Inter Quartile Range for ungrouped data

Inter quartile range is given by

IQR=Q3Q1

where

  • Q1 is the first quartile
  • Q3 is the third quartile

The formula for ith quartile is

Qi= Value of (i(n+1)4)th observation, i=1,2,3

where n is the total number of observations.

Example 1

A random sample of 15 patients yielded the following data on the length of stay (in days) in the hospital.

5, 6, 9, 10, 15, 10, 14, 12, 10, 13, 13, 9, 8, 10, 12.

Find quartiles.

Solution

The formula for ith quartile is

Qi= Value of (i(n+1)4)th observation, i=1,2,3

where n is the total number of observations.

Arrange the data in ascending order

5, 6, 8, 9, 9, 10, 10, 10, 10, 12, 12, 13, 13, 14, 15

First Quartile Q1

The first quartle Q1 can be computed as follows:

Q1=Value of (1(n+1)4)th observation=Value of (1(15+1)4)th observation=Value of (4)th observation=9 days.

Thus, 25 % of the patients had length of stay in the hospital less than or equal to 9 days.

Third Quartile Q3

The third quartile Q3 can be computed as follows:

Q3=Value of (3(n+1)4)th observation=Value of (3(15+1)4)th observation=Value of (12)th observation=13 days. Thus, 75 % of the patients had length of stay in the hospital less than or equal to 13 days.

Inter-Quartile Range

The inter-quartile range is

IQR=Q3Q1=139=4 days.

Example 2

Blood sugar level (in mg/dl) of a sample of 20 patients admitted to the hospitals are as follows:

75,89,72,78,87, 85, 73, 75, 97, 87, 84, 76,73,79,99,86,83,76,78,73.

Find the value of Q1, Q2 and Q3.

Solution

The formula for ith quartile is

Qi= Value of (i(n+1)4)th observation, i=1,2,3

where n is the total number of observations.

Arrange the data in ascending order

72, 73, 73, 73, 75, 75, 76, 76, 78, 78, 79, 80, 82, 83, 84, 85, 86, 87, 97, 99

First Quartile Q1

The first quartle Q1 can be computed as follows:

Q1=Value of (1(n+1)4)th observation=Value of (1(20+1)4)th observation= Value of (5.25)th observation=Value of (5)th obs.+0.25(Value of (6)th obs.Value of (5)th obs.)=75+0.25(7575)=75 mg/dl. Thus, 25 % of the patients had blood sugar level less than or equal to 75 mg/dl.

Third Quartile Q3

The third quartile Q3 can be computed as follows:

Q3=Value of (3(n+1)4)th observation=Value of (3(20+1)4)th observation= Value of (15.75)th observation=Value of (15)th obs.+0.75(Value of (16)th obs.Value of (15)th obs.)=84+0.75(8584)=84.75 mg/dl.

Thus, 75 % of the patients had blood sugar level less than or equal to 84.75 mg/dl.

Inter-Quartile Range

The inter-quartile range is

IQR=Q3Q1=84.7575=9.75 mg/dl.

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