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Find the interquartile range for the following two data sets and compare them.

Test Scores for Class A
69; 96; 81; 79; 65; 76; 83; 99; 89; 67; 90; 77; 85; 98; 66; 91; 77; 69; 80; 94
Test Scores for Class B
90; 72; 80; 92; 90; 97; 92; 75; 79; 68; 70; 80; 99; 95; 78; 73; 71; 68; 95; 100

Class A

Order the data from smallest to largest.

  • 65
  • 66
  • 67
  • 69
  • 69
  • 76
  • 77
  • 77
  • 79
  • 80
  • 81
  • 83
  • 85
  • 89
  • 90
  • 91
  • 94
  • 96
  • 98
  • 99

Median = 80 + 81 2 = 80.5

Q 1 = 69 + 76 2 = 72.5

Q 3 = 90 + 91 2 = 90.5

IQR = 90.5 – 72.5 = 18

Class B

Order the data from smallest to largest.

  • 68
  • 68
  • 70
  • 71
  • 72
  • 73
  • 75
  • 78
  • 79
  • 80
  • 80
  • 90
  • 90
  • 92
  • 92
  • 95
  • 95
  • 97
  • 99
  • 100

Median = 80 + 80 2 = 80

Q 1 = 72 + 73 2 = 72.5

Q 3 = 92 + 95 2 = 93.5

IQR = 93.5 – 72.5 = 21

The data for Class B has a larger IQR , so the scores between Q 3 and Q 1 (middle 50%) for the data for Class B are more spread out and not clustered about the median.

Fifty statistics students were asked how much sleep they get per school night (rounded to the nearest hour). The results were:

AMOUNT OF SLEEP PER SCHOOL NIGHT (HOURS) FREQUENCY RELATIVE FREQUENCY CUMULATIVE RELATIVE FREQUENCY
4 2 0.04 0.04
5 5 0.10 0.14
6 7 0.14 0.28
7 12 0.24 0.52
8 14 0.28 0.80
9 7 0.14 0.94
10 3 0.06 1.00

Find the 28 th percentile . Notice the 0.28 in the "cumulative relative frequency" column. Twenty-eight percent of 50 data values is 14 values. There are 14 values less than the 28 th percentile. They include the two 4s, the five 5s, and the seven 6s. The 28 th percentile is between the last six and the first seven. The 28 th percentile is 6.5.

Find the median . Look again at the "cumulative relative frequency" column and find 0.52. The median is the 50 th percentile or the second quartile. 50% of 50 is 25. There are 25 values less than the median. They include the two 4s, the five 5s, the seven 6s, and eleven of the 7s. The median or 50 th percentile is between the 25 th , or seven, and 26 th , or seven, values. The median is seven.

Find the third quartile . The third quartile is the same as the 75 th percentile. You can "eyeball" this answer. If you look at the "cumulative relative frequency" column, you find 0.52 and 0.80. When you have all the fours, fives, sixes and sevens, you have 52% of the data. When you include all the 8s, you have 80% of the data. The 75 th percentile, then, must be an eight . Another way to look at the problem is to find 75% of 50, which is 37.5, and round up to 38. The third quartile, Q 3 , is the 38 th value, which is an eight. You can check this answer by counting the values. (There are 37 values below the third quartile and 12 values above.)

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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.

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Source:  OpenStax, Statistics i - math1020 - red river college - version 2015 revision a - draft 2015-10-24. OpenStax CNX. Oct 24, 2015 Download for free at http://legacy.cnx.org/content/col11891/1.8
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