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 The Percentile – Explanation & Examples
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The Percentile – Explanation & Examples
The definition of percentile is:
“The percentile is the value below which a certain percent of numerical data falls.”
In this topic, we will discuss the percentile from the following aspects:
 What does percentile mean in statistics?
 How to find the percentile?
 Percentile formula.
 Practical questions.
 Answers.
What does percentile mean in statistics?
The percentile is the value below which a certain percent of numerical data falls.
For example, if you score 90 out of 100 on a certain test. That score has no meaning unless you know what percentile you fall into.
If your score (90 out of 100) is the 90th percentile. This means that you score better than 90% of the test takers.
If your score (90 out of 100) is the 60th percentile. This means that you score better than only 60% of the test takers.
The 25th percentile is the first quartile or Q1.
The 50th percentile is the second quartile or Q2.
The 75th percentile is the third quartile or Q3.
How to find the percentile?
We will go through several examples.
– Example 1
For the 10 numbers,10,20,30,40,50,60,70,80,90,100. Find the 30th, 40th, 50th and 100th percentiles.
1. Order the numbers from smallest to largest number.
The data is already ordered, 10,20,30,40,50,60,70,80,90,100.
2. Assign a rank to each value of your data.
values  rank 
10  1 
20  2 
30  3 
40  4 
50  5 
60  6 
70  7 
80  8 
90  9 
100  10 
3. Calculate the ordinal rank for each required percentile. Round the obtained number to the next integer.
Ordinal rank = (percentile/100) X total number of data points.
4. The value with the next rank to the ordinal rank is the required percentile.
The ordinal rank for the 30th percentile = (30/100) X 10 = 3. The next rank is 4 with 40 data value, so 40 is the 30th percentile.
We note that 40 is higher than 10,20,30 or 3 data values/10 data values = 0.3 or 30% of the data.
The ordinal rank for the 40th percentile = (40/100) X 10 = 4. The next rank is 5 with 50 data value, so 50 is the 40th percentile.
We note that 50 is higher than 10,20,30,40 or 4/10 = 0.4 or 40% of the data.
The ordinal rank for the 50th percentile = (50/100) X 10 = 5. The next rank is 6 with 60 data value, so 60 is the 50th percentile.
We note that 60 is higher than 10,20,30,40,50 or 5/10 = 0.5 or 50% of the data.
The ordinal rank for the 100th percentile = (100/100) X 10 = 10. The next rank is 11 with no data value.
In that case, we assume that 100 is the 100th percentile, although it is also the 90th percentile.
It is always that the 100th percentile is the maximum value and the 0th percentile is the minimum value.
– Example 2
The following is the age in years for 20 participants from a certain survey.
26 48 67 39 25 25 36 44 44 47 53 52 52 51 52 40 77 44 40 45.
Find the 10th, 30th, 60th, 80th percentiles.
1. Order the numbers from smallest to largest number.
25 25 26 36 39 40 40 44 44 44 45 47 48 51 52 52 52 53 67 77.
2. Assign a rank to each value of your data.
values  rank 
25  1 
25  2 
26  3 
36  4 
39  5 
40  6 
40  7 
44  8 
44  9 
44  10 
45  11 
47  12 
48  13 
51  14 
52  15 
52  16 
52  17 
53  18 
67  19 
77  20 
Note that repeated values or ties are ranked sequentially as usual.
3. Calculate the ordinal rank for each required percentile. Round the obtained number to the next integer.
Ordinal rank = (percentile/100) X total number of data points.
4. The value with the next rank to the ordinal rank is the required percentile.
The ordinal rank for the 10th percentile = (10/100) X 20 = 2. The next rank is 3 with 26 data value, so 26 is the 10th percentile.
We note that 26 is higher than 25,25 or 2 data values/20 data values = 0.1 or 10% of the data.
The ordinal rank for the 30th percentile = (30/100) X 20 = 6. The next rank is 7 with 40 data value, so 40 is the 30th percentile.
We note that 40 is higher than 25,25,26,36,39,40 or 6 data values/20 data values = 0.3 or 30% of the data.
The ordinal rank for the 60th percentile = (60/100) X 20 = 12. The next rank is 13 with 48 data value, so 48 is the 60th percentile.
We note that 48 is higher than 25,25,26,36,39,40,40,44,44,44,45,47 or 12 data values/20 data values = 0.6 or 60% of the data.
The ordinal rank for the 80th percentile = (80/100) X 20 = 16. The next rank is 17 with 52 data value, so 52 is the 80th percentile.
We note that 52 is higher (in rank) than 25,25,26,36,39,40,40,44,44,44,45,47,48,51,52,52 or 16 data values/20 data values = 0.8 or 80% of the data.
– Example 2
The following is the daily temperature measurements for 50 days in New York, May to September 1973.
67 72 74 62 56 66 65 59 61 69 74 69 66 68 58 64 66 57 68 62 59 73 61 61 57 58 57 67 81 79 76 78 74 67 84 85 79 82 87 90 87 93 92 82 80 79 77 72 65 73.
Find the 10th, 20th, 30th, 40th, 50th, 60th, 70th, 80th, 90th percentiles.
1. Order the numbers from smallest to largest number.
56 57 57 57 58 58 59 59 61 61 61 62 62 64 65 65 66 66 66 67 67 67 68 68 69 69 72 72 73 73 74 74 74 76 77 78 79 79 79 80 81 82 82 84 85 87 87 90 92 93.
2. Assign a rank to each value of your data.
values  rank 
56  1 
57  2 
57  3 
57  4 
58  5 
58  6 
59  7 
59  8 
61  9 
61  10 
61  11 
62  12 
62  13 
64  14 
65  15 
65  16 
66  17 
66  18 
66  19 
67  20 
67  21 
67  22 
68  23 
68  24 
69  25 
69  26 
72  27 
72  28 
73  29 
73  30 
74  31 
74  32 
74  33 
76  34 
77  35 
78  36 
79  37 
79  38 
79  39 
80  40 
81  41 
82  42 
82  43 
84  44 
85  45 
87  46 
87  47 
90  48 
92  49 
93  50 
3. Calculate the ordinal rank for each required percentile. Round the obtained number to the next integer.
Ordinal rank = (percentile/100) X total number of data points.
4. The value with the next rank to the ordinal rank is the required percentile.
The ordinal rank for the 10th percentile = (10/100) X 50 = 5. The next rank is 6 with 58 data value, so 58 is the 10th percentile.
The ordinal rank for the 20th percentile = (20/100) X 50 = 10. The next rank is 11 with 61 data value, so 61 is the 20th percentile.
The ordinal rank for the 30th percentile = (30/100) X 50 = 15. The next rank is 16 with 65 data value, so 65 is the 30th percentile.
The ordinal rank for the 40th percentile = (40/100) X 50 = 40. The next rank is 21 with 67 data value, so 67 is the 40th percentile.
The ordinal rank for the 50th percentile = (50/100) X 50 = 25. The next rank is 26 with 69 data value, so 69 is the 50th percentile.
The ordinal rank for the 60th percentile = (60/100) X 50 = 30. The next rank is 31 with 74 data value, so 74 is the 60th percentile.
The ordinal rank for the 70th percentile = (70/100) X 50 = 35. The next rank is 36 with 78 data value, so 78 is the 70th percentile.
The ordinal rank for the 80th percentile = (80/100) X 50 = 40. The next rank is 41 with 81 data value, so 81 is the 80th percentile.
The ordinal rank for the 90th percentile = (90/100) X 50 = 45. The next rank is 46 with 87 data value, so 87 is the 90th percentile.
We can add this to the above table.
values  rank  percentile 
56  1 

57  2  
57  3 

57  4  
58  5 

58  6  10th 
59  7 

59  8  
61  9  
61  10 

61  11  20th 
62  12 

62  13  
64  14 

65  15  
65  16  30th 
66  17  
66  18 

66  19  
67  20 

67  21  40th 
67  22 

68  23  
68  24 

69  25  
69  26  50th 
72  27  
72  28 

73  29  
73  30 

74  31  60th 
74  32 

74  33  
76  34 

77  35  
78  36  70th 
79  37  
79  38 

79  39  
80  40 

81  41  80th 
82  42 

82  43  
84  44 

85  45  
87  46  90th 
87  47  
90  48 

92  49  
93  50 
We can plot this data as a box plot with lines for different percentiles.
Percentile formula
To calculate the percentile for a certain number (x) in your data, use the formula:
percentile = (number of ranks below x/total number of ranks) X 100.
For example, in the table above, the number 58 with a rank = 6.
Number of ranks below 58 = 5, total number of ranks = 50.
The percentile for 58 = (5/50)X 100 = 10th.
Using that formula, we can calculate the percentiles for all numbers in our data.
Generally speaking, the 0th percentile is the minimum value and the 100th percentile is the maximum value.
values  rank  percentile 
56  1  0th 
57  2  2th 
57  3  4th 
57  4  6th 
58  5  8th 
58  6  10th 
59  7  12th 
59  8  14th 
61  9  16th 
61  10  18th 
61  11  20th 
62  12  22th 
62  13  24th 
64  14  26th 
65  15  28th 
65  16  30th 
66  17  32th 
66  18  34th 
66  19  36th 
67  20  38th 
67  21  40th 
67  22  42th 
68  23  44th 
68  24  46th 
69  25  48th 
69  26  50th 
72  27  52th 
72  28  54th 
73  29  56th 
73  30  58th 
74  31  60th 
74  32  62th 
74  33  64th 
76  34  66th 
77  35  68th 
78  36  70th 
79  37  72th 
79  38  74th 
79  39  76th 
80  40  78th 
81  41  80th 
82  42  82th 
82  43  84th 
84  44  86th 
85  45  88th 
87  46  90th 
87  47  92th 
90  48  94th 
92  49  96th 
93  50  98th 
Although 93 is the 98th percentile, it is also considered the 100th percentile as there is no value in our data that is larger than all our data values.
Practical questions
1. The following are some percentiles for some daily ozone measurements in New York, May to September 1973.
percentile  value 
10%  11.00 
30%  20.00 
70%  49.50 
75%  63.25 
What percentage of data is less than 20?
What is the third quartile of this data or Q3?
2. The following are daily solar radiation measurements for 20 days in New York, May to September 1973.
236 259 238 24 112 237 224 27 238 201 238 14 139 49 20 193 145 191 131 223.
Construct a table with the rank and percentile for each value.
3. The following are murder rates per 100,000 population for 50 states of the United States of America in 1976.
state  value 
Alabama  15.1 
Alaska  11.3 
Arizona  7.8 
Arkansas  10.1 
California  10.3 
Colorado  6.8 
Connecticut  3.1 
Delaware  6.2 
Florida  10.7 
Georgia  13.9 
Hawaii  6.2 
Idaho  5.3 
Illinois  10.3 
Indiana  7.1 
Iowa  2.3 
Kansas  4.5 
Kentucky  10.6 
Louisiana  13.2 
Maine  2.7 
Maryland  8.5 
Massachusetts  3.3 
Michigan  11.1 
Minnesota  2.3 
Mississippi  12.5 
Missouri  9.3 
Montana  5.0 
Nebraska  2.9 
Nevada  11.5 
New Hampshire  3.3 
New Jersey  5.2 
New Mexico  9.7 
New York  10.9 
North Carolina  11.1 
North Dakota  1.4 
Ohio  7.4 
Oklahoma  6.4 
Oregon  4.2 
Pennsylvania  6.1 
Rhode Island  2.4 
South Carolina  11.6 
South Dakota  1.7 
Tennessee  11.0 
Texas  12.2 
Utah  4.5 
Vermont  5.5 
Virginia  9.5 
Washington  4.3 
West Virginia  6.7 
Wisconsin  3.0 
Wyoming  6.9 
Construct a table with the rank and percentile for each value.
4. The following are some percentiles of temperature in certain months.
Month  10th  90th 
5  57.0  74.0 
6  72.9  87.3 
7  81.0  89.0 
8  77.0  94.0 
9  67.9  91.1 
For August or Month 8, what percent of temperatures are less than 94?
Which month has the highest spread in its temperatures?
5. The following are some percentiles of per capita income in 1974 for the 4 regions of the US.
region  10th  90th 
Northeast  3864.4  5259.2 
South  3461.5  4812.0 
North Central  4274.4  5053.4 
West  4041.4  5142.0 
Which region has the highest 90th percentile?
Which region has the highest 10th percentile?
Answers
1. The percentage of data that is less than 20 is 30% because 20 is 30% percentile.
The third quartile of this data or Q3 is 75% percentile or 63.25.
2. Following the above steps, we can construct the following table:
values  rank  percentile 
14  1  0th 
20  2  5th 
24  3  10th 
27  4  15th 
49  5  20th 
112  6  25th 
131  7  30th 
139  8  35th 
145  9  40th 
191  10  45th 
193  11  50th 
201  12  55th 
223  13  60th 
224  14  65th 
236  15  70th 
237  16  75th 
238  17  80th 
238  18  85th 
238  19  90th 
259  20  95th 
3. Following the above steps, we can construct the following table:
state  value  rank  percentile 
North Dakota  1.4  1  0th 
South Dakota  1.7  2  2th 
Iowa  2.3  3  4th 
Minnesota  2.3  4  6th 
Rhode Island  2.4  5  8th 
Maine  2.7  6  10th 
Nebraska  2.9  7  12th 
Wisconsin  3.0  8  14th 
Connecticut  3.1  9  16th 
Massachusetts  3.3  10  18th 
New Hampshire  3.3  11  20th 
Oregon  4.2  12  22th 
Washington  4.3  13  24th 
Kansas  4.5  14  26th 
Utah  4.5  15  28th 
Montana  5.0  16  30th 
New Jersey  5.2  17  32th 
Idaho  5.3  18  34th 
Vermont  5.5  19  36th 
Pennsylvania  6.1  20  38th 
Delaware  6.2  21  40th 
Hawaii  6.2  22  42th 
Oklahoma  6.4  23  44th 
West Virginia  6.7  24  46th 
Colorado  6.8  25  48th 
Wyoming  6.9  26  50th 
Indiana  7.1  27  52th 
Ohio  7.4  28  54th 
Arizona  7.8  29  56th 
Maryland  8.5  30  58th 
Missouri  9.3  31  60th 
Virginia  9.5  32  62th 
New Mexico  9.7  33  64th 
Arkansas  10.1  34  66th 
California  10.3  35  68th 
Illinois  10.3  36  70th 
Kentucky  10.6  37  72th 
Florida  10.7  38  74th 
New York  10.9  39  76th 
Tennessee  11.0  40  78th 
Michigan  11.1  41  80th 
North Carolina  11.1  42  82th 
Alaska  11.3  43  84th 
Nevada  11.5  44  86th 
South Carolina  11.6  45  88th 
Texas  12.2  46  90th 
Mississippi  12.5  47  92th 
Louisiana  13.2  48  94th 
Georgia  13.9  49  96th 
Alabama  15.1  50  98th 
4. For August or Month 8, the percent of temperatures that are less than 94 is 90% since 94 is the 90th percentile.
To see the spread of temperatures for each month, we can see the difference between 90th and 10th percentiles.
Month  10th  90th  difference 
5  57.0  74.0  17.0 
6  72.9  87.3  14.4 
7  81.0  89.0  8.0 
8  77.0  94.0  17.0 
9  67.9  91.1  23.2 
The highest difference is for Month 9 or September, so September has the highest spread in its temperatures.
5. Northeast has the highest 90th percentile of 5259.2.
North Central has the highest 10th percentile of 4274.4.