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2.1 Measures of the location of the data  (Page 4/15)

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Try it

Refer to the [link] . Find the third quartile. What is another name for the third quartile?

The third quartile is the 75 th percentile, which is four. The 65 th percentile is between three and four, and the 90 th percentile is between four and 5.75. The third quartile is between 65 and 90, so it must be four.

Collaborative statistics

Your instructor or a member of the class will ask everyone in class how many sweaters they own. Answer the following questions:

  1. How many students were surveyed?
  2. What kind of sampling did you do?
  3. Construct two different histograms. For each, starting value = _____ ending value = ____.
  4. Find the median, first quartile, and third quartile.
  5. Construct a table of the data to find the following:
    1. the 10 th percentile
    2. the 70 th percentile
    3. the percent of students who own less than four sweaters

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

  • 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.

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.
  • 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, Statistics 1. OpenStax CNX. Feb 24, 2014 Download for free at http://legacy.cnx.org/content/col11633/1.1
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