Transcript No Slide Title

Distributions of Variables
I.
II.
III.
IV.
V.
Properties of Variables
Nominal Data & Bar Charts
Ordinal Data
Interval & Ratio Data, Histograms &
Frequency Distributions
Cumulative Frequency Distributions &
Percentile Ranks
Anthony J Greene
1
Variables
Variable: A characteristic that takes on
multiple values. I.e.,varies from one person
or thing to another.
Anthony J Greene
2
Variables
Cause and Effect
• The Independent Variable
• The Dependent Variable
Anthony J Greene
3
Distributions
• The distribution of population data is called
the population distribution or the
distribution of the variable.
• The distribution of sample data is called a
sample distribution.
Anthony J Greene
4
Variables
Anthony J Greene
5
Variables
Kinds of Variables (any of which can be an
independent or dependent variable)
• Qualitative variable: A nonnumerically valued
variable.
• Quantitative variable: A numerically valued
variable.
• Discrete Variable: A quantitative variable whose
possible values form a finite (or countably
infinite) set of numbers.
• Continuous variable: A quantitative variable
whose possible values form some interval of
numbers.
Anthony J Greene
6
Quantitative Variables
• Discrete data: Data obtained by observing
values of a discrete variable.
• Continuous data: Data obtained by
observing values of a continuous variable.
Anthony J Greene
7
The Four Scales
•
•
•
•
Nominal: Categories
Ordinal: Sequence
Interval: Mathematical Scale w/o a true zero
Ratio: Mathematical Scale with a true zero
Anthony J Greene
8
The Four Scales
• Nominal: Classes or Categories. Also called a
Categorical scale. E.g., Catholic, Methodist,
Jewish, Hindu, Buddhist, …
Qualitative Data
Anthony J Greene
9
The Four Scales
• Ordinal: Sequential Categories. e.g., 1st,
2nd, 3rd, … with no indication of the
distance between classes
Discrete Data
Anthony J Greene
10
The Four Scales
• Interval: Data where equal spacing in the
variable corresponds to equal spacing in
the scale. E.g., 1940s, 1950s, 1960s… : or
SAT Scores.
Discrete or Continuous
Anthony J Greene
11
The Four Scales
• Ratio: An interval scale with a
mathematically meaningful zero. e.g.,
latencies of 1252 ms, 1856 ms, ….: mg of
Prozac
Discrete or Continuous
Anthony J Greene
12
The Four Scales
Nominal: No mathematical operations
Ordinal: <, >, =
Interval: +, -, and ordinal operations
Ratio: , , and interval operations
Anthony J Greene
13
Nominal Variables
• Classes: Categories for grouping data.
• Frequency: The number of observations that
fall in a class.
• Frequency distribution: A listing of all
classes along with their frequencies.
• Relative frequency: The ratio of the
frequency of a class to the total number of
observations.
• Relative-frequency distribution: A listing of
all classes along with their relative
frequencies.
Anthony J Greene
14
Frequencies of
Nominal Variables
Party Affiliation of
College Students
Republican
Frequency Relative
Frequency
130
0.325
Democrat
180
0.450
Other
90
0.225
Total
400
1.000
Anthony J Greene
15
Sample Pie Charts and Bar
Charts of Nominal Data
23%
44%
33%
Democratic
Republican
Other
50
45
40
35
30
25
20
15
10
5
0
Republican Democratic
Anthony J Greene
Other
16
Frequency Bar Charts
• Frequency bar chart: A graph that displays
the independent variable on the horizontal
axis -- categories -- and the frequencies -dependent variable -- on the vertical axis.
The frequency is represented by a vertical
bar whose height is equal to the frequency
of cases that fall within a given class of the
I.V.
Anthony J Greene
17
Frequency Charts of
Nominal Data
200
180
130
90
160
140
120
100
80
180
60
40
20
Democratic
Republican
Other
0
Republican
Anthony J Greene
Democratic
Other
18
Relative Frequency Bar Charts
• Relative-frequency bar chart: A graph that displays
the I.V. on the horizontal axis -- categories -- and the
relative frequencies -- D.V. -- on the vertical axis. The
relative frequency of each class is represented by a
vertical bar whose height is equal to the relative
frequency of the class.
• The difference between this and a frequency bar chart
is that the proportion or percentage (always between
zero and one) is listed instead of the numbers that fall
into a given class.
Anthony J Greene
19
Relative Frequency Charts
of Nominal Data
44%
33%
Democratic
Republican
Other
%
23%
50
45
40
35
30
25
20
15
10
5
0
R
Anthony J Greene
ub
p
e
n
il ca
D
tic
a
r
oc
em
O
er
h
t
20
Probability Distribution and
Probability Bar Chart
Frequency Distributions and Charts for a whole population
Probability distribution: A listing of the possible values and
corresponding probabilities of a discrete random variable; or a
formula for the probabilities.
Probability bar chart: A graph of the probability distribution
that displays the possible values of a discrete random variable on
the horizontal axis and the probabilities of those values on the
vertical axis. The probability of each value is represented by a
vertical bar whose height is equal to the probability.
Anthony J Greene
21
Probability Charts
of Nominal Data
Anthony J Greene
th
er
O
em
oc
ra
tic
R
Democratic
Republican
Other
ep
ub
l
ica
n
33%
D
44%
%
23%
50
45
40
35
30
25
20
15
10
5
0
22
Bar
Chart
Anthony J Greene
23
The Bar Graph: Nominal Data
Anthony J Greene
24
Sum of the Probabilities of a
Discrete Random Variable
For any discrete random variable, X, the sum of the probabilities of
its possible values equals 1; in symbols, we have
S P(X = x) = 1.
For example Republicans: 32.5%, Democrats 45.0%, Other 22.5%
0.325 + 0.450 + 0.225 = 1.00 or 100%
Anthony J Greene
25
Ordinal Variables
Note that “Rank” is the ordinal variable. “Mortality” is a
ratio variable but can easily be downgraded to an ordinal
variable with a loss of information
Anthony J Greene
26
Distributions and Charts
for Ordinal Data
• Frequency distributions, relative
frequency distribution, and probability
distributions are done exactly as they
were for Nominal Data
• Bar charts are used.
Anthony J Greene
27
Distribution of Education Level
0.5
Level
P(x)
Elementary
0.03
High School
0.45
0.45
0.4
0.35
0.3
0.25
Associates
0.12
Bachelors
0.28
Masters
0.10
0.2
0.15
0.1
0.05
Anthony J Greene
D
oc
to
ra
te
er
s
as
t
M
Ba
ch
elo
rs
oc
ia
te
s
ss
A
H
ig
h
Sc
ho
ar
y
0.02
El
em
en
t
Doctorate
ol
0
28
Interval and Ratio Data
• Frequency: The number of observations that fall
in a class.
• Frequency distribution: A listing of all classes
along with their frequencies.
• Relative frequency: The ratio of the frequency of
a class to the total number of observations.
• Relative-frequency distribution: A listing of all
classes along with their relative frequencies.
Anthony J Greene
29
Histograms
• Frequency histogram: A graph that
displays the independent variable on the
horizontal axis and the frequencies -dependent variable -- on the vertical axis.
The frequency is represented by a vertical
bar whose height is equal to the frequency
of cases that fall within a given range of the
I.V.
Anthony J Greene
30
Interval and Ratio Variables
80
70
60
Avg.
Income
50
(in thousands)
30
40
20
10
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Years of Education
Anthony J Greene
31
Enrollment in
Milwaukee
Public
Elementary
Schools
Anthony J Greene
32
Relative Frequency distribution
of Enrollments in MPS
Anthony J Greene
33
Probability distribution of a
randomly selected elementaryschool student
Anthony J Greene
34
Probability distribution of the age
of a randomly selected student
Anthony J Greene
35
Probability Histogram
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
19
20
21
22
23
Anthony J Greene
24
25
26
27
36
Another Example
Anthony J Greene
37
Frequency vs. Relative Frequency
Anthony J Greene
38
Frequency vs. Relative Frequency
This is also the
Probability Distribution
Anthony J Greene
39
More Examples:
Frequency Histogram
Anthony J Greene
40
More Examples:
Grouped Frequency Histogram
Anthony J Greene
41
Grouped Frequency Histogram
Anthony J Greene
42
Anthony J Greene
43
Proportions and Frequency
Anthony J Greene
44
Frequency
Groupings
9 intervals with
each interval 5
points wide. The
frequency column
(f) lists the number
of individuals with
scores in each of
the class intervals.
Anthony J Greene
45
70
60
Groupings:
There had
to be a
catch
50
40
30
20
10
0
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
• What to do with the in-betweens?
• Only a concern for continuous variables
• Real Limits -- those in the “14” bar are really from
13.5 to 14.5
• Upper Real Limits & Lower Real Limits: For the case
of whole numbers, simply add 0.5 to the high score
and subtract 0.5 from the lowest observed score (these
observed scores are the “apparent limits”)
Understanding Real Limits
Anthony J Greene
47
Real Limits & Apparent Limits
Anthony J Greene
48
Frequency & Cumulative Frequency
I.Q. Range
< 52
52-67
68-84
85-100
101-116
117-132
133-148
>148
Real Limits
0 – 52.5
52.5-67.5
67.5-84.5
84.5-100.5
100.5-116.5
116.5-132.5
132.5-148.5
148.5 +
Frequency
1
4
11
34
34
11
4
1
Anthony J Greene
Cuml. Freq.
1
5
16
50
84
95
99
100
49
Frequency (Normal Distribution)
40
35
30
25
20
15
10
5
0
>52
52-68
68-84
84-100
100-116
116-132
132-148
>148
Cumulative Frequency (Ogive)
100
90
80
70
60
50
40
30
20
10
0
>52
52-68
68-84
84-100
100-116
116-132
132-148
>148
Computing Percentile Ranks
Pounds Real
x
Limits
0
0-0.5
Freq
f
8
Relative Cuml.
Freq.
Freq.
%ile
0.200
0.200
20.0
1
0.5-1.5
17
0.425
0.625
62.5
2
1.5-2.5
11
0.275
0.900
90.0
3
2.5-3.5
3
0.075
0.975
97.5
4
3.5-4.5
1
0.025
1.000
100
Anthony J Greene
52
Pounds
x
Real
Limits
Freq
f
Relative
Freq.
Cuml.
Freq.
%ile
0
0-0.5
8
0.200
0.200
20.0
1
0.5-1.5
17
0.425
0.625
62.5
2
1.5-2.5
11
0.275
0.900
90.0
3
2.5-3.5
3
0.075
0.975
97.5
4
3.5-4.5
1
0.025
1.000
100
Computing
Percentile Ranks
• Remember that each value has real limits
• What is the 90th %ile?
Anthony J Greene
53
Pounds
x
Real
Limits
Freq
f
Relative
Freq.
Cuml.
Freq.
%ile
0
0-0.5
8
0.200
0.200
20.0
1
0.5-1.5
17
0.425
0.625
62.5
2
1.5-2.5
11
0.275
0.900
90.0
3
2.5-3.5
3
0.075
0.975
97.5
4
3.5-4.5
1
0.025
1.000
100
Computing
Percentile Ranks
• Remember that each value has real limits
• What is the 90th %ile? 2.5 because at or below “2”
are 90% of the scores, but “2” includes all from
1.5 to 2.5
Anthony J Greene
54
Pounds
x
Real
Limits
Freq
f
Relative
Freq.
Cuml.
Freq.
%ile
0
0-0.5
8
0.200
0.200
20.0
1
0.5-1.5
17
0.425
0.625
62.5
2
1.5-2.5
11
0.275
0.900
90.0
3
2.5-3.5
3
0.075
0.975
97.5
4
3.5-4.5
1
0.025
1.000
100
Computing
Percentile Ranks
• Remember that each value has real limits
• What is the 90th %ile? 2.5 because at or below “2”
are 90% of the scores, but “2” includes all from
1.5 to 2.5
• What is the 20th %ile?
Anthony J Greene
55
Pounds
x
Real
Limits
Freq
f
Relative
Freq.
Cuml.
Freq.
%ile
0
0-0.5
8
0.200
0.200
20.0
1
0.5-1.5
17
0.425
0.625
62.5
2
1.5-2.5
11
0.275
0.900
90.0
3
2.5-3.5
3
0.075
0.975
97.5
4
3.5-4.5
1
0.025
1.000
100
Computing
Percentile Ranks
• Remember that each value has real limits
• What is the 90th %ile? 2.5 because at or below “2”
are 90% of the scores, but “2” includes all from
1.5 to 2.5
• What is the 20th %ile? 0.5
Anthony J Greene
56
Pounds
x
Real
Limits
Freq
f
Relative
Freq.
Cuml.
Freq.
%ile
0
0-0.5
8
0.200
0.200
20.0
1
0.5-1.5
17
0.425
0.625
62.5
2
1.5-2.5
11
0.275
0.900
90.0
3
2.5-3.5
3
0.075
0.975
97.5
4
3.5-4.5
1
0.025
1.000
100
Computing
Percentile Ranks
• What about the in-betweens?
• What is the 80th %ile?
• What %ile corresponds to 2 lbs?
Anthony J Greene
57
Linear
Interpolation
Anthony J Greene
58
Pounds
x
Real
Limits
Freq
f
Relative
Freq.
Cuml.
Freq.
%ile
0
0-0.5
8
0.200
0.200
20.0
1
0.5-1.5
17
0.425
0.625
62.5
2
1.5-2.5
11
0.275
0.900
90.0
3
2.5-3.5
3
0.075
0.975
97.5
4
3.5-4.5
1
0.025
1.000
100
Linear
Interpolation
And
Percentiles
• What is the 80th %ile?
Anthony J Greene
59
Pounds
x
Real
Limits
Freq
f
Relative
Freq.
Cuml.
Freq.
%ile
0
0-0.5
8
0.200
0.200
20.0
1
0.5-1.5
17
0.425
0.625
62.5
2
1.5-2.5
11
0.275
0.900
90.0
3
2.5-3.5
3
0.075
0.975
97.5
4
3.5-4.5
1
0.025
1.000
100
Linear
Interpolation
And
Percentiles
• What is the 80th %ile?
Where’s the 80th %ile? 17.5/27.5 = 0.63. The interval is 1.0 lb,
so 1.5 + 1(0.63) = 2.13
Anthony J Greene
60
Pounds
x
Real
Limits
Freq
f
Relative
Freq.
Cuml.
Freq.
%ile
0
0-0.5
8
0.200
0.200
20.0
1
0.5-1.5
17
0.425
0.625
62.5
2
1.5-2.5
11
0.275
0.900
90.0
3
2.5-3.5
3
0.075
0.975
97.5
4
3.5-4.5
1
0.025
1.000
100
Linear
Interpolation
And
Percentiles
• What is the 80th %ile?
Where’s the 80th %ile? 17.5/27.5 = 0.63. The interval is 1.0 lb,
so 1.5 + 1(0.63) = 2.13
• What %ile corresponds to 2 lbs?
Anthony J Greene
61
Pounds
x
Real
Limits
Freq
f
Relative
Freq.
Cuml.
Freq.
%ile
0
0-0.5
8
0.200
0.200
20.0
1
0.5-1.5
17
0.425
0.625
62.5
2
1.5-2.5
11
0.275
0.900
90.0
3
2.5-3.5
3
0.075
0.975
97.5
4
3.5-4.5
1
0.025
1.000
100
Linear
Interpolation
And
Percentiles
• What is the 80th %ile?
Where’s the 80th %ile? 17.5/27.5 = 0.63. The interval is 1.0 lb,
so 1.5 + 1(0.63) = 2.13
• What %ile corresponds to 2 lbs?
2 lbs. Is halfway into the interval (0.5). So its halfway between 62.5 90.0. Since 27.5% of the scores are in this interval we need to go up
0.5(27.5%) = 13.75%.
62.5% + 13.75% = 76.25%
Anthony J Greene
62
The Stem & Leaf Diagram
Anthony J Greene
63
Stem & Leaf Plots
Anthony J Greene
64
Comparison of Frequency
Histogram vs.
Stem & Leaf Diagram
Anthony J Greene
65
The Blocked Frequency
Histogram
Anthony J Greene
66
The Frequency Distribution
Polygon –or– Line Graph
Anthony J Greene
67
Grouped Frequency Polygon
Anthony J Greene
68
The Normal Distribution
Anthony J Greene
69
Variants on the Normal
Distribution
Anthony J Greene
70
Comparing Two Distributions
Number of Sentences recalled
from each category
Anthony J Greene
71
Comparing Distributions
Anthony J Greene
72
Distributions
Anthony J Greene
73
Variables and Distributions
In Class Exercise
Anthony J Greene
74
The Math You’ll Need To Know
Calculate:
ΣX =
ΣX2 =
(ΣX)2 =
X
1
2
0
4
Anthony J Greene
75
The Math You’ll Need To Know
Calculate:
ΣX = 7
ΣX2 =
(ΣX)2 =
X
1
2
0
4
Anthony J Greene
76
The Math You’ll Need To Know
Calculate:
ΣX = 7
ΣX2 = 21
(ΣX)2 =
X
1
2
0
4
Anthony J Greene
77
The Math You’ll Need To Know
Calculate:
ΣX = 7
ΣX2 = 21
(ΣX)2 = 49
X
1
2
0
4
Anthony J Greene
78
The Math You’ll Need To Know
Calculate:
ΣX =
ΣY =
ΣX ΣY =
ΣXY =
ΣX2 =
(ΣX)2 =
ΣY2 =
(ΣY)2 =
Anthony J Greene
X
Y
1
3
3
1
0
-2
2
-4
79
The Math You’ll Need To Know
Calculate:
ΣX = 6
ΣY =
ΣX ΣY =
ΣXY =
ΣX2 =
(ΣX)2 =
ΣY2 =
(ΣY)2 =
Anthony J Greene
X
Y
1
3
3
1
0
-2
2
-4
80
The Math You’ll Need To Know
Calculate:
ΣX = 6
ΣY = -2
ΣX ΣY =
ΣXY =
ΣX2 =
(ΣX)2 =
ΣY2 =
(ΣY)2 =
Anthony J Greene
X
Y
1
3
3
1
0
-2
2
-4
81
The Math You’ll Need To Know
Calculate:
ΣX = 6
ΣY = -2
ΣX ΣY = -12
ΣXY =
ΣX2 =
(ΣX)2 =
ΣY2 =
(ΣY)2 =
Anthony J Greene
X
Y
1
3
3
1
0
-2
2
-4
82
The Math You’ll Need To Know
Calculate:
ΣX = 6
ΣY = -2
ΣX ΣY = -12
ΣXY = -2
ΣX2 =
(ΣX)2 =
ΣY2 =
(ΣY)2 =
Anthony J Greene
X
Y
1
3
3
1
0
-2
2
-4
83
The Math You’ll Need To Know
Calculate:
ΣX = 6
ΣY = -2
ΣX ΣY = -12
ΣXY = -2
ΣX2 = 14
(ΣX)2 =
ΣY2 =
(ΣY)2 =
Anthony J Greene
X
Y
1
3
3
1
0
-2
2
-4
84
The Math You’ll Need To Know
Calculate:
ΣX = 6
ΣY = -2
ΣX ΣY = -12
ΣXY = -2
ΣX2 = 14
(ΣX)2 = 36
ΣY2 =
(ΣY)2 =
Anthony J Greene
X
Y
1
3
3
1
0
-2
2
-4
85
The Math You’ll Need To Know
Calculate:
ΣX = 6
ΣY = -2
ΣX ΣY = -12
ΣXY = -2
ΣX2 = 14
(ΣX)2 = 36
ΣY2 = 30
(ΣY)2 =
Anthony J Greene
X
Y
1
3
3
1
0
-2
2
-4
86
The Math You’ll Need To Know
Calculate:
ΣX = 6
ΣY = -2
ΣX ΣY = -12
ΣXY = -2
ΣX2 = 14
(ΣX)2 = 36
ΣY2 = 30
(ΣY)2 = 4
Anthony J Greene
X
Y
1
3
3
1
0
-2
2
-4
87
The Math You’ll Need To Know
The Mean
Σx/n = M
where n = sample size
X
1
4
8
3
Anthony J Greene
88
The Math You’ll Need To Know
Calculate:
Σ(x-M) =
Σ(x-M)2 =
Σ(x2 –M2) =
X
M=4
1
4
8
3
Anthony J Greene
89
The Math You’ll Need To Know
Calculate:
Σ(x-M) = 0
Σ(x-M)2 =
Σ(x2 –M2) =
X
M=4
1
4
8
3
Anthony J Greene
90
The Math You’ll Need To Know
Calculate:
Σ(x-M) = 0
Σ(x-M)2 = 26
Σ(x2 –M2) =
X
M=4
1
4
8
3
Anthony J Greene
91
The Math You’ll Need To Know
Calculate:
Σ(x-M) = 0
Σ(x-M)2 = 26
Σ(x2 –M2) = 26
X
M=4
1
4
8
3
Anthony J Greene
92
The Math You’ll Need To Know
Calculate:
sp = 13
s
2
p
n1

s
2
p
n2
n1= 8

n2= 10
Anthony J Greene
93
The Math You’ll Need To Know
Calculate:
169 169


8
10
sp = 13
n1= 8
n2= 10
21.125  16.9 
38.025  6.17
Anthony J Greene
94
The Math You’ll Need To Know
Calculate:
sp = 13
s 2p
n1

s 2p
n2
n1= 8
 6.17
n2= 10
Anthony J Greene
95
What Type of Data?
Years Spent in the Military
Anthony J Greene
96
What Type of Data?
Military Rank:
Lieutenant
Captain
Major
Lt. Colonel
Colonel
General
Anthony J Greene
97
What Type of Data?
Branch of Service:
Army
Air Force
Navy
Marine Corps
Coast Guard
Anthony J Greene
98
What Type of Data?
Time taken to complete a 30 mile bicycle race
Anthony J Greene
99
What Type of Data?
Finishing place in a 30 mile bicycle race
Anthony J Greene
100
Frequency Dist. & Percentile
Raw Scores:
15, 18, 21, 23, 27, 33, 33, 35, 36, 36, 39, 41
44, 47, 49, 50
Anthony J Greene
101
Frequency Dist. & Percentile
X
f
10-19 2
20-29 3
30-39 6
40-49 4
50-59 1
Anthony J Greene
102
Frequency Dist. & Percentile
Compute the 52%ile
X
10-19
20-29
30-39
40-49
50-59
Anthony J Greene
f
2
3
6
4
1
103
Frequency Dist. & Percentile
Compute the 52%ile
X
10-19
20-29
30-39
40-49
50-59
Anthony J Greene
f
2
3
6
4
1
Cum f
2
5
11
15
16
104
Frequency Dist. & Percentile
X
Compute the 52%ile
10-19
• The 52%ile is
20-29
somewhere between
30-39
30-39.
40-49
50-59
Anthony J Greene
f
2
3
6
4
1
Cum f
2 0.125
5 0.3125
11 0.6875
15 0.9375
16 1.0
105
Frequency Dist. & Percentile
X
Compute the 52%ile
10-19
• The 52%ile is
20-29
somewhere between
30-39
30-39.
40-49
• That interval is from
50-59
0.3125 – 0.6875
Anthony J Greene
f
2
3
6
4
1
Cum f
2 0.125
5 0.3125
11 0.6875
15 0.9375
16 1.0
106
Frequency
Dist.
&
Percentile
Compute the 52%ile
X
• The 52%ile is
somewhere between 30- 10-19
39.
20-29
• That interval is from
30-39
0.3125 – 0.6875
40-49
• That interval is 0.375 50-59
wide
Anthony J Greene
f
2
3
6
4
1
Cum f
2 0.125
5 0.3125
11 0.6875
15 0.9375
16 1.0
107
Frequency
Dist.
&
Percentile
Compute the 52%ile
X
• The 52%ile is
somewhere between 20- 10-19
29.
20-29
• That interval is from
30-39
0.3125 – 0.6875
40-49
• That interval is 0.375 50-59
wide
• To get from 0.3125 to
0.52 we go 0.2075 into
the interval
Anthony J Greene
f
2
3
6
4
1
Cum f
2 0.125
5 0.3125
11 0.6875
15 0.9375
16 1.0
108
•
Frequency
Dist.
&
Percentile
That interval is from
X
0.3125 – 0.6875
• That interval is 0.375 10-19
wide
20-29
• To get from 0.3125 to 30-39
0.52 we go 0.2075 into 40-49
the interval
50-59
• That’s 0.553 of the way
into the interval
(0.2075/0.375)
Anthony J Greene
f
2
3
6
4
1
Cum f
2 0.125
5 0.3125
11 0.6875
15 0.9375
16 1.0
109
Frequency Dist. & Percentile
• That’s 0.553 of the
way into the interval
(0.2075/0.375)
• The real limits are
from 19.5 to 29.5
(a range of 10)
• 52%ile is
29.5 + 5.53 = 35.03
This Process is called
Linear Interpolation
X
10-19
20-29
30-39
40-49
50-59
Anthony J Greene
f
2
3
6
4
1
Cum f
2 0.125
5 0.3125
11 0.6875
15 0.9375
16 1.0
110