Quartile and interquartile range calculator
Quartiles cut sorted data into four equal groups. See each group coloured, and see which values fall outside the outlier fences.
Your sorted data, split into quarters
Outliers (1.5 × IQR rule): none. Every value is between the fences.
Interpolated quartiles (Excel QUARTILE.INC, R default): Q1 = 68.5, Q3 = 81.5.
How quartiles are found
- Sort the data from smallest to largest.
- Find the median (Q2). It splits the data into a lower half and an upper half.
- Q1 is the median of the lower half; Q3 is the median of the upper half. With an odd count, the middle value belongs to neither half.
The coloured chips above show the result: each colour is one quarter of your data, and each quarter holds about the same number of values, even though the quarters can span very different ranges. A wide quarter means the data are spread out there.
IQR and the outlier fences
The interquartile range (IQR) is the width of the middle half. John Tukey's rule, from his 1977 book Exploratory Data Analysis, flags anything more than 1.5 IQRs outside the box as a potential outlier. Outliers are outlined in red above.
See the same numbers as a picture with the box plot maker.
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Questions students ask
How do you find Q1 and Q3?
Sort the data and find the median. Q1 is the median of the lower half and Q3 is the median of the upper half. With an odd count, this calculator leaves the median out of both halves (Tukey's method, used by the TI-84).
What is the interquartile range?
IQR = Q3 − Q1. It is the spread of the middle 50% of the data and ignores the extremes, so it is resistant to outliers.
How do you find outliers with the IQR?
Any value below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR is flagged as an outlier. This is the rule box plots use to draw separate dots.
Why does my textbook give a different Q1?
Some methods include the median in both halves, and software such as Excel interpolates instead. The interpolated values are shown too, so you can match your course.