Chapters 2 and 3 : Frequency Distributions, Histograms,
Percentiles and Percentile Ranks and their Graphical Representations
Note: we’ll be skipping book sections:
2.4 (apparent and real limits)
2.8, 2.9 (percentile and percentile ranks for grouped data)
Chapter 2: Frequency Distributions, Histograms,
Percentiles and Percentile Ranks
How can we represent or summarize a list of values?
frequency distribution: shows the number of observations for the possible categories or score
values in a set of data. Can be done on any scale (nominal, ordinal, interval, or ratio).
Often represented as a bar graph (Chapter 3).
Example of a frequency distribution for nominal scale data:
2008 Auto sales by country:
Japan:
11,563,629
China:
9,345,101
US:
8,705,239
Germany:
6,040,582
South Korea:
3,806,682
Brazil:
3,220,475
Car sales drawn as a histogram
12
Japan:
11,563,629
China:
9,345,101
US:
8,705,239
Germany:
6,040,582
South Korea: 3,806,682
Brazil:
3,220,475
Car Sales in 2008 (millions)
10
8
6
4
2
0
Japan
China
US
Germany South Korea
Brazil
This histogram shows the proportion of members for each category.
Distribution of all M&M's.
Ice Dancing , compulsory
dance scores, 4 Winter
Olympics
111.15
108.55
106.6
103.33
100.06
97.38
96.67
96.12
92.75
89.62
85.36
84.58
83.89
83.12
80.47
80.3
79.31
76.73
74.25
72.01
68.87
63.73
59.64
Making histograms from interval and ratio data
We need to bin the raw scores into a set of class intervals.
How do we decide these class intervals?
Be sure the intervals don’t overlap, have the same
width, and cover the entire range of scores.
Use around 10 to 20 intervals.
Use a ‘sensible’ width (like 5, and not 2.718285)
Make the lower score a multiple of the width (e.g. if
the width is 5, a lower score should be 50, not 48)
If a score lands on the border, put it in the lower class
interval.
Ice Dancing , compulsory
dance scores,
Winter Olympics
n=23
111.15
108.55
106.6
103.33
100.06
97.38
96.67
96.12
92.75
89.62
85.36
84.58
83.89
83.12
80.47
80.3
79.31
76.73
74.25
72.01
68.87
63.73
59.64
Let’s use a class interval width of 5
points, with a lowest score of 55.
Class Intervals
Frequency (f)
110-115
105-110
100-105
95-100
90-95
85-90
80-85
75-80
70-75
65-70
60-65
55-60
1
2
2
3
1
2
5
2
2
1
1
1
Count the number of scores in each bin to get the frequency
Histogram of Ice Dancing Scores (frequency)
Frequency (f)
110-115
105-110
100-105
95-100
90-95
85-90
80-85
75-80
70-75
65-70
60-65
55-60
1
2
2
3
1
2
5
2
2
1
1
1
5
4
Frequency
Class Intervals
3
2
1
0
55
60
65
70
75 80 85 90 95 100 105 110 115
Ice Dancing Score
Relative frequency
n=23
111.15
108.55
106.6
103.33
100.06
97.38
96.67
96.12
92.75
89.62
85.36
84.58
83.89
83.12
80.47
80.3
79.31
76.73
74.25
72.01
68.87
63.73
59.64
Class Intervals
Frequency (f)
110-115
105-110
100-105
95-100
90-95
85-90
80-85
75-80
70-75
65-70
60-65
55-60
1
2
2
3
1
2
5
2
2
1
1
1
Relative
frequency
(prop)
.0435
.0870
.0870
.1304
.0435
.0870
.2174
.0870
.0870
.0435
.0435
.0435
Relative
frequency
(%)
4.35
8.70
8.70
13.04
4.35
8.70
21.74
8.70
8.70
4.35
4.35
4.35
Divide by the total number of scores to get relative frequency in proportion
Then multiply by 100 to get relative frequency in percent
Relative frequency histogram of Ice Dancing Scores (frequency)
25
Relative Frequency (%)
Relative
Class Intervals frequency (%)
110-115
4.35
105-110
8.70
100-105
8.70
95-100
13.04
90-95
4.35
85-90
8.70
80-85
21.74
75-80
8.70
70-75
8.70
65-70
4.35
60-65
4.35
55-60
4.35
20
15
10
5
0
55
60
65
70
75 80 85 90 95 100 105 110 115
Ice Dancing Score
Choosing your class intervals can have an influence on the way your histogram looks
interval width 10
interval width 5
5
Frequency
Frequency
7
6
5
4
3
2
1
0
4
3
2
1
60
70 80 90 100 110 120
Ice Dancing Score
0
60
interval width 3
interval width 1
2
Frequency
Frequency
3
2
1
0
80
100
Ice Dancing Score
60
80
100
Ice Dancing Score
1
0
60
80
100
Ice Dancing Score
These three graphs have the same class intervals on the same scores!
5
5
3
4
2
Frequency
Frequency
4
1
Frequency
0
5
4
3
2
1
0
60
70
80
90
100
Ice Dancing Score
110
3
2
1
60
70
80
90
Ice Dancing Score
100
110
0
60 80 100
Ice Dancing Score
When possible, include zero on your y-axis.
Not like this
When possible, include zero on your y-axis.
y-axis:
Like this
Enrollment (Millions)
8
6
4
2
0
As of March 27
March 31 Goal
As of March 27
March 31 Goal
Not like this
Enrollment (Millions)
7
6.8
6.6
6.4
6.2
6
“Fox News Apologizes For Obamacare Graphic,
Corrects Its 'Mistake‘”
Percentile ranks and percentile point:
Percentile Point: A point on the measurement scale below which a specific
percentage of scores fall.
Percentile Rank: The percentage of cases that fall below a given point on the
measurement scale.
Percentile ranks are always between zero and 100.
Growth charts convert percentile points to percentile ranks
At 30 mos.
P95 = 36lbs
Percentile ranks and percentile point:
What is the percentile rank for a percentile point of 100?
In other words,
What proportion of scores fall below a score of 100?
Class
interval
110-115
105-110
100-105
95-100
90-95
85-90
80-85
75-80
70-75
65-70
60-65
55-60
f
1
2
2
3
1
2
5
2
2
1
1
1
rel f(%)
4.35
8.7
8.7
13.04
4.35
8.7
21.74
8.7
8.7
4.35
4.35
4.35
Cumulative f
23
22
20
18
15
14
12
7
5
3
2
1
Cumulative %
100
95.65
86.96
78.26
65.22
60.87
52.17
30.43
21.74
13.04
8.7
4.35
78.26% of the scores
fall below 100
The number 78.26 is the percentile rank
The number 100 is the corresponding percentile point
We write P78.26 =100
Ice Dancing , compulsory dance scores, Winter Olympics
Percentile ranks and percentile point:
Class
interval
110-115
105-110
100-105
95-100
90-95
85-90
80-85
75-80
70-75
65-70
60-65
55-60
f
1
2
2
3
1
2
5
2
2
1
1
1
rel f(%)
4.35
8.7
8.7
13.04
4.35
8.7
21.74
8.7
8.7
4.35
4.35
4.35
Cumulative f
23
22
20
18
15
14
12
7
5
3
2
1
Cumulative %
100
95.65
86.96
78.26
65.22
60.87
52.17
30.43
21.74
13.04
8.7
4.35
21.74% of the scores are below 75
or
P21.74 = 75
or
100-21.74=78.26% of the scores are above 75.
Ice Dancing , compulsory dance scores, Winter Olympics
The Cumulative Percentage Curve
Cumulative %
100
95.65
86.96
78.26
65.22
60.87
52.17
30.43
21.74
13.04
8.7
4.35
100
90
Cumulative Percentage
Class
interval
110-115
105-110
100-105
95-100
90-95
85-90
80-85
75-80
70-75
65-70
60-65
55-60
80
70
60
50
40
30
20
10
0
60
65
70
75
80
85
90
95
100 105 110
Ice Dancing Score
21.74% of the scores fall below a score of 75
The number 21.74 is the percentile rank
The number 75 is the corresponding percentile point
We write P21.74 = 75
115
The Cumulative Percentage Curve
Cumulative %
100
95.65
86.96
78.26
65.22
60.87
52.17
30.43
21.74
13.04
8.7
4.35
100
90
Cumulative Percentage
Class
interval
110-115
105-110
100-105
95-100
90-95
85-90
80-85
75-80
70-75
65-70
60-65
55-60
80
70
60
50
40
30
20
10
0
60
65
70
75
80
85
90
95
100 105 110
Ice Dancing Score
78.26% of the scores fall below a score of 100
The number 78.26is the percentile rank
The number 100 is the corresponding percentile point
We write P78.26 = 100
115
The Cumulative Percentage Curve
Cumulative %
100
95.65
86.96
78.26
65.22
60.87
52.17
30.43
21.74
13.04
8.7
4.35
100
90
Cumulative Percentage
Class
interval
110-115
105-110
100-105
95-100
90-95
85-90
80-85
75-80
70-75
65-70
60-65
55-60
80
70
60
50
40
30
20
10
0
60
65
70
75
80
85
90
95
100 105 110
Ice Dancing Score
50% of the scores fall below a score of about 84
The number 50 is the percentile rank
The number 84 is an estimate of the percentile point
We write P50 = 84
115
Cumulative frequency distribution
Class
interval
110-115
105-110
100-105
95-100
90-95
85-90
80-85
75-80
70-75
65-70
60-65
55-60
f
1
2
2
3
1
2
5
2
2
1
1
1
rel f(%)
4.35
8.7
8.7
13.04
4.35
8.7
21.74
8.7
8.7
4.35
4.35
4.35
Cumulative f
23
22
20
18
15
14
12
7
5
3
2
1
Cumulative %
100
95.65
86.96
78.26
65.22
60.87
52.17
30.43
21.74
13.04
8.7
4.35
What is the percentile point for a percentile rank of 21.74%?
Answer: 75 points (21.75% of the scores fall below 75)
Ice Dancing , compulsory dance scores, Winter Olympics
Cumulative frequency distribution
Class Intervals Frequency (f)
110-115
105-110
100-105
95-100
90-95
85-90
80-85
75-80
70-75
65-70
60-65
55-60
1
2
2
3
1
2
5
2
2
1
1
1
Cumulative
frequency
23
22
20
18
15
14
12
7
5
3
2
1
Cumulative
proportion
1.00
.96
.87
.78
.65
.61
.52
.30
.22
.13
.09
.04
Cumulative
percent
100
96
87
78
65
61
52
30
22
13
8
4
What is the percentile point for a percentile rank of 50? (Or what is P50?)
We know it’s between 80 and 85, since 52% fall below 85 and 30% fall below 80.
Ice Dancing , compulsory dance scores, Winter Olympics
Here’s how to calculate the percentile rank for each raw score:
note this is different from the book!
Score
111.15
108.55
106.6
103.33
100.06
97.38
96.67
96.12
92.75
89.62
85.36
84.58
83.89
83.12
80.47
80.3
79.31
76.73
74.25
72.01
68.87
63.73
59.64
Rank order
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
Subtract 1/2
22.5
21.5
20.5
19.5
18.5
17.5
16.5
15.5
14.5
13.5
12.5
11.5
10.5
9.5
8.5
7.5
6.5
5.5
4.5
3.5
2.5
1.5
0.5
Divide by n (23) Multiply by 100
0.98
98
0.93
93
0.89
89
0.85
85
0.80
80
0.76
76
0.72
72
0.67
67
0.63
63
0.59
59
0.54
54
0.50
50
0.46
46
0.41
41
0.37
37
0.33
33
0.28
28
0.24
24
0.20
20
0.15
15
0.11
11
0.07
7
0.02
2
The percentile point
for a percentile rank of
50 is 84.58
( P50 = 84.58)
Ice Dancing, compulsory dance scores, Winter Olympics
Here’s how to calculate the percentile rank for each raw score:
Score
111.15
108.55
106.6
103.33
100.06
97.38
96.67
96.12
92.75
89.62
85.36
84.58
83.89
83.12
80.47
80.3
79.31
76.73
74.25
72.01
68.87
63.73
59.64
Rank order
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
Subtract 1/2
22.5
21.5
20.5
19.5
18.5
17.5
16.5
15.5
14.5
13.5
12.5
11.5
10.5
9.5
8.5
7.5
6.5
5.5
4.5
3.5
2.5
1.5
0.5
Divide by 23
0.98
0.93
0.89
0.85
0.80
0.76
0.72
0.67
0.63
0.59
0.54
0.50
0.46
0.41
0.37
0.33
0.28
0.24
0.20
0.15
0.11
0.07
0.02
Multiply by 100
98
93
89
85
80
76
72
67
63
59
54
50
46
41
37
33
28
24
20
15
11
7
2
The percentile point
for a percentile rank of
80 is 100.6
(P80 = 100.6)
Ice Dancing , compulsory dance scores, Winter Olympics
How do we calculate the percentile point for all the other ranks?
Example: What is the percentile point for the percentile rank of 90%?
Score
111.15
108.55
106.6
103.33
100.06
97.38
96.67
Rank order
23
22
21
20
19
18
17
Subtract 1/2
22.5
21.5
20.5
19.5
18.5
17.5
16.5
Divide by 23
0.98
0.93
0.89
0.85
0.80
0.76
0.72
Multiply by 100
98
93
89
85
80
76
72
We know it’s between 106.6 and 108.55
In fact, it’s ¼ of the way between 106.6 and 108.55 (90-89)/(93-89) = 1/4
That means that P90 = 106.6 + 1/4(108.55-106.6) = 107.09
How do we calculate the percentile point for other ranks?
Example, what is the percentile point for the percentile rank of P75?
Score
111.15
108.55
106.6
103.33
100.06
97.38
96.67
Rank order
23
22
21
20
19
18
17
Subtract 1/2
22.5
21.5
20.5
19.5
18.5
17.5
16.5
Divide by 23
0.98
0.93
0.89
0.85
0.80
0.76
0.72
Multiply by 100
98
93
89
85
80
76
72
We know it’s ¾ of the way between 96.67 and 97.38
96.67 + 3/4(97.38-96.67) = 97.2
How do we calculate the percentile point for other ranks?
Example, what is the percentile score for the percentile rank of P25?
Score
80.47
80.3
79.31
76.73
74.25
72.01
68.87
63.73
59.64
Rank order
9
8
7
6
5
4
3
2
1
Subtract 1/2
8.5
7.5
6.5
5.5
4.5
3.5
2.5
1.5
0.5
Divide by 23
0.37
0.33
0.28
0.24
0.20
0.15
0.11
0.07
0.02
Multiply by 100
37
33
28
24
20
15
11
7
2
We know it’s 1/4 of the way between 76.73 and 79.31
76.73 + 1/4(79.31-76.73) = 77.37
General formula for calculating percentile points:
Example, what is the percentile point for the percentile rank of 81?
Score
111.15
108.55
106.6
103.33
100.06
97.38
96.67
1)
2)
3)
4)
5)
Rank order
23
22
21
20
19
18
17
Subtract 1/2
22.5
21.5
20.5
19.5
18.5
17.5
16.5
Divide by 23
0.98
0.93
0.89
0.85
0.80
0.76
0.72
Multiply by 100
98
93
89
85
80
76
72
Make a chart like the one above
Find the two rows that fall above and below the percentile rank
Let PH and PL be the high and low cumulative percentiles (85 and 80 in this example)
Let SH and SL be the high and low scores (103.33 and 100.06 in this example)
If p is the percentile rank (81 in our example), then the percentile point is:
 p  PL 
SL  ( SH  SL)

 PH  PL 
 81  80 
100.06  (103.33  100.06)
  100.71
85

80


Going the other way: from percentile ranks to percentile points
Example: What is the percentile rank for the percentile point of 103.33?
Score
111.15
108.55
106.6
103.33
100.06
97.38
96.67
Rank order
23
22
21
20
19
18
17
Subtract 1/2
22.5
21.5
20.5
19.5
18.5
17.5
16.5
Divide by 23
0.98
0.93
0.89
0.85
0.80
0.76
0.72
Multiply by 100
98
93
89
85
80
76
72
This is easy, since 103.33 is one of the scores. The percentile rank is 85%.
85% of the scores fall below 103.33
Going the other way: from percentile ranks to percentile points
Example: What is the percentile rank for the percentile point of 100?
Score
111.15
108.55
106.6
103.33
100.06
97.38
96.67
Rank order
23
22
21
20
19
18
17
Subtract 1/2
22.5
21.5
20.5
19.5
18.5
17.5
16.5
Divide by 23
0.98
0.93
0.89
0.85
0.80
0.76
0.72
Multiply by 100
98
93
89
85
80
76
72
This is not as easy, since 100 is not one of the scores. We do know that it is between
76 and 80. In fact, we know it must be really close to 80, since P80 is 100.06
Here’s how to do it. After finding the two rows that bracket the percentile point, if S is
the percentile point, then the percentile rank is:
 S  SL 
PL  ( PH  PL)

 SH  SL 
 100  97.38 
76  (80  76)
  79.91
 100.06  97.38 
79.91% o the scores fall below 100
Another Example: integer valued data
Scores on Professor Flans’ Midterm (n = 20)
Raw Test Scores
94
93
92
91
87
86
85
84
84
83
82
81
81
80
80
77
73
73
68
59
We’ll choose a class interval width of 3. An odd
number for width is good for integer data because
the middle value will be a whole number.
Class
interval
58-61
61-64
64-67
67-70
70-73
73-76
76-79
79-82
82-85
85-88
88-91
91-94
94-97
97-100
f
1
0
0
1
2
0
1
5
4
2
1
3
0
0
Remember, scores that land
on the border are assigned to
the lower class interval.
So 85 lands in the interval
82-85.
Bins labeled by the centers of the class intervals
5
f
1
0
0
1
2
0
1
5
4
2
1
3
0
0
4
Frequency
Class
interval
58-61
61-64
64-67
67-70
70-73
73-76
76-79
79-82
82-85
85-88
88-91
91-94
94-97
97-100
3
2
1
0
60 63 66 69 72 75 78 81 84 87 90 93 96 99
Test Score
You can also show the whole interval on the x-axis labels
5
Frequency
4
3
2
1
0
58-61
61-64
64-67
67-70
70-73
73-76
76-79
79-82
Test Score
82-85
85-88
88-91
91-94
94-97 97-100
The Cumulative Percentage Curve
Class
Interval
97-100
94-97
91-94
88-91
85-88
82-85
79-82
76-79
73-76
70-73
67-70
64-67
61-64
58-61
Cumulative
frequency frequency
0
20
0
20
3
20
1
17
2
16
4
14
5
10
1
5
0
4
2
4
1
2
0
1
0
1
1
1
Relative
frequency(%)
0
0
15
5
10
20
25
5
0
10
5
0
0
5
cumulative
frequency %
100
100
100
85
80
70
50
25
20
20
10
5
5
5
The Cumulative Percentage Curve for Professor Flans’ Midterm
Estimate the percentile point for a percentile rank of 50%
Cumulative
frequency%
100
100
100
85
80
70
50
25
20
20
10
5
5
5
100
Cumulative Frequency (%)
Class
Interval
97-100
94-97
91-94
88-91
85-88
82-85
79-82
78-79
73-76
70-73
67-70
64-67
61-64
58-61
90
80
70
60
50
40
30
20
10
0
61 64 67 70 73 76 79 82 85 88 91 94 97 100
Test Score
About 50% of the scores fall below 82. (So P50 is about 82)
Estimating percentile points and percentile ranks from the cumulative percentage curve
Estimate the percentile point for a percentile rank of 90%
Cumulative Frequency (%)
100
90
80
70
60
50
40
30
20
10
0
61 64 67 70 73 76 79 82 85 88 91 94 97 100
Test Score
90% of the scores fall below a score of about 92. (P90 is about 92)
Calculating percentile points from raw data.
What is the percentile point for a percentile rank of 50%?
Test score
94
93
92
91
87
86
85
84
84
83
82
81
81
80
80
77
73
73
68
59
Rank order
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
Subtract 1/2
19.5
18.5
17.5
16.5
15.5
14.5
13.5
12.5
11.5
10.5
9.5
8.5
7.5
6.5
5.5
4.5
3.5
2.5
1.5
0.5
Divide by 20
0.975
0.925
0.875
0.825
0.775
0.725
0.675
0.625
0.575
0.525
0.475
0.425
0.375
0.325
0.275
0.225
0.175
0.125
0.075
0.025
Multiply by 100
97.5
92.5
87.5
82.5
77.5
72.5
It’s between 82 and 83
67.5
 p  PL 
62.5


SL

(
SH

SL
)
57.5
pH

pL


52.5
47.5
42.5 82  (83  82) 50  47.5   82.5


37.5
 52.5  47.5 
32.5
27.5
P50 = 82.5
22.5
17.5
12.5
7.5
2.5
Calculating percentile points from raw data.
What is the percentile point for a percentile rank of 90%?
Test score
94
93
92
91
87
86
85
84
84
83
82
81
81
80
80
77
73
73
68
59
Rank order
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
Subtract 1/2
19.5
18.5
17.5
16.5
15.5
14.5
13.5
12.5
11.5
10.5
9.5
8.5
7.5
6.5
5.5
4.5
3.5
2.5
1.5
0.5
Divide by 20
0.975
0.925
0.875
0.825
0.775
0.725
0.675
0.625
0.575
0.525
0.475
0.425
0.375
0.325
0.275
0.225
0.175
0.125
0.075
0.025
Multiply by 100
97.5
92.5
87.5
82.5
77.5
72.5
It’s between 92 and 93
67.5
 p  PL 
62.5


SL

(
SH

SL
)
57.5
pH

pL


52.5
47.5
42.5 92  (93  92) 90  87.5   92.5


37.5
 92.5  87.5 
32.5
27.5
It’s exactly halfway
22.5
between 92 and 93
17.5
12.5
7.5
2.5
Going the other way: from percentile ranks to percentile points
Example, what is the percentile rank for the percentile point of 90?
Test score
94
93
92
91
87
86
85
84
84
83
82
81
81
80
80
77
73
73
68
59
Rank order
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
Subtract 1/2
19.5
18.5
17.5
16.5
15.5
14.5
13.5
12.5
11.5
10.5
9.5
8.5
7.5
6.5
5.5
4.5
3.5
2.5
1.5
0.5
Divide by 23
0.975
0.925
0.875
0.825
0.775
0.725
0.675
0.625
0.575
0.525
0.475
0.425
0.375
0.325
0.275
0.225
0.175
0.125
0.075
0.025
Multiply by 100
97.5
92.5
87.5
82.5
77.5
72.5
67.5
62.5
57.5
52.5
47.5
42.5
37.5
32.5
27.5
22.5
17.5
12.5
7.5
2.5
It’s between 77.5 and 82.5
 S  SL 
pL  ( pH  pL)

SH

SL


 90  87 
77.5  (82.5  77.5)
  81.25
91

87


81.25% of the scores fall
below 90 points
More stuff about frequency distributions:
Frequency polygon
5
5
4
4
Frequency
Frequency
Frequency histogram
3
2
1
0
3
2
1
60 63 66 69 72 75 78 81 84 87 90 93 96 99
Test Score
0
60 63 66 69 72 75 78 81 84 87 90 93 96 99
Test Score
Properties of frequency distributions
‘normal’ or bell-shaped
Negatively skewed
positively skewed
Example of a negatively skewed distribution
40
35
Frequency
30
25
20
15
10
5
0
300
350
400
450
500 550 600 650
GRE quant scores
700
750
800
Example of positively skewed distribution: Household annual income
Household income distribution as of 2006:
•P0-89 (bottom 90%) — income below $104,696 (average income, $30,374*)
•P90-100 (top 10%) —
income above $104,696 (average income, $269,658*)
•P90-95 (next 5%) —
income between $104,696 and $148,423 (average income, $122,429*)
•P95-99 (next 4%) —
income between $148,423 and $382,593 (average income, $210,597*)
•P99-100 (top 1%) —
income above $382,593 (average income, $1,243,516*)
•P99.5-100 (top 0.5%) — income above $597,584 (average income, $2,022,315*)
•P99.9-100 (top 0.1%) — income above $1,898,200 (average income, $6,289,800*)
•P99.99-100 (top .01%) —income above $10,659,283 (average income, $29,638,027*)
So the ‘top 1%’ can be described as:
P99 = $382,593
http://www.wealthandwant.com/issues/income/income_distribution.html
Two (of many) ways that frequency distributions differ
Shift in central tendency
0
20
40
60
Scores
80
100
80
100
Shift in variability
0
20
40
60
Scores