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Descriptive Statistics: Measuring the Location of Data explains percentiles and quartiles and is part of the collection col10555 written by Barbara Illowsky and Susan Dean. Roberta Bloom contributed the section "Interpreting Percentiles, Quartile and the Median."

The common measures of location are quartiles and percentiles (%iles). Quartiles are special percentiles. The first quartile, Q 1 is the same as the 25th percentile (25th %ile) and the third quartile, Q 3 , is the same as the 75th percentile (75th %ile). The median, M , is called both the second quartile and the 50th percentile (50th %ile).

Quartiles are given special attention in the Box Plots module in this chapter.

To calculate quartiles and percentiles, the data must be ordered from smallest to largest. Recall that quartiles divide ordered data into quarters. Percentiles divide ordered data into hundredths. To score in the 90th percentile of an exam does not mean, necessarily, that you received 90% on a test. It means that 90% of test scores are the same or less than your score and 10% of the test scores are the same or greater than your test score.

Percentiles are useful for comparing values. For this reason, universities and colleges use percentiles extensively.

Percentiles are mostly used with very large populations. Therefore, if you were to say that 90% of the test scores are less (and not the same or less) than your score, it would be acceptable because removing one particular data value is not significant.

The interquartile range is a number that indicates the spread of the middle half or the middle 50% of the data. It is the difference between the third quartile ( Q 3 ) and the first quartile ( Q 1 ).

IQR = Q 3 - Q 1

The IQR can help to determine potential outliers . A value is suspected to be a potential outlier if it is less than (1.5)(IQR) below the first quartile or more than (1.5)(IQR) above the third quartile . Potential outliers always need further investigation.

For the following 13 real estate prices, calculate the IQR and determine if any prices are outliers. Prices are in dollars. ( Source: San Jose Mercury News )

  • 389,950
  • 230,500
  • 158,000
  • 479,000
  • 639,000
  • 114,950
  • 5,500,000
  • 387,000
  • 659,000
  • 529,000
  • 575,000
  • 488,800
  • 1,095,000

Order the data from smallest to largest.

  • 114,950
  • 158,000
  • 230,500
  • 387,000
  • 389,950
  • 479,000
  • 488,800
  • 529,000
  • 575,000
  • 639,000
  • 659,000
  • 1,095,000
  • 5,500,000

M = 488,800

Q 1 = 230500 + 387000 2 = 308750

Q 3 = 639000 + 659000 2 = 649000

IQR = 649000 - 308750 = 340250

( 1.5 ) ( IQR ) = ( 1.5 ) ( 340250 ) = 510375

Q 1 - ( 1.5 ) ( IQR ) = 308750 - 510375 = - 201625

Q 3 + ( 1.5 ) ( IQR ) = 649000 + 510375 = 1159375

No house price is less than -201625. However, 5,500,000 is more than 1,159,375. Therefore, 5,500,000 is a potential outlier .

For the two data sets in the test scores example , find the following:

  • The interquartile range. Compare the two interquartile ranges.
  • Any outliers in either set.
  • The 30th percentile and the 80th percentile for each set. How much data falls below the 30th percentile? Above the 80th percentile?

For the IQRs, see the answer to the test scores example . The first data set has the larger IQR, so the scores between Q3 and Q1 (middle 50%) for the first data set are more spread out and not clustered about the median.

First data set

  • ( 3 2 )  ⋅  ( IQR )  =  ( 3 2 )  ⋅  ( 26.5 )  =  39.75
  • Xmax  -  Q3  =  99  -  82.5  =  16.5
  • Q1  -  Xmin  =  56  -  32  =  24
( 3 2 ) ( IQR ) = 39.75 is larger than 16.5 and larger than 24, so the first set has no outliers.

Second data set

  • ( 3 2 ) ( IQR ) = ( 3 2 ) ( 11 ) = 16.5
  • Xmax - Q3 = 98 - 89 = 9
  • Q1 - Xmin = 78 - 25.5 = 52.5
( 3 2 ) ( IQR ) = 16.5 is larger than 9 but smaller than 52.5, so for the second set 45 and 25.5 are outliers.

To find the percentiles, create a frequency, relative frequency, and cumulative relative frequency chart (see "Frequency" from the Sampling and Data Chapter ). Get the percentiles from that chart.

    First data set

  • 30th %ile (between the 6th and 7th values)  =  ( 56  +  59 ) 2  =  57.5
  • 80th %ile (between the 16th and 17th values)  =  ( 84  +  84.5 ) 2  =  84.25

    Second data set

  • 30th %ile (7th value) = 78
  • 80th %ile (18th value) = 90

30% of the data falls below the 30th %ile, and 20% falls above the 80th %ile.

Practice Key Terms 4

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Source:  OpenStax, Collaborative statistics-parzen remix. OpenStax CNX. Jul 15, 2009 Download for free at http://legacy.cnx.org/content/col10732/1.2
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