Lower and upper quartile

Your observation must be sorted with the smallest number first.

If we split our data set in the middle, we get 5 observations on each side.

 

 

Lower observations:

 

Name	Height
Benjamin	156 cm
Charlotte	158 cm
Mia	160 cm
Emma	160 cm
Sophia	161 cm

 

 

Upper observations:

 

Name	Height
Ava	167 cm
Lucas	168 cm
William	170 cm
Oliver	172 cm
James	176 cm

 

 

Lower quartile

If you take the lower observations, you can find the lower quartile right in the middle. In this case the lower quartile is 160 (Mia).

25% of the observations are less than or equal to the lower quartile.

 

Upper quartile

If you take the upper observations, you can find the upper quartile right in the middle. In this case the upper quartile is 170 (William).

75% of the observations are less than or equal to the upper quartile.

 

Quartile set

A quartile set consists of three numbers: upper quartile, median, and lower quartile.

In our example, the quartile set is:

• Upper quartile = 170 cm
• Median = 164 cm
• Lower quartile = 160 cm

 

If there is no number in the middle

If there is no number in the middle. It could be that there are 6 numbers on each side of the median and the upper observations then look like this:

 

Name	Height
Ava	167 cm
Lucas	168 cm
William	170 cm
Oliver	172 cm
Emily	174 cm
James	176 cm

 

To find the upper quartile, we must take the average of the third and fourth observation (William and Oliver):

 

$$ Upper\ quartile=\frac{170+172}{2}=171 $$

 

The upper quartile is equal to 171

 

Percentiles

Quartiles are special percentiles. The lower quartile corresponds to the 25 th percentile, the median to the 50 th percentile, and the upper quartile to the 75 th percentile.

In general, any percentile can be found, for example the 90 th percentile, which is the value where 90 % of the observations are less than or equal to it.

Percentiles are often used to describe the distribution in larger data sets, for example in test results or health surveys.