Quartiles

Quartiles

The collective term for the first quartile and third quartile of a set of data. That is, the 25th and 75th percentiles.

Worked Example

Problem: Find Q1, Q2, and Q3 for this data set: 3, 7, 8, 12, 15, 18, 21, 24, 28, 30, 35.

Step 1: Arrange the data in order (already done) and find the median (Q2). There are 11 values, so the median is the 6th value.

Q_2 = 18

Step 2: Find Q1: take the lower half of the data (all values below the median). The lower half is 3, 7, 8, 12, 15. The median of these 5 values is the 3rd value.

Q_1 = 8

Step 3: Find Q3: take the upper half of the data (all values above the median). The upper half is 21, 24, 28, 30, 35. The median of these 5 values is the 3rd value.

Q_3 = 28

Answer: Q1 = 8, Q2 = 18, Q3 = 28. The data is split into four groups: {3, 7}, {8, 12, 15}, {18, 21, 24}, {28, 30, 35}.

Another Example

Problem: Find the quartiles of this even-sized data set: 5, 10, 14, 20, 25, 30.

Step 1: Find Q2 (the median). With 6 values, the median is the average of the 3rd and 4th values.

Q_2 = \frac{14 + 20}{2} = 17

Step 2: Find Q1: the lower half is 5, 10, 14. The median of three values is the middle one.

Q_1 = 10

Step 3: Find Q3: the upper half is 20, 25, 30. The median of three values is the middle one.

Q_3 = 25

Answer: Q1 = 10, Q2 = 17, Q3 = 25.

Frequently Asked Questions

How do you find quartiles when the data set has an even number of values?

When the data set has an even number of values, the median (Q2) is the average of the two middle values. Then split the data into two equal halves — the lower half and the upper half — and find the median of each half to get Q1 and Q3.

What is the difference between quartiles and percentiles?

Quartiles are specific percentiles. Q1 is the 25th percentile, Q2 is the 50th percentile, and Q3 is the 75th percentile. Percentiles divide data into 100 equal parts, while quartiles divide it into just 4 parts. Every quartile is a percentile, but most percentiles are not quartiles.

Quartiles vs. Quintiles

Quartiles divide data into 4 equal parts using 3 cut points (Q1, Q2, Q3). Quintiles divide data into 5 equal parts using 4 cut points. Quartiles are far more commonly used in introductory statistics courses and appear in box plots and the five-number summary.

Why It Matters

Quartiles are the foundation of box-and-whisker plots, one of the most common ways to visualize the spread and skewness of data. The interquartile range (IQR = Q3 − Q1) measures how spread out the middle 50% of your data is, and it is used to identify outliers. Quartiles appear constantly in standardized test score reports, salary data, and medical research.

Common Mistakes

Mistake: Forgetting to sort the data before finding quartiles.

Correction: Quartiles only make sense for ordered data. Always arrange your values from smallest to largest before you begin.

Mistake: Including the median in both the lower and upper halves when finding Q1 and Q3 with an odd-sized data set.

Correction: When the data set has an odd number of values, the median itself is excluded from both halves. Find Q1 from the values below the median and Q3 from the values above it.

Related Terms

First Quartile — Q1, the 25th percentile boundary
Third Quartile — Q3, the 75th percentile boundary
Percentile — Divides data into 100 equal parts
Five Number Summary — Uses min, Q1, median, Q3, and max
Interquartile Range — Spread of the middle 50%, equals Q3 − Q1
Quintiles — Divides data into 5 equal parts
Deciles — Divides data into 10 equal parts