Calculate Q1, Q2 (median), Q3, IQR, midhinge, trimean, and five-number summary using 3 quartile methods. Box plot, method comparison, and decile table included.
The quartile calculator divides your data into four equal parts using Q1 (25th percentile), Q2 (median), and Q3 (75th percentile). Quartiles are the foundation of the five-number summary, box plots, and the interquartile range (IQR) — essential tools for understanding data distribution.
This tool computes quartiles using three standard methods (exclusive/Tukey, inclusive, and interpolated/Excel), shows a method comparison table, generates a box plot, and provides derived measures including IQR, semi-IQR, midhinge, trimean, and quartile coefficient of dispersion.
Enter your data, select a quartile method, and get a complete positional analysis with box plot and decile table. Check the example with realistic values before reporting. Use the steps shown to verify rounding and units. Cross-check this output using a known reference case. Use the example pattern when troubleshooting unexpected results. Validate that outputs match your chosen standards. Check the example with realistic values before reporting. Use the steps shown to verify rounding and units.
This calculator provides all three standard quartile methods side-by-side, so you can see exactly how they differ for your data. With the five-number summary, box plot, derived measures (IQR, midhinge, trimean, QCD), and a full decile table, it's the most comprehensive quartile analysis available online.
Whether you're constructing a box plot, computing IQR for outlier detection, or comparing quartile methods for a homework problem, this tool gives you everything in one place.
Exclusive: Q1 = median of lower half (median excluded). Inclusive: Q1 = median of lower half (median included). Interpolated: Q1 = value at rank 0.25 × (n−1). IQR = Q3 − Q1. Midhinge = (Q1 + Q3) / 2. Trimean = (Q1 + 2 × Median + Q3) / 4.
Result: Q1 = 73, Median = 82.5, Q3 = 89, IQR = 16
Sorted: 68,70,72,74,76,81,84,85,88,90,92,95. Lower half: 68,70,72,74,76,81, median = 73. Upper half: 84,85,88,90,92,95, median = 89. IQR = 89 − 73 = 16.
Hyndman and Fan (1996) documented nine different methods for computing sample quantiles, including quartiles. They differ in three ways: (1) whether the data is treated as continuous or discrete, (2) how interpolation is performed between adjacent order statistics, and (3) how the plotting position formula works. Most software uses method 7 (Excel) or method 6 (Minitab) by default.
John Tukey's "Exploratory Data Analysis" (1977) built an entire analytical framework around quartiles. The letter-value plot extends box plots using additional quantiles (eighths, sixteenths) for large datasets. The letter values (median, fourths, eighths, sixteenths) give increasingly detailed views of a distribution's shape — particularly useful for detecting long tails.
Bowley skewness = (Q3 + Q1 − 2 × Median) / (Q3 − Q1) measures asymmetry using only quartiles. It ranges from −1 to +1, with 0 indicating symmetry. Unlike Fisher skewness (based on the third moment), Bowley skewness is robust to outliers. A positive value means the upper quartile is farther from the median than the lower quartile.
Quartiles divide sorted data into four equal groups. Q1 (first quartile) is the 25th percentile — 25% of values fall below it. Q2 is the median (50th percentile). Q3 (third quartile) is the 75th percentile — 75% of values fall below it. Together with min and max, they form the five-number summary.
There are at least 9 different methods for computing quartiles (Hyndman & Fan, 1996). The three most common are exclusive (Tukey), inclusive, and interpolated (Excel). They differ in how they handle the median when splitting data into halves, and whether they interpolate between adjacent values. For large datasets, the differences are usually tiny.
The exclusive method splits the data at the median, excluding the median value if n is odd. Q1 is the median of the lower half, Q3 is the median of the upper half. This is the method used in most introductory statistics textbooks and is the standard for constructing box plots.
The trimean = (Q1 + 2 × Median + Q3) / 4 was introduced by John Tukey as a resistant measure of central tendency. It gives 50% weight to the median and 25% each to Q1 and Q3. Unlike the mean, it resists outliers. Unlike the median alone, it uses information about the shape of the middle 50%.
In a box plot, the box spans from Q1 to Q3 (the IQR). A line inside the box marks the median (Q2). Whiskers extend to the farthest non-outlier values (within 1.5×IQR of Q1 and Q3). Points beyond the whiskers are plotted individually as outliers. The entire box plot is built from quartiles.
The five-number summary consists of Min, Q1, Median, Q3, Max. These five values describe the center (median), spread (IQR = Q3−Q1), range (max−min), and symmetry of a distribution. It's the basis for box plots and is preferred over mean/SD for skewed or outlier-prone data.