1. Percentiles and Percentile Ranks and their Graphical Representations
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)

2. Percentiles and Percentile Ranks
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

3. Car sales drawn as a histogram
Japan
China US
Germany
South Korea
Brazil 2 4 6 8 10 12
Car Sales in 2008 (millions)
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

4. This histogram shows the proportion of members for each category.
Distribution of all M&M's.

5. Ice Dancing , compulsory dance scores, 4 Winter Olympics
Making histograms from interval and ratio data
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
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 )
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.

6. Ice Dancing , compulsory dance scores,
Winter Olympics
Let’s use a class interval width of 5 points, with a lowest score of 55.
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)
95-100
90-95
85-90
80-85
75-80
70-75
65-70
60-65
55-60 1 2 3 5
n=23
Count the number of scores in each bin to get the frequency

7. Histogram of Ice Dancing Scores (frequency)
Class Intervals
Frequency (f) 55 60 65 70 75 80 85 90 95
100
105
110
115 1 2 3 4 5
Ice Dancing Score
Frequency
95-100
90-95
85-90
80-85
75-80
70-75
65-70
60-65
55-60 1 2 3 5

8. Relative frequency .0435 .0870 .1304 .2174 Relative frequency (prop)
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
.0435
.0870
.1304
.2174
Relative
frequency
(prop)
Relative
frequency
(%)
4.35
8.70
13.04
21.74
Class Intervals
Frequency (f)
95-100
90-95
85-90
80-85
75-80
70-75
65-70
60-65
55-60 1 2 3 5
n=23
Divide by the total number of scores to get relative frequency in proportion
Then multiply by 100 to get relative frequency in percent

9. Relative frequency histogram of Ice Dancing Scores (frequency)
Class Intervals 55 60 65 70 75 80 85 90 95
100
105
110
115 5 10 15 20 25
Ice Dancing Score
Relative Frequency (%)
95-100
90-95
85-90
80-85
75-80
70-75
65-70
60-65
55-60
4.35
8.70
13.04
21.74

10. Choosing your class intervals can have an influence on the way your histogram looks
interval width 10
interval width 5 60 70 80 90
100
110
120 1 2 3 4 5 6 7
Ice Dancing Score
Frequency 60 80
100 1 2 3 4 5
Ice Dancing Score
Frequency
interval width 3
interval width 1 60 80
100 1 2 3
Ice Dancing Score
Frequency 60 80
100 1 2
Ice Dancing Score
Frequency

11. These three graphs have the same class intervals on the same scores!60 80
100 1 2 3 4 5
Ice Dancing Score
Frequency 60 70 80 90
100
110 1 2 3 4 5
Ice Dancing Score
Frequency 60 70 80 90
100
110 1 2 3 4 5
Ice Dancing Score
Frequency

12. When possible, include zero on your y-axis.
Not like this

13. When possible, include zero on your y-axis:
As of March 27
March 31 Goal 2 4 6 8
Enrollment (Millions)
Like this
Not like this

14. “Fox News Apologizes For Obamacare Graphic, Corrects Its 'Mistake‘”

15. 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.

16. Growth charts convert percentile points to percentile ranks
At 30 mos.
P95 = 36lbs

17. Ice Dancing , compulsory dance scores, Winter Olympics
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 f
rel f(%)
Cumu-lative f
Cumu-lative % 1
4.35 23
100 2
8.7 22
95.65 20
86.96
95-100 3
13.04 18
78.26
90-95 15
65.22
85-90 14
60.87
80-85 5
21.74 12
52.17
75-80 7
30.43
70-75
65-70
60-65
55-60
78.26% of the scores fall below 100
The number is the percentile rank
The number 100 is the corresponding percentile point
We write P78.26 =100
Ice Dancing , compulsory dance scores, Winter Olympics

18. Percentile ranks and percentile point:
Class interval f
rel f(%)
Cumu-lative f
Cumu-lative % 1
4.35 23
100 2
8.7 22
95.65 20
86.96
95-100 3
13.04 18
78.26
90-95 15
65.22
85-90 14
60.87
80-85 5
21.74 12
52.17
75-80 7
30.43
70-75
65-70
60-65
55-60
21.74% of the scores are below 75 or
P21.74 = 75
=78.26% of the scores are above 75.
Ice Dancing , compulsory dance scores, Winter Olympics

19. The Cumulative Percentage Curve
Class interval
Cumu-lative %
100
95.65
86.96
95-100
78.26
90-95
65.22
85-90
60.87
80-85
52.17
75-80
30.43
70-75
21.74
65-70
13.04
60-65
8.7
55-60
4.35
100 90 80 70 60
Cumulative Percentage 50 40 30 20 10 60 65 70 75 80 85 90 95
100
105
110
115
Ice Dancing Score
21.74% of the scores fall below a score of 75
The number is the percentile rank
The number 75 is the corresponding percentile point
We write P21.74 = 75

20. The Cumulative Percentage Curve
Class interval
Cumu-lative %
100
95.65
86.96
95-100
78.26
90-95
65.22
85-90
60.87
80-85
52.17
75-80
30.43
70-75
21.74
65-70
13.04
60-65
8.7
55-60
4.35
100 90 80 70 60
Cumulative Percentage 50 40 30 20 10 60 65 70 75 80 85 90 95
100
105
110
115
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

21. The Cumulative Percentage Curve
Class interval
Cumu-lative %
100
95.65
86.96
95-100
78.26
90-95
65.22
85-90
60.87
80-85
52.17
75-80
30.43
70-75
21.74
65-70
13.04
60-65
8.7
55-60
4.35
100 90 80 70 60
Cumulative Percentage 50 40 30 20 10 60 65 70 75 80 85 90 95
100
105
110
115
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

22. Ice Dancing , compulsory dance scores, Winter Olympics
Cumulative frequency distribution
Class interval f
rel f(%)
Cumu-lative f
Cumu-lative % 1
4.35 23
100 2
8.7 22
95.65 20
86.96
95-100 3
13.04 18
78.26
90-95 15
65.22
85-90 14
60.87
80-85 5
21.74 12
52.17
75-80 7
30.43
70-75
65-70
60-65
55-60
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

23. Ice Dancing , compulsory dance scores, Winter Olympics
Cumulative frequency distribution
Cumulative
frequency
Cumulative
proportion
Cumulative
percent
Class Intervals
Frequency (f)
95-100
90-95
85-90
80-85
75-80
70-75
65-70
60-65
55-60 1 2 3 5 23 22 20 18 15 14 12 7 5 3 2 1
1.00
.96
.87
.78
.65
.61
.52
.30
.22
.13
.09
.04
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

24. note this is different from the book!
Here’s how to calculate the percentile rank for each raw score:
note this is different from the book!
Score
Rank order
Subtract 1/2
Divide by n (23)
Multiply by 100
111.15 23
22.5
0.98 98
108.55 22
21.5
0.93 93
106.6 21
20.5
0.89 89
103.33 20
19.5
0.85 85
100.06 19
18.5
0.80 80
97.38 18
17.5
0.76 76
96.67 17
16.5
0.72 72
96.12 16
15.5
0.67 67
92.75 15
14.5
0.63 63
89.62 14
13.5
0.59 59
85.36 13
12.5
0.54 54
84.58 12
11.5
0.50 50
83.89 11
10.5
0.46 46
83.12 10
9.5
0.41 41
80.47 9
8.5
0.37 37
80.3 8
7.5
0.33 33
79.31 7
6.5
0.28 28
76.73 6
5.5
0.24 24
74.25 5
4.5
0.20
72.01 4
3.5
0.15
68.87 3
2.5
0.11
63.73 2
1.5
0.07
59.64 1
0.5
0.02
The percentile point for a percentile rank of 50 is 84.58
( P50 = 84.58)
Ice Dancing, compulsory dance scores, Winter Olympics

25. Ice Dancing , compulsory dance scores, Winter Olympics
Here’s how to calculate the percentile rank for each raw score:
Score
Rank order
Subtract 1/2
Divide by 23
Multiply by 100
111.15 23
22.5
0.98 98
108.55 22
21.5
0.93 93
106.6 21
20.5
0.89 89
103.33 20
19.5
0.85 85
100.06 19
18.5
0.80 80
97.38 18
17.5
0.76 76
96.67 17
16.5
0.72 72
96.12 16
15.5
0.67 67
92.75 15
14.5
0.63 63
89.62 14
13.5
0.59 59
85.36 13
12.5
0.54 54
84.58 12
11.5
0.50 50
83.89 11
10.5
0.46 46
83.12 10
9.5
0.41 41
80.47 9
8.5
0.37 37
80.3 8
7.5
0.33 33
79.31 7
6.5
0.28 28
76.73 6
5.5
0.24 24
74.25 5
4.5
0.20
72.01 4
3.5
0.15
68.87 3
2.5
0.11
63.73 2
1.5
0.07
59.64 1
0.5
0.02
The percentile point for a percentile rank of 80 is 100.6
(P80 = 100.6)
Ice Dancing , compulsory dance scores, Winter Olympics

26. 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
Rank order
Subtract 1/2
Divide by 23
Multiply by 100
111.15 23
22.5
0.98 98
108.55 22
21.5
0.93 93
106.6 21
20.5
0.89 89
103.33 20
19.5
0.85 85
100.06 19
18.5
0.80 80
97.38 18
17.5
0.76 76
96.67 17
16.5
0.72 72
We know it’s between and
In fact, it’s ¼ of the way between and (90-89)/(93-89) = 1/4
That means that P90 = /4( ) =

27. How do we calculate the percentile point for other ranks?
Example, what is the percentile point for the percentile rank of P75?
Score
Rank order
Subtract 1/2
Divide by 23
Multiply by 100
111.15 23
22.5
0.98 98
108.55 22
21.5
0.93 93
106.6 21
20.5
0.89 89
103.33 20
19.5
0.85 85
100.06 19
18.5
0.80 80
97.38 18
17.5
0.76 76
96.67 17
16.5
0.72 72
We know it’s ¾ of the way between and 97.38
/4( ) = 97.2

28. How do we calculate the percentile point for other ranks?
Example, what is the percentile score for the percentile rank of P25?
Score
Rank order
Subtract 1/2
Divide by 23
Multiply by 100
80.47 9
8.5
0.37 37
80.3 8
7.5
0.33 33
79.31 7
6.5
0.28 28
76.73 6
5.5
0.24 24
74.25 5
4.5
0.20 20
72.01 4
3.5
0.15 15
68.87 3
2.5
0.11 11
63.73 2
1.5
0.07
59.64 1
0.5
0.02
We know it’s 1/4 of the way between and 79.31
/4( ) = 77.37

29. General formula for calculating percentile points:
Example, what is the percentile point for the percentile rank of 81?
Score
Rank order
Subtract 1/2
Divide by 23
Multiply by 100
111.15 23
22.5
0.98 98
108.55 22
21.5
0.93 93
106.6 21
20.5
0.89 89
103.33 20
19.5
0.85 85
100.06 19
18.5
0.80 80
97.38 18
17.5
0.76 76
96.67 17
16.5
0.72 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 ( and in this example)
If p is the percentile rank (81 in our example), then the percentile point is:

30. Going the other way: from percentile ranks to percentile points
Example: What is the percentile rank for the percentile point of ?
Score
Rank order
Subtract 1/2
Divide by 23
Multiply by 100
111.15 23
22.5
0.98 98
108.55 22
21.5
0.93 93
106.6 21
20.5
0.89 89
103.33 20
19.5
0.85 85
100.06 19
18.5
0.80 80
97.38 18
17.5
0.76 76
96.67 17
16.5
0.72 72
This is easy, since is one of the scores. The percentile rank is 85%.
85% of the scores fall below

31. Going the other way: from percentile ranks to percentile points
Example: What is the percentile rank for the percentile point of 100?
Score
Rank order
Subtract 1/2
Divide by 23
Multiply by 100
111.15 23
22.5
0.98 98
108.55 22
21.5
0.93 93
106.6 21
20.5
0.89 89
103.33 20
19.5
0.85 85
100.06 19
18.5
0.80 80
97.38 18
17.5
0.76 76
96.67 17
16.5
0.72 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
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:
79.91% o the scores fall below 100

32. Another Example: integer valued data
Scores on Professor Flans’ Midterm (n = 20)
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.
Raw Test Scores 94 93 92 91 87 86 85 84 83 82 81 80 77 73 68 59
Class interval f
58-61 1
61-64
64-67
67-70
70-73 2
73-76
76-79
79-82 5
82-85 4
85-88
88-91
91-94 3
94-97
97-100
Remember, scores that land on the border are assigned to the lower class interval.
So 85 lands in the interval
82-85.

33. Bins labeled by the centers of the class intervals
58-61 1
61-64
64-67
67-70
70-73 2
73-76
76-79
79-82 5
82-85 4
85-88
88-91
91-94 3
94-97
97-100 5 4 3
Frequency 2 1 60 63 66 69 72 75 78 81 84 87 90 93 96 99
Test Score

34. You can also show the whole interval on the x-axis labels5 4 3
Frequency 2 1
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
Test Score

35. The Cumulative Percentage Curve
Class Interval
frequency
97-100
94-97
91-94 3
88-91 1
85-88 2
82-85 4
79-82 5
76-79
73-76
70-73
67-70
64-67
61-64
58-61
Cumulative
frequency 20 17 16 14 10 5 4 2 1
Relative frequency(%) 15 5 10 20 25
cumulative frequency %
100 85 80 70 50 25 20 10 5

36. Cumulative frequency%
The Cumulative Percentage Curve for Professor Flans’ Midterm
Estimate the percentile point for a percentile rank of 50%
Class Interval
Cumulative frequency%
97-100
100
94-97
91-94
88-91 85
85-88 80
82-85 70
79-82 50
78-79 25
73-76 20
70-73
67-70 10
64-67 5
61-64
58-61
100 90 80 70 60
Cumulative Frequency (%) 50 40 30 20 10 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)

37. Estimate the percentile point for a percentile rank of 90%
Estimating percentile points and percentile ranks from the cumulative percentage curve
Estimate the percentile point for a percentile rank of 90%
100 90 80 70 60
Cumulative Frequency (%) 50 40 30 20 10 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)

38. Calculating percentile points from raw data.
What is the percentile point for a percentile rank of 50%?
Test score
Rank order
Subtract 1/2
Divide by 20
Multiply by 100 94 20
19.5
0.975
97.5 93 19
18.5
0.925
92.5 92 18
17.5
0.875
87.5 91 17
16.5
0.825
82.5 87 16
15.5
0.775
77.5 86 15
14.5
0.725
72.5 85 14
13.5
0.675
67.5 84 13
12.5
0.625
62.5 12
11.5
0.575
57.5 83 11
10.5
0.525
52.5 82 10
9.5
0.475
47.5 81 9
8.5
0.425
42.5 8
7.5
0.375
37.5 80 7
6.5
0.325
32.5 6
5.5
0.275
27.5 77 5
4.5
0.225
22.5 73 4
3.5
0.175 3
2.5
0.125 68 2
1.5
0.075 59 1
0.5
0.025
It’s between 82 and 83
P50 = 82.5

39. It’s exactly halfway between 92 and 93
Calculating percentile points from raw data.
What is the percentile point for a percentile rank of 90%?
Test score
Rank order
Subtract 1/2
Divide by 20
Multiply by 100 94 20
19.5
0.975
97.5 93 19
18.5
0.925
92.5 92 18
17.5
0.875
87.5 91 17
16.5
0.825
82.5 87 16
15.5
0.775
77.5 86 15
14.5
0.725
72.5 85 14
13.5
0.675
67.5 84 13
12.5
0.625
62.5 12
11.5
0.575
57.5 83 11
10.5
0.525
52.5 82 10
9.5
0.475
47.5 81 9
8.5
0.425
42.5 8
7.5
0.375
37.5 80 7
6.5
0.325
32.5 6
5.5
0.275
27.5 77 5
4.5
0.225
22.5 73 4
3.5
0.175 3
2.5
0.125 68 2
1.5
0.075 59 1
0.5
0.025
It’s between 92 and 93
It’s exactly halfway between 92 and 93

40. Going the other way: from percentile ranks to percentile points
Example, what is the percentile rank for the percentile point of 90?
Test score
Rank order
Subtract 1/2
Divide by 23
Multiply by 100 94 20
19.5
0.975
97.5 93 19
18.5
0.925
92.5 92 18
17.5
0.875
87.5 91 17
16.5
0.825
82.5 87 16
15.5
0.775
77.5 86 15
14.5
0.725
72.5 85 14
13.5
0.675
67.5 84 13
12.5
0.625
62.5 12
11.5
0.575
57.5 83 11
10.5
0.525
52.5 82 10
9.5
0.475
47.5 81 9
8.5
0.425
42.5 8
7.5
0.375
37.5 80 7
6.5
0.325
32.5 6
5.5
0.275
27.5 77 5
4.5
0.225
22.5 73 4
3.5
0.175 3
2.5
0.125 68 2
1.5
0.075 59 1
0.5
0.025
It’s between 77.5 and 82.5
81.25% of the scores fall below 90 points

41. More stuff about frequency distributions:
Frequency polygon
Frequency histogram 60 63 66 69 72 75 78 81 84 87 90 93 96 99 1 2 3 4 5
Test Score
Frequency 5 4 3
Frequency 2 1 60 63 66 69 72 75 78 81 84 87 90 93 96 99
Test Score

42. Properties of frequency distributions
‘normal’ or bell-shaped
positively skewed
Negatively skewed

43. Example of a negatively skewed distribution40 35 30 25
Frequency 20 15 10 5
300
350
400
450
500
550
600
650
700
750
800
GRE quant scores

44. Example of positively skewed distribution: Household annual income

45. Household income distribution as of 2006:
P0-89 (bottom 90%) —   income below $104,696 (average income, $30,374*)
P (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*)
P (top 1%) —         income above $382,593 (average income, $1,243,516*)
P (top 0.5%) —   income above $597,584 (average income, $2,022,315*)
P (top 0.1%) —   income above $1,898,200 (average income, $6,289,800*)
P (top .01%) —income above $10,659,283 (average income, $29,638,027*)
So the ‘top 1%’ can be described as:
P99 = $382,593

46. Two (of many) ways that frequency distributions differ
Shift in central tendency 20 40 60 80
100
Scores
Shift in variability 20 40 60 80
100
Scores