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

Range

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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

-**Q**_{2} (the middle quartile) is the median.

-**Q**_{1} (the lower quartile) is the median of the numbers to the left of, or below Q_{2}.

-**Q**_{3} (the upper quartile) is the median of the numbers to the right of, or above Q_{2}.

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, Q_{2}= 22

The median of the numbers to the left of Q_{2}: 12, 14, **16**, 18, 20

16 is the median, therefore, Q_{1} = 16

The median of the numbers to the right of Q_{2}: 24, 26, **28**, 30, 32

28 is the median, therefore, Q_{3} = 28

Interquartile Range

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

That is, interquartile range = Q_{3} – Q_{1}

Therefore using the example above, the interquartile range is:

Interquartile range = Q_{3} – Q_{1}

Since,

Q_{3} = 28

Q_{1} = 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