Q1, Q2 and Q3 explained, with how to find the lower quartile
Quick answer: Quartiles split sorted data into four equal parts. The lower quartile (Q1) is the 25th percentile, the second quartile (Q2) is the median (50th percentile), and the upper quartile (Q3) is the 75th percentile.
Quartiles are three cut points that divide an ordered data set into four groups of roughly equal size. About a quarter of the values lie below the first quartile (Q1), half lie below the second quartile (Q2) — which is just the median — and three quarters lie below the third quartile (Q3). Q1 and Q3 are often called the lower and upper quartiles.
Sort the data and find the median, which divides it into a lower half and an upper half. The lower quartile is the median of the lower half. The upper quartile (Q3) is found the same way using the upper half.
Data: 4, 9, 2, 7, 12, 5, 15, 8 → sorted 2, 4, 5, 7, 8, 9, 12, 15 (n = 8).
Median = (7 + 8) ÷ 2 = 7.5. Lower half: 2, 4, 5, 7 → Q1 = (4 + 5) ÷ 2 = 4.5. Upper half: 8, 9, 12, 15 → Q3 = (9 + 12) ÷ 2 = 10.5.
When the count is odd, the median is an actual data value, and methods disagree on whether to include it in the halves. Tukey excludes it, Moore & McCabe includes it, and Excel interpolates. For even counts (like the example above) the methods usually agree. The quartile calculator supports all three so you can match your textbook.
The gap between Q1 and Q3 is the interquartile range (IQR), a robust measure of spread, and the quartiles are the building blocks of the box plot. Together with the minimum and maximum they form the five-number summary.
Yes. Q2 is the 50th percentile, which is exactly the median of the data.
The lower quartile (Q1) is the 25th percentile — the median of the lower half of the sorted data.
Different accepted methods (Tukey, Moore & McCabe, interpolation) treat the median differently for odd-sized data sets, which can shift Q1 and Q3 slightly.