Understanding Quartiles: Definitions, Calculations, and Examples

What Is a Quartile?

Quartiles are statistical measures that divide a data set into four equal parts, each representing 25% of the observations. By arranging data points in increasing order, you can identify three quartile values: the lower quartile, median quartile, and upper quartile. These measures help to analyze the spread and distribution of data, offering insight into how individual values compare to the overall dataset.

The Role of Quartiles in Data Analysis

To understand the quartile, it's important to understand the median as a measure of central tendency. The median in statistics is the middle value of a set of numbers. It's the point at which exactly half the data lies below and above the central value.

The median is a robust estimator of location, but it says nothing about how the data on either side of its value is spread or dispersed. That's where the quartile steps in. The quartile measures the spread of values above and below the median by dividing the distribution into four groups. They're grouped into four sections of 25% of the data, with the second and third groups representing the interquartile range.

The median splits the data in half, with 50% below and 50% above. Similarly, quartiles divide data into quarters: 25% below the lower quartile, 50% below the median, and 75% below the upper quartile.

There are three quartile values: a lower quartile, a median, and an upper quartile. They divide the data set into four ranges, each containing 25% of the data points:

• First quartile: The set of data points between the minimum value and the first quartile.
• Second quartile: The set of data points between the lower quartile and the median.
• Third quartile: The set of data between the median and the upper quartile.
• Fourth quartile: The set of data points between the upper quartile and the maximum value of the data set.

How to Calculate Quartiles Using a Spreadsheet

Suppose you have a distribution of math scores in a class of 19 students. You'd want to enter them into a spreadsheet in ascending order in a row. You could also use a column:

Use the MEDIAN function to get the median value:

• =MEDIAN (A2:R2)

Then use the quartile function to return the values for each quartile, where the second variable in the function is the quartile you're calculating for:

• =QUARTILE (A2:R2, 1)
• =QUARTILE (A2:R2, 2)
• =QUARTILE (A2:R2, 3)

You should end up with the values for each quartile. There's no need to calculate the fourth quartile because it's the last value in your dataset:

• Median = 75
• Q1 = 68.25
• Q2 = 75
• Q3 = 81.75
  

You can see that the first quartile contains scores between 59 and 68.5, and the second quartile contains scores between 68.5 and 75. The third quartile contains scores between 75 and 81.75. It can help to visualize it:

Manual Method for Calculating Quartiles

Quartile manual calculation requires more effort because there are formulas involved. Using the same values as in the spreadsheet example would look like this:

• 59, 60, 65, 65, 68, 69, 70, 72, 75, 75, 76, 77, 81, 82, 84, 87, 90, 95, 98

You can calculate each quartile using the following formula:

• First Quartile (Q1) = (n + 1) x 1/4
• Second Quartile (Q2), or the median = (n + 1) x 2/4
• Third Quartile (Q3) = (n + 1) x 3/4

n is the number of integers in your dataset, and the result is the position of the number in the sequence dataset. So:

• First Quartile (Q1) = 20 x 1/4 = 5
• Second Quartile (Q2) = 20 x 2/4 = 10
• Third Quartile (Q3) = 20 x 3/4 = 15

Here, we have the Q1 (fifth) value of 68, the Q2 (tenth and the median) value of 75, and the Q3 (fifteenth) value of 84. The results differ slightly from the spreadsheet results because the spreadsheet calculates them differently. Your graph would then look like this:

Quartiles help compute the interquartile range, indicating variability around the median. It’s the range between the first and third quartiles.

You'd have an interquartile range of 68 to 84 in this example, the fifth value to the tenth value in the dataset.

Important Considerations in Quartile Analysis

You can say there's a greater dispersion among the smaller values of the dataset than among the larger values if the data point for Q1 is farther away from the median than Q3 is from the median. The same logic applies if Q3 is farther away from Q2 than Q1 is from the median. This is referred to as quartile skewness.

Another aspect to consider is whether there's an even number of data points. You'd use the average of the middle two numbers to get the median in this case. The median of their scores would be the arithmetic average of the tenth and eleventh numbers in the above example if you had 20 students instead of 19.

The Bottom Line

Quartiles are values that split lists of datasets into quarters, resulting in lower, middle, and upper segments. The purpose of quartiles is to give shape to a distribution, primarily indicating whether a distribution is skewed, which can be used to determine the consistency of a fund's performance.