 # 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.

## 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?

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.