Quartile

Quartile – is a term used in statistics for dividing a set of observations into four more or less equal parts. Therefore, each part takes around 25% of information. There are a lower quartile and an upper one, as well as a median in the center, which divides the information into two equal parts.

Quartiles help to organize the information and define a central tendency or dispersion, which are integral parts of descriptive statistics. It complements the data given by the median.

Relations between the Quartile and the median

It’s important to pay attention to the term “median” first, and then explain the term “quartile”. As was mentioned before, these terms are correlated and even complement each other in some way.

Basically, the median divides a set of information by 50%. In other words, there is 50% of observations above the median and the same amount below it. Let’s illustrate with an example. If you have a set of numbers from 1 to 7, the median will be 4 (because there are 3 numbers before and after it):

1, 2, 3, 4, 5, 6, 7 

The median is more frequently used in statistics than a mean, because it’s more stable and representative. It shows a more realistic distribution of observations, rather than the mean. The latter is another term of statistics that demonstrates the average value within a dataset. Let’s also illustrate it. For example, you have the same 7 numbers. The mean will be calculated the following way:

(1+2+3+4+5+6+7) / 7 = 4

In these examples, the mean and the median are equal, but it’s not always happen like this. Let’s consider another example as well.

We have the following set of numbers: 13, 15, 19, 38, 44

The median is 19, but the mean is 25.8.

Now, when the term “median” is more clear, we can define its relation with quartiles. Actually, while the median is a measure of strict division by 50%, the quartile is a measure of the information’s distribution around it. Otherwise, it gives us a more particular analysis of the data and shows a tendency by dividing it above and below the median into four groups altogether. Overall, there are three quartiles – a lower one (Q1), the median itself (Q2), and the one above it, or an upper one (Q3).

Characteristics of the Quartiles and intervals between them

Let’s summarize some of the main characteristics of the quartiles:

• There are three quartiles (Q1, Q2, Q3), which divide the observations into four groups, or intervals.
• Each interval contains around 25% of observations, therefore, the data is equally distributed between all of them.
• The second quartile (Q2) is a median, which divides the information into two equal parts.

Also, let’s pay some attention to the intervals that were mentioned before. As we said, there are four of them:

1. The first interval lies between the lowest point of the dataset and the first quartile.
2. The second interval is between the first quartile and the median.
3. The third interval is between the median and the third quartile.
4. The fourth interval is between the third quartile and up to the top.

Example of the Quartile

As an example, we might consider prices for the same product in different shops. Imagine that we compare five shops and get the following set of data (starting with the lowest price):

$2.3, $2.5, $2.9, $3.4, $5

First, we may determine a median. It is $2.9, because it divides the prices into two equal groups. However, if we take an even set of numbers, the median will be counted another way. We just have to count the arithmetic average of the two numbers in the middle. Basically, we have to sum up these two numbers and divide them by two:

($2.9 + $3.4) / 2 = $3.15

So, $3.15 will be the median if we gather prices from six shops. But let’s go back to our example, where we have five shops to compare.

A lower quartile lies in the middle of the lowest price and the median. Therefore, Q1 = $2.5. We count an upper quartile the same way as the lower one. Consequently, Q3 = $4, as the middle price between the median and the highest price in our set. 

When all the quartiles are defined, we might analyze the data:

• Q1 = $2.5, which means that there are 25% of shops where the price of the particular product is below $2.5.
• Q2 = $2.9, which means that 50% of shops’ prices are below $2.9, and 50% of shops where the price for the same product is above $2.9.
• Q3 = $3.4, which means that in 75% of shops, the price for the product is lower than $3.4.

In order to determine the interquartile range, we have to subtract Q1 from Q3:

$3.4 - $2.5 = $0.9

We consider a simplified example, that usually doesn’t look like real cases. Therefore, different programs for analyzing databases, and calculating quartiles specifically, were developed. For instance, you even can count quartiles in Microsoft Excel by using a QUARTILE function.