The first quartile, Q1, marks the boundary of the bottom quarter of the data. If Q1 of a set of exam marks is 75, then roughly 25% of students scored 75 or less. It is used to describe the low end of a distribution, to set thresholds (“the bottom 25%”), and as one end of the box in a box plot.
To find Q1 by hand, sort the data and take the median of the lower half – the values below the overall median. With an even number of values the lower half is simply the first n/2 values; with an odd number the median itself is left out in the most common method. Software often interpolates instead: Excel’s QUARTILE.INC reads the value at position 1 + (n − 1)/4 in the sorted list.
This calculator gives Q1 with whichever method you select and shows the working, so you can match the answer in your textbook or spreadsheet. The median, Q3 and IQR are included for context.
How to use the First Quartile Calculator
- Data set
- Quartile method
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How it works
Sort the data and split it at the median. Q1 is the median of the lower half (or the interpolated value at the 25% position for the Excel and Minitab methods).
Worked example
Ten exam marks: 68, 72, 75, 79, 81, 84, 86, 90, 93, 97. The median is (81 + 84)/2 = 82.5. The lower half is 68, 72, 75, 79, 81, and its middle value is Q1 = 75. Excel’s QUARTILE.INC instead interpolates at position 3.25: 75 + 0.25 × (79 − 75) = 76, and QUARTILE.EXC at position 2.75: 72 + 0.75 × 3 = 74.25.
How to interpret Q1
Q1 tells you where the bottom quarter ends. In the example, a mark of 75 or less puts a student in the lowest quarter of the class. Compare Q1 with the median: the gap 82.5 − 75 = 7.5 is the spread of the second quarter of the data; a much larger gap than Q3 − median signals a long lower tail.
Formula
Q1 = median of the values below the median · or Q1 at position (n + 1)/4 (exclusive) or 1 + (n − 1)/4 (inclusive)
Worked example
Inputs — Data set: 68, 72, 75, 79, 81, 84, 86, 90, 93, 97; Quartile method: Median of halves, median excluded (TI-84, most textbooks).
Results — First quartile (Q1, 25th percentile) = 75; Median (Q2) = 82.5; Third quartile (Q3) = 90; Interquartile range (IQR = Q3 − Q1) = 15; Minimum = 68; Maximum = 97.
Key terms
- Lower quartile
- Another name for Q1.
- Lower half
- The sorted values below the median.
- 25th percentile
- The value below which 25% of the data lie.
Frequently asked questions
What is the first quartile?
The value that separates the lowest 25% of the data from the rest; also called the lower quartile or 25th percentile.
How do I calculate Q1?
Sort the data, find the median, and take the median of the values below it. For 68, 72, 75, 79, 81, 84, 86, 90, 93, 97 the lower half is 68, 72, 75, 79, 81 and Q1 = 75.
What is the formula for the position of Q1?
(n + 1)/4 for the exclusive method, 1 + (n − 1)/4 for the inclusive (Excel QUARTILE.INC) method. For n = 10 those are 2.75 and 3.25.
How do I find Q1 in Excel?
=QUARTILE.INC(range, 1) or =QUARTILE.EXC(range, 1). They give 76 and 74.25 for the example data.
Is Q1 the same as the 25th percentile?
Yes, by definition, although percentile formulas vary in the same way quartile formulas do.
Can Q1 be equal to the minimum?
Yes, with small or heavily tied data sets, e.g. 2, 2, 2, 5, 9.