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2.2 Measures of the location of the data  (Page 3/12)

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Forty bus drivers were asked how many hours they spend each day running their routes (rounded to the nearest hour). Find the 65 th percentile.

Amount of time spent on route (hours) Frequency Relative Frequency Cumulative Relative Frequency
2 12 0.30 0.30
3 14 0.35 0.65
4 10 0.25 0.90
5 4 0.10 1.00

The 65 th percentile is between the last three and the first four.

The 65 th percentile is 3.5.

Using [link] :

  1. Find the 80 th percentile.
  2. Find the 90 th percentile.
  3. Find the first quartile. What is another name for the first quartile?

Using the data from the frequency table, we have:

  1. The 80 th percentile is between the last eight and the first nine in the table (between the 40 th and 41 st values). Therefore, we need to take the mean of the 40 th an 41 st values. The 80 th percentile = 8 + 9 2 = 8.5
  2. The 90 th percentile will be the 45 th data value (location is 0.90(50) = 45) and the 45 th data value is nine.
  3. Q 1 is also the 25 th percentile. The 25 th percentile location calculation: P 25 = 0.25(50) = 12.5 ≈ 13 the 13 th data value. Thus, the 25th percentile is six.

A formula for finding the k Th percentile

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

k = the k th percentile. It may or may not be part of the data.

i = the index (ranking or position of a data value)

n = the total number of data points, or observations

  • Order the data from smallest to largest.
  • Calculate i = k 100 ( n + 1 )
  • If i is an integer, then the k th percentile is the data value in the i th position in the ordered set of data.
  • If i is not an integer, then round i up and round i 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.

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

  1. Find the 70 th percentile.
  2. Find the 83 rd percentile.
    • k = 70
    • i = the index
    • n = 29
    i = k 100 ( n + 1) = ( 70 100 )(29 + 1) = 21. Twenty-one is an integer, and the data value in the 21 st position in the ordered data set is 64. The 70 th percentile is 64 years.
    • k = 83 rd percentile
    • i = the index
    • n = 29
    i  = k 100 ( n + 1) = ) 83 100 )(29 + 1) = 24.9, which is NOT an integer. Round it down to 24 and up to 25. The age in the 24 th position is 71 and the age in the 25 th position is 72. Average 71 and 72. The 83 rd percentile is 71.5 years.

Try it

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

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
Calculate the 20 th percentile and the 55 th percentile.

k = 20. Index = i = k 100 ( n + 1 ) = 20 100 (29 + 1) = 6. The age in the sixth position is 27. The 20 th percentile is 27 years.

k = 55. Index = i = k 100 ( n + 1 ) = 55 100 (29 + 1) = 16.5. Round down to 16 and up to 17. The age in the 16 th position is 52 and the age in the 17 th position is 55. The average of 52 and 55 is 53.5. The 55 th percentile is 53.5 years.

A formula for finding the percentile of a value in a data set

  • Order the data from smallest to largest.
  • x = 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.
  • y = the number of data values equal to the data value for which you want to find the percentile.
  • n = the total number of data.
  • Calculate x + 0.5 y n (100). Then round to the nearest integer.
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Source:  OpenStax, Introductory statistics. OpenStax CNX. Aug 09, 2016 Download for free at http://legacy.cnx.org/content/col11776/1.26
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