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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:

range = highest value lowest value

range = highest value lowest value

Theory

Interquartile Range

The interquartile range tells us the difference between the first quartile Q1 and third quartile Q3 of the data:

Interquartile range = third quartile first quartile = Q3 Q1

Interquartile range = third quartile first quartile = Q3 Q1

How do you find Q1 and Q3? You find them like this:

Rule

How to Find Q1 and Q3

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 Q1.
3.
Finally, you find the median value of the upper group. This median represents Q3.

Theory

The Semi-Interquartile Range

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

Q3 Q1 2 = interquartile range 2

Semi-interquartile range = Q3 Q1 2 = interquartile range 2

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:

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

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 gives that the median is the 8th observation.
Median = 12
2.
Now you find Q1 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).
Q1 = 6
3.
Now you find Q3 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).
Q3 = 14
4.
At last, you calculate the interquartile range Q3 Q1. Q3 Q1 = 14 6 = 8 years

The interquartile range is 8 years.

Semi-interquartile range

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

Q3 Q1 2 = 8 2 = 4

The semi-interquartile range is 4 years.

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