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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.

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