores of
-0.44 and 0.44
Interval Example
• Show that 34% of a normally distributed
population lies between the z scores of
-0.44 and 0.44
• Table 9.4, page 585
• 33% to the left of -0.44
• 67% to the left of 0.44
• 67% - 33% = 34%
Day 5
Homework Questions
Page 576
Normal Distribution Lab
Day 6
Lab Questions
Statistics Review
M&M Lab