Interquartile Range and Semi-interquartile Range

The interquartile range is connected to the range of data. It’s sometimes called the fourth spread.

Here is a short reminder about range:

Theory

Range

The range tells us the difference between the highest and lowest values in a data set:

\begin{array}{l} range & = highest value \\ - lowest value \end{array}

range = highest value - lowest value

Theory

Interquartile Range

The interquartile range tells us the difference between the first quartile, $Q_{1}$ , and third quartile, $Q_{3}$ , of the data:

\begin{array}{l} Interquartile range & = third quartile \\ - first quartile \\ = Q_{3} - Q_{1} \end{array}

\begin{array}{l} Interquartile range & = third quartile - first quartile \\ = Q_{3} - Q_{1} \end{array}

How do you find $Q_{1}$ and $Q_{3}$ ? You find them like this:

Rule

How to Find $Q_{1}$ and $Q_{3}$

1.: First you order the values in your data set from low to high. Then you find the median value of this data set. Let this median value split the data set into two subgroups: The values below the median comprise the lower group of data, and the values above the median make up the upper group of data.
2.: Then, you find the median value of the lower group. This median represents $Q_{1}$ .
3.: Finally, you find the median value of the upper group. This median represents $Q_{3}$ .

Theory

The Semi-Interquartile Range

The semi-interquartile range measures half of the interquartile range:

\frac{Q_{3} - Q_{1}}{2} = \frac{interquartile range}{2}

\begin{array}{l} Semi-interquartile range & = \frac{Q_{3} - Q_{1}}{2} \\ = \frac{interquartile range}{2} \end{array}

Example 1

The age of a group of young gymnasts are 4, 5, 6, 3, 6, 12, 12, 14, 15, 13, 12, 12, 13, 14 and 15. Find the range, the interquartile range, and the semi-interquartile range.

First you arrange the data in ascending order:

\begin{array}{l} 3, 4, 5, 6, 6, 12, 12, 12, \\ 12, 13, 13, 14, 14, 15, 15 \end{array}

3, 4, 5, 6, 6, 12, 12, 12, 12, 13, 13, 14, 14, 15, 15

Range

You find the range from the formula above:

range = 15 - 3 = 12 years

The range is 12 years.

Interquartile range

This is how you find the interquartile range:

1.

First you find the median of the data set. There are 15 observations, which means that the median is the 8^th observation.

Median = 12

2.

Now you find

Q_{1}

as the median in the lower group. There are 7 observations in the lower group which means that the median is the 4^th observation. (You count the fourth number starting from the lowest end of the data set).

Q_{1} = 6

3.

Now you find

Q_{3}

as the median in the upper group. There are 7 observations in the lower group which means that the median is the 4^th observation. (You count the fourth number starting from the highest end of the data set).