Range, Interquartile and Semi-interquartile Ranges (Raw Data)

Range

The range of a set of numbers is the difference between the largest and the smallest number.

Example:

Calculate the range of the following numbers:

204, 210, 215, 220, 225, 234, 238, 240

The range

= the largest number – the smallest number

=  240 – 204

=  36

Ungrouped Frequency Table-Range

The range of a frequency distribution with ungrouped events is calculated using the formula below.

The range = the upper boundary limit of the largest event – the lower boundary of the smallest event

Example:

Find the range of the points in the table above.

Firstly identify the largest and smallest points.

Largest point = 13

Smallest point = 7

Find the upper boundary limit of the largest and the lower boundary limit of the smallest.

Upper boundary limit of 13 is, 13.5

Lower boundary limit of 7 is, 6.5

The range

= Upper boundary limit of 13 – lower boundary limit of 7

=             13.5 – 6.5

=             7

Quartiles

–Q₂ (the middle quartile) is the median.

–Q₁(the lower quartile) is the median of the numbers to the left of, or below Q₂.

–Q₃ (the upper quartile) is the median of the numbers to the right of, or above Q₂.

Example:

12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32

Find the lower, middle and upper quartiles of the data above.

Since the data is already in ascending order, identify the median.

12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32

22 is the median, therefore, Q₂= 22

The median of the numbers to the left of Q₂: 12, 14, 16, 18, 20

16 is the median, therefore, Q₁ = 16

The median of the numbers to the right of Q₂: 24, 26, 28, 30, 32

28 is the median, therefore, Q₃ = 28

Interquartile Range

The interquartile range of a distribution is the difference between the upper and lower quartiles.

That is, interquartile range = Q₃ – Q₁

Therefore using the example above, the interquartile range is:

Interquartile range = Q₃ – Q₁

Since,

Q₃ = 28

Q₁ = 16

Interquartile range

= 28 – 16

= 12

Semi-Interquartile Range

The semi-interquartile range of a distribution is half the difference between the upper and lower quartiles, or half the interquartile range.

Therefore, from the example above, it was determined that the interquartile range = 12.

Therefore,          semi-interquartile range