acalculator

What is the percentile?

Enter your numbers to find the value at any percentile, and the percentile rank of any value, with the method stated.

Your numbers

Read as: 12; 15; 18; 20; 22; 25; 28; 30; 35; 40
Percentile method
Value at the percentile
39.5

Percentile 90 of your 10 numbers is 39.5, and the value 25 is at percentile rank 55.

Position in the sorted list
9.9
Percentile rank of the value
55
Percent below the value
50%
Percent at or below the value
60%
Count (N)
10

Value at the percentile: 39.5. Percentile 90 of your 10 numbers is 39.5, and the value 25 is at percentile rank 55.

Where are your numbers?

How to calculate

Computes the value at any percentile of a list of numbers, by the NIST, Excel inclusive, or nearest-rank method, and the percentile rank of a value.

Example with the default inputs (Your numbers [12, 15, 18, 20, 22, 25, 28, 30, 35, 40], Find the value at percentile 90, Percentile method NIST (N + 1), Find the percentile of value 25): Percentile 90 of your 10 numbers is 39.5, and the value 25 is at percentile rank 55.

Method: Sort the N numbers. Position = (N + 1)p (NIST), 1 + (N − 1)p (Excel inclusive), or ⌈Np⌉ (nearest rank); value = x[k] + d × (x[k + 1] − x[k]) for position k + d. Percentile rank of x = (B + 0.5E) ÷ N × 100.

  • Positions below 1 give the smallest number, and positions at or above N give the largest (NIST §7.2.6.2).
  • In the percentile rank, B counts the numbers below the value and E the numbers equal to it.
  • The arithmetic is exact on the decimals you type, then rounded once for display.

Machine-readable copies: Markdown, JSON.

Worked examples

Each example is checked against the calculator on every build.

  1. Your numbers 95.1772, 95.1567, 95.1937, 95.1959, 95.1442, 95.061, 95.1591, 95.1195, 95.1065, 95.0925, 95.199, 95.1682, Find the value at percentile 90, Percentile method NIST (N + 1) gives Value at the percentile 95.19807, Position in the sorted list 11.7.Source: NIST/SEMATECH e-Handbook of Statistical Methods, §7.2.6.2 Percentiles. https://www.itl.nist.gov/div898/handbook/prc/section2/prc262.htm (95.1981)
  2. Your numbers 95.1772, 95.1567, 95.1937, 95.1959, 95.1442, 95.061, 95.1591, 95.1195, 95.1065, 95.0925, 95.199, 95.1682, Find the value at percentile 90, Percentile method Excel inclusive gives Value at the percentile 95.19568, Position in the sorted list 10.9.Source: NIST/SEMATECH e-Handbook of Statistical Methods, §7.2.6.2 Percentiles. https://www.itl.nist.gov/div898/handbook/prc/section2/prc262.htm (95.1957 for the Excel definition)
  3. Your numbers 18, 21, 22, 25, 26, 27, 29, 30, 31, 33, 36, 37, 41, 42, 47, 52, 55, 57, 58, 62, 64, 67, 69, 71, 72, 73, 74, 76, 77, Find the value at percentile 70, Percentile method NIST (N + 1), Find the percentile of value 58 gives Value at the percentile 64, Position in the sorted list 21, Percentile rank of the value 63.793103, Percent below the value 62.068966%, Percent at or below the value 65.517241%.Source: OpenStax, Introductory Statistics 2e, §2.3 Measures of the Location of the Data. https://openstax.org/books/introductory-statistics-2e/pages/2-3-measures-of-the-location-of-the-data (70th percentile 64; 58 is at (18 + 0.5) ÷ 29 × 100 = 63.80)
  4. Your numbers 12, 15, 18, 20, 22, 25, 28, 30, 35, 40, Find the value at percentile 90, Percentile method NIST (N + 1), Find the percentile of value 25 gives Value at the percentile 39.5, Position in the sorted list 9.9, Percentile rank of the value 55, Percent below the value 50%, Percent at or below the value 60%.Source: NIST/SEMATECH e-Handbook of Statistical Methods, §7.2.6.2 Percentiles. https://www.itl.nist.gov/div898/handbook/prc/section2/prc262.htm
  5. Your numbers 12, 15, 18, 20, 22, 25, 28, 30, 35, 40, Find the value at percentile 90, Percentile method Nearest rank gives Value at the percentile 35, Position in the sorted list 9.Source: NIST/SEMATECH e-Handbook of Statistical Methods, §7.2.6.2 Percentiles. https://www.itl.nist.gov/div898/handbook/prc/section2/prc262.htm
  6. Your numbers 12, 15, 18, 20, 22, 25, 28, 30, 35, 40, Find the value at percentile 95, Percentile method NIST (N + 1) gives Value at the percentile 40, Position in the sorted list 10.45.Source: NIST/SEMATECH e-Handbook of Statistical Methods, §7.2.6.2 Percentiles. https://www.itl.nist.gov/div898/handbook/prc/section2/prc262.htm (for k ≥ N, the largest value)

How it works

Sort your N numbers from smallest to largest: x[1] ≤ x[2] ≤ … ≤ x[N]. Let p be the percentile divided by 100 (0.9 for the 90th).

The value at a percentile

  1. Find the position in the sorted list:
    • NIST (N + 1): position = (N + 1) × p. This is the NIST handbook's method and definition 6 of Hyndman and Fan.
    • Excel inclusive: position = 1 + (N − 1) × p, as in Excel's PERCENTILE.INC (definition 7).
    • Nearest rank: position = ⌈N × p⌉, the smallest whole number at or above N × p; for p = 0 the position is 1.
  2. Split the position into a whole part k and a fraction d.
  3. Value = x[k] + d × (x[k + 1] − x[k]). A position of 1 or less gives x[1]; a position of N or more gives x[N].

The page shows the value at the percentile and the position in the sorted list (the position from step 1, before it is held at 1 or N).

The percentile rank of a value

For a value v, count B, the numbers below v, and E, the numbers equal to v:

  • Percentile rank = (B + 0.5 × E) ÷ N × 100, a number from 0 to 100 (OpenStax §2.3).
  • Percent below = B ÷ N × 100%.
  • Percent at or below = (B + E) ÷ N × 100%.

The value does not have to be in the list.

The page also shows the count N.

Rules

  • The list holds 1 to 1,000 numbers. The percentile is from 0 to 100.
  • Fill in the percentile, the value, or both. With neither there is no answer. With only one, the page shows the results for that one.
  • The positions, values, and ranks are exact on the decimals as typed, then rounded once to the nearest 64-bit float.
  • Results show up to 6 decimal places (6 significant digits below 0.0001), with halves rounded up; the two percent results show up to 4 decimal places with a % sign.

Worked examples by hand

NIST resistivity data, 90th percentile. The 12 measurements sorted end with x[10] = 95.1937, x[11] = 95.1959, x[12] = 95.1990.

  • NIST: position = 13 × 0.9 = 11.7, so k = 11, d = 0.7: 95.1959 + 0.7 × (95.1990 − 95.1959) = 95.1959 + 0.00217 = 95.19807 (NIST shows 95.1981).
  • Excel inclusive: position = 1 + 11 × 0.9 = 10.9: 95.1937 + 0.9 × 0.0022 = 95.19568 (NIST shows 95.1957).

Ages of 29 Academy Award winners (OpenStax §2.3), sorted: 18, 21, 22, 25, 26, 27, 29, 30, 31, 33, 36, 37, 41, 42, 47, 52, 55, 57, 58, 62, 64, 67, 69, 71, 72, 73, 74, 76, 77.

  • 70th percentile (NIST): position = 30 × 0.7 = 21, the 21st age: 64.
  • Rank of 58: B = 18, E = 1, so (18 + 0.5) ÷ 29 × 100 = 63.793103. Percent below = 18 ÷ 29 = 62.069%; at or below = 19 ÷ 29 = 65.5172%.

The default list 12, 15, 18, 20, 22, 25, 28, 30, 35, 40 (N = 10).

  • 90th percentile (NIST): position = 11 × 0.9 = 9.9: 35 + 0.9 × (40 − 35) = 39.5.
  • 90th percentile (nearest rank): ⌈10 × 0.9⌉ = 9: the 9th number, 35.
  • 95th percentile (NIST): position = 11 × 0.95 = 10.45, past the 10th number, so the value is the largest, 40.
  • Rank of 25: B = 5, E = 1, so (5 + 0.5) ÷ 10 × 100 = 55. Below: 50%; at or below: 60%.

Other questions people ask

What is a percentile?

The pth percentile is a value with about p percent of the data at or below it. The median is the 50th percentile, and the 90th percentile is a value that about 90% of the numbers do not exceed.

How do I calculate a percentile?

Sort the numbers and find the position (N + 1) × p, where p is the percentile as a fraction. For the 90th percentile of 10 numbers the position is 11 × 0.9 = 9.9: 90% of the way from the 9th number to the 10th. If the 9th is 35 and the 10th is 40, the 90th percentile is 35 + 0.9 × 5 = 39.5.

Why do different calculators give different percentiles?

There is no single definition. The NIST handbook uses the position (N + 1)p, Excel’s PERCENTILE.INC uses 1 + (N − 1)p, and the nearest-rank method takes the ⌈Np⌉th number with no interpolation. For large lists they agree closely; for short lists they can differ. Pick the method your course or software uses.

What is a percentile rank?

The percentile rank of a value is the percent of the data below it, counting values equal to it as half: (B + 0.5E) ÷ N × 100, where B is the count below and E the count equal. With 29 ages, 18 of them below 58 and one equal to 58, the rank of 58 is (18 + 0.5) ÷ 29 × 100 = 63.8.

What is the difference between a percentile and a percentage?

A percentage is a share of a whole, like 85% of the questions right. A percentile compares one result with everyone else’s: scoring at the 85th percentile means doing as well as or better than about 85% of the group, whatever your raw score.

Is the 25th percentile the same as the first quartile?

They measure the same idea, but quartile methods and percentile methods do not always agree on short lists. The NIST percentile at 25 and the TI-84 first quartile can differ; for large lists they are close.