Understanding Quartiles: Definitions, Calculations, and Examples

What is a quartile?

Quartiles are values that divide a sorted data set into four equal parts, each containing 25% of the observations. The three quartile values are:
* Q1 (first or lower quartile) — 25th percentile
* Q2 (second quartile) — 50th percentile (the median)
* Q3 (third or upper quartile) — 75th percentile

Quartiles summarize the distribution and show how values are spread around the center.

Why quartiles matter

Quartiles:
* Describe the spread and central tendency beyond the median.
* Help identify skewness by comparing distances from the median to Q1 and Q3.
* Are used to compute the interquartile range (IQR), a robust measure of variability that ignores extreme values.

Calculating quartiles in a spreadsheet

In most spreadsheets you can compute quartiles quickly:
* Median: =MEDIAN(range)
* Q1: =QUARTILE(range, 1)
* Q2 (median): =QUARTILE(range, 2)
* Q3: =QUARTILE(range, 3)

Example (19 scores in A2:R2):
* =MEDIAN(A2:R2) → 75
* =QUARTILE(A2:R2,1) → 68.25
* =QUARTILE(A2:R2,2) → 75
* =QUARTILE(A2:R2,3) → 81.75

Note: Different spreadsheet functions or versions (e.g., QUARTILE.INC vs QUARTILE.EXC) may use different interpolation methods and give slightly different results.

Manual calculation (position method)

A common manual method uses positions in the sorted list:
Position for Qk = (n + 1) × k / 4, where n is the number of observations and k = 1, 2, 3.

Example — sorted scores:
59, 60, 65, 65, 68, 69, 70, 72, 75, 75, 76, 77, 81, 82, 84, 87, 90, 95, 98 (n = 19; n + 1 = 20)
* Q1 position = 20 × 1/4 = 5 → Q1 = 5th value = 68
* Q2 position = 20 × 2/4 = 10 → Q2 = 10th value = 75
* Q3 position = 20 × 3/4 = 15 → Q3 = 15th value = 84

This method places quartiles at actual data positions; spreadsheet interpolation can yield fractional values between observations.

Interquartile range (IQR)

IQR = Q3 − Q1. It captures the middle 50% of the data and is less sensitive to outliers than the full range.
* Using the manual positions above: IQR = 84 − 68 = 16
* Using the spreadsheet interpolation example: IQR = 81.75 − 68.25 = 13.5

Important considerations

• Multiple legitimate methods exist for computing quartiles (different interpolation/inclusion rules). Expect small differences between software and manual methods.
• For even n, the median is the average of the two middle values; quartile calculations may split the data differently depending on the method.
• Quartile skewness: if |Q2−Q1| ≠ |Q3−Q2|, the distribution is asymmetric. Larger distance on one side indicates skew toward the opposite tail.

Quick answers

• How do I find the lower quartile? Use a spreadsheet: =QUARTILE(range, 1).
• How do I find the upper quartile? Use a spreadsheet: =QUARTILE(range, 3).
• What is the IQR? The range between Q3 and Q1; it represents the middle 50% of values.

Bottom line

Quartiles break a distribution into four parts and, together with the IQR, provide simple but powerful summaries of spread and skewness. Use spreadsheet functions for speed and be aware that different calculation conventions can produce slightly different quartile values.