A Formula for Finding the kth Percentile

Learning Outcomes

  • Given a data set, find the [latex]k^{th}[/latex] percentile

A Formula for Finding the [latex]k[/latex] th Percentile

If you were to do a little research, you would find several formulas for calculating the [latex]k[/latex]th percentile. Here is one of them.

[latex]k[/latex] = the [latex]k[/latex]th percentile. It may or may not be part of the data.

[latex]i[/latex] = the index (ranking or position of a data value)

[latex]n[/latex] = the total number of data

  • Order the data from smallest to largest
  • Calculate [latex]\displaystyle{i}=\frac{{k}}{{100}}{({n}+{1})}[/latex]
  • If [latex]i[/latex] is an integer, then the [latex]k[/latex]th percentile is the data value in the [latex]i[/latex]th position in the ordered set of data.
  • If [latex]i[/latex] is not an integer, then round [latex]i[/latex] up and round [latex]i[/latex] down to the nearest integers. Average the two data values in these two positions in the ordered data set. This is easier to understand in an example.

Example

Listed are 29 ages for trees found in the Saint Louis Botanical Garden in order from smallest to largest.

[latex]18[/latex]; [latex]21[/latex]; [latex]22[/latex]; [latex]25[/latex]; [latex]26[/latex]; [latex]27[/latex]; [latex]29[/latex]; [latex]30[/latex]; [latex]31[/latex]; [latex]33[/latex]; [latex]36[/latex]; [latex]37[/latex]; [latex]41[/latex]; [latex]42[/latex]; [latex]47[/latex]; [latex]52[/latex]; [latex]55[/latex]; [latex]57[/latex]; [latex]58[/latex]; [latex]62[/latex]; [latex]64[/latex]; [latex]67[/latex]; [latex]69[/latex]; [latex]71[/latex]; [latex]72[/latex]; [latex]73[/latex]; [latex]74[/latex]; [latex]76[/latex]; [latex]77[/latex]

  1. Find the [latex]70[/latex]th percentile.
  2. Find the [latex]83[/latex]rd percentile.

Try It

Listed are [latex]29[/latex] ages for Academy Award winning best actors in order from smallest to largest.

[latex]18[/latex]; [latex]21[/latex]; [latex]22[/latex]; [latex]25[/latex]; [latex]26[/latex]; [latex]27[/latex]; [latex]29[/latex]; [latex]30[/latex]; [latex]31[/latex]; [latex]33[/latex]; [latex]36[/latex]; [latex]37[/latex]; [latex]41[/latex]; [latex]42[/latex]; [latex]47[/latex]; [latex]52[/latex]; [latex]55[/latex]; [latex]57[/latex]; [latex]58[/latex]; [latex]62[/latex]; [latex]64[/latex]; [latex]67[/latex]; [latex]69[/latex]; [latex]71[/latex]; [latex]72[/latex]; [latex]73[/latex]; [latex]74[/latex]; [latex]76[/latex]; [latex]77[/latex]

Calculate the [latex]20[/latex]th percentile and the [latex]55[/latex]th percentile.

NOTE

You can calculate percentiles using calculators and computers. There are a variety of online calculators.

A Formula for Finding the Percentile of a Value in a Data Set

  • Order the data from smallest to largest
  • [latex]x[/latex] = the number of data values counting from the bottom of the data list up to but not including the data value for which you want to find the percentile
  • [latex]y[/latex] = the number of data values equal to the data value for which you want to find the percentile
  • [latex]n[/latex] = the total number of data
  • Calculate [latex]\displaystyle\frac{{{x}+{0.5}{y}}}{{n}}{({100})}[/latex]. Then round to the nearest integer.

Example

Listed are [latex]29[/latex] ages for Academy Award winning best actors in order from smallest to largest.

[latex]18[/latex]; [latex]21[/latex]; [latex]22[/latex]; [latex]25[/latex]; [latex]26[/latex]; [latex]27[/latex]; [latex]29[/latex]; [latex]30[/latex]; [latex]31[/latex]; [latex]33[/latex]; [latex]36[/latex]; [latex]37[/latex]; [latex]41[/latex]; [latex]42[/latex]; [latex]47[/latex]; [latex]52[/latex]; [latex]55[/latex]; [latex]57[/latex]; [latex]58[/latex]; [latex]62[/latex]; [latex]64[/latex]; [latex]67[/latex]; [latex]69[/latex]; [latex]71[/latex]; [latex]72[/latex]; [latex]73[/latex]; [latex]74[/latex]; [latex]76[/latex]; [latex]77[/latex]

  1. Find the percentile for [latex]58[/latex].
  2. Find the percentile for [latex]25[/latex].

Try It

Listed are [latex]30[/latex] ages for New York Times published columnists in order from smallest to largest.

[latex]18[/latex]; [latex]21[/latex]; [latex]22[/latex]; [latex]25[/latex]; [latex]26[/latex]; [latex]27[/latex]; [latex]29[/latex]; [latex]30[/latex]; [latex]31[/latex], [latex]31[/latex]; [latex]33[/latex]; [latex]36[/latex]; [latex]37[/latex]; [latex]41[/latex]; [latex]42[/latex]; [latex]47[/latex]; [latex]52[/latex]; [latex]55[/latex]; [latex]57[/latex]; [latex]58[/latex]; [latex]62[/latex]; [latex]64[/latex]; [latex]67[/latex]; [latex]69[/latex]; [latex]71[/latex]; [latex]72[/latex]; [latex]73[/latex]; [latex]74[/latex]; [latex]76[/latex]; [latex]77[/latex]

Find the percentiles for [latex]47[/latex] and [latex]31[/latex].