Quartile Calculator

What Are Quartiles?

Quartiles divide a sorted dataset into four equal parts. Each part contains 25% of the data. They are a fundamental tool in descriptive statistics for understanding how data is distributed.

• Q1 (First Quartile / 25th percentile): 25% of data falls below this value
• Q2 (Second Quartile / Median / 50th percentile): The middle value; 50% falls below
• Q3 (Third Quartile / 75th percentile): 75% of data falls below this value

How to Calculate Quartiles

Step 1: Sort the data from smallest to largest. Step 2: Find Q2 (the median of the full dataset). Step 3: Find Q1 (the median of the lower half, not including Q2 if N is odd). Step 4: Find Q3 (the median of the upper half, not including Q2 if N is odd).

Interquartile Range (IQR)

IQR = Q3 − Q1

The IQR is the range of the middle 50% of the data. It is a robust measure of spread — unlike range or standard deviation, it is not affected by extreme outliers.

The Five-Number Summary

The five-number summary provides a complete picture of a distribution:

1. Minimum — smallest value (excluding outliers)
2. Q1 — 25th percentile
3. Median (Q2) — 50th percentile
4. Q3 — 75th percentile
5. Maximum — largest value (excluding outliers)

These five values define a box plot (box-and-whisker plot).

Identifying Outliers: Tukey’s Method

Any value outside the following fences is considered an outlier:

Lower Fence = Q1 − 1.5 × IQR Upper Fence = Q3 + 1.5 × IQR

Values beyond 3.0 × IQR are called extreme outliers. John Tukey introduced this method in 1977.

Why Use Quartiles Instead of Mean/Standard Deviation?

Quartiles are robust to outliers. If your dataset includes a few extreme values (like income data, where billionaires skew the mean), the median and IQR give a more representative picture of the typical value than the mean and standard deviation.

Real-World Uses

• Test scores: Find what score is in the top 25% (above Q3)
• Income data: Median income is far more informative than mean income
• Stock returns: IQR shows the typical range of variation without outlier distortion
• Quality control: Track whether manufacturing measurements stay within quartile boundaries
• Medical data: Reference ranges in blood tests are often expressed as percentiles

Worked Example

Data: 4, 7, 8, 9, 12, 15, 18, 22, 23, 30

• Sorted: 4, 7, 8, 9, 12, 15, 18, 22, 23, 30
• Q2 (median): (12 + 15) / 2 = 13.5
• Q1 (median of lower half: 4,7,8,9,12): 8
• Q3 (median of upper half: 15,18,22,23,30): 22
• IQR = 22 − 8 = 14
• Fences: Lower = 8 − 21 = −13 | Upper = 22 + 21 = 43 — no outliers