Quartile Calculator

Quartiles divide a sorted dataset into four equal parts.

• Q1 (First Quartile) — 25th percentile: 25% of values fall below this point
• Q2 (Second Quartile / Median) — 50th percentile: the middle value
• Q3 (Third Quartile) — 75th percentile: 75% of values fall below this point

IQR (Interquartile Range): IQR = Q3 − Q1

The IQR measures the spread of the middle 50% of your data. A large IQR means data is spread out; a small IQR means it is tightly clustered.

Outlier detection — Tukey Fences: Lower fence = Q1 − 1.5 × IQR Upper fence = Q3 + 1.5 × IQR

Any value outside these fences is considered an outlier. Values beyond Q1 − 3×IQR or Q3 + 3×IQR are called extreme outliers.

How quartiles are calculated (exclusive method):

1. Sort the data in ascending order
2. Q2 = median of the full dataset
3. For even n: lower half = first n/2 values, upper half = last n/2 values
4. For odd n: exclude the median value, then find medians of each half
5. Q1 = median of the lower half; Q3 = median of the upper half

Box-and-whisker plot: A box plot visualises the 5-number summary: Min, Q1, Median, Q3, Max. The box spans Q1 to Q3 (the IQR). Whiskers extend to the min/max within the fences. Points outside the fences are plotted individually as outliers.

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

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

• Q2 (median of 10 values): (12 + 15) / 2 = 13.5
• Lower half: 4, 7, 8, 9, 12 → Q1 = 8
• Upper half: 15, 18, 22, 23, 30 → Q3 = 22
• IQR = 22 − 8 = 14
• Fences: −13 to 43 — no outliers in this dataset

Real-world uses:

• Test scores: find what score puts someone in the top 25%
• Income data: median income is more informative than mean income
• Stock returns: IQR shows typical variation without outlier distortion
• Medical data: reference ranges in blood tests are often expressed as percentiles