# 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:

Theory

### InterquartileRange

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

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

Rule

### HowtoFind${Q}_{1}$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 is the lower group of data, the values above the median is 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

### TheSemi-InterquartileRange

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

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:

$\begin{array}{llll}\hfill & 3,4,5,6,6,12,12,12,\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & 12,13,13,14,14,15,15\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

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

### Range

You find the range from the formula above:

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 gives that the median is the 8th observation.
 $\text{Median}=12$
2.
Now you find ${Q}_{1}$ as the median in the lower group. There are 7 observations in the lower group which gives that the median is the 4th observation. (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 gives that the median is the 4th observation. (Count the fourth number starting from the highest end of the data set).
 ${Q}_{3}=14$
4.
At last, you calculate the interquartile range ${Q}_{3}-{Q}_{1}$.

The interquartile range is 8 years.

### Semi-interquartile range

Put into the formula of semi-interquartile range and calculate:

 $\frac{{Q}_{3}-{Q}_{1}}{2}=\frac{8}{2}=4$

The semi-interquartile range is 4 years.