The third quartile, Q3, is the boundary of the top quarter of a data set. If Q3 of delivery times is 3.35 days, three quarters of deliveries arrive within 3.35 days and a quarter take longer. Q3 is often used as a “high but typical” benchmark, and together with Q1 it defines the box in a box plot and the interquartile range.
By hand, Q3 is the median of the upper half of the sorted data, leaving out the overall median when the count is odd. Some textbooks include the median in both halves (Tukey’s hinges), and spreadsheets interpolate at a fractional position, so it is worth checking which convention you need – the answers differ most for small odd-sized samples.
Enter your values, pick the method and read Q3 together with the upper half or position that produced it. The upper fence Q3 + 1.5 × IQR is included, since it is the usual cut-off for high outliers.
How to use the Third Quartile Calculator
- Data set
- Quartile method
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How it works
Sort the data and split it at the median. Q3 is the median of the upper half (or the interpolated value at the 75% position for the Excel and Minitab methods).
Worked example
Nine delivery times in days: 1.2, 1.5, 1.9, 2.3, 2.4, 2.8, 3.1, 3.6, 4.0. The median is the 5th value, 2.4. The upper half without the median is 2.8, 3.1, 3.6, 4.0, so Q3 = (3.1 + 3.6)/2 = 3.35. Q1 = (1.5 + 1.9)/2 = 1.7, giving IQR = 1.65 and an upper fence of 3.35 + 2.475 = 5.825 days. Tukey’s hinges and Excel’s QUARTILE.INC both give Q3 = 3.1 for this odd-sized sample.
How to interpret Q3
Q3 is a robust “high end” of normal. A quarter of the observations exceed it, so a value above Q3 is high but not unusual. A value above the upper fence is unusual. Comparing Q3 − median with median − Q1 shows whether the upper part of the data is more stretched than the lower part.
Formula
Q3 = median of the values above the median · or Q3 at position 3(n + 1)/4 (exclusive) or 1 + 3(n − 1)/4 (inclusive)
Worked example
Inputs — Data set: 1.2, 1.5, 1.9, 2.3, 2.4, 2.8, 3.1, 3.6, 4.0; Quartile method: Median of halves, median excluded (TI-84, most textbooks).
Results — Third quartile (Q3, 75th percentile) = 3.35; Median (Q2) = 2.4; First quartile (Q1) = 1.7; Interquartile range (IQR = Q3 − Q1) = 1.65; Maximum = 4; Minimum = 1.2.
Key terms
- Upper quartile
- Another name for Q3.
- Upper half
- The sorted values above the median.
- Upper fence
- Q3 + 1.5 × IQR, beyond which values are flagged as high outliers.
Frequently asked questions
What is the third quartile?
The value below which 75% of the data lie; also called the upper quartile or 75th percentile.
How do I calculate Q3?
Sort the data, find the median and take the median of the values above it. For 1.2, 1.5, 1.9, 2.3, 2.4, 2.8, 3.1, 3.6, 4.0 the upper half is 2.8, 3.1, 3.6, 4.0 and Q3 = 3.35.
What is the position of Q3?
3(n + 1)/4 for the exclusive method and 1 + 3(n − 1)/4 for Excel's inclusive method; for n = 9 those are 7.5 and 7.
Why does Tukey's method give a different Q3?
It includes the median in the upper half: 2.4, 2.8, 3.1, 3.6, 4.0, whose median is 3.1.
How do I find Q3 in Excel?
=QUARTILE.INC(range, 3) gives 3.1 for the example; =QUARTILE.EXC(range, 3) gives 3.35.
What is Q3 used for?
Box plots, the IQR, upper outlier fences and "top 25%" thresholds.