Interquartile range and interpretation
IQR is the distance from the first to the third quartile. The two fences are numeric thresholds; the listed observations are strictly outside them and are retained in the calculation.
To compare the arithmetic mean, median, and mode of observations, use the Mean, Median & Mode Calculator and follow its own input conventions.
The formula
Compute the 25th and 75th percentiles with type-seven interpolation, subtract them, then extend each quartile by 1.5 times that difference to form the lower and upper fences.
Verified worked example
For integers 1 through 8, inclusive interpolated quartiles are Q₁ = 2.75 and Q₃ = 6.25. The IQR is 3.5 and the fences are −2.5 and 11.5; no values exceed them.
How to use this calculator
- Enter at least two numeric observations in any order.
- Keep duplicates and confirm that inclusive interpolation matches the quartile convention you need.
- Read Q1, Q3, IQR, and the fences; investigate listed extreme values without automatically deleting them.
Interquartile Range input conventions
The interquartile range measures the distance between the 25th and 75th percentiles. It concentrates on the central half of an ordered dataset rather than relying on the minimum and maximum. The quartiles here use inclusive linear interpolation at one-based position 1 + (n − 1)p. A classroom method that takes medians of separate halves can yield different quartiles.
Interpreting interquartile range
The fences are screening thresholds. An observation beyond them can deserve investigation, but the label alone does not establish an error or justify deleting data. A strongly skewed distribution can contain legitimate extreme observations. This calculator leaves every entry in the dataset and reports values strictly outside the fences; an entry exactly on a fence is not listed. Zero IQR is possible when the middle values are tied.
Assumptions & limitations
What this calculation assumes
- Quartiles use inclusive interpolation and the outlier screen uses a fixed factor of 1.5.
What to keep in mind
- Unweighted type-seven quartiles and a fixed 1.5-IQR rule; no automatic outlier removal.
Common questions
Are the fences the minimum and maximum whiskers?
Not necessarily. Many box plots put whiskers at the most extreme observed values still inside the fences. This page displays the numeric thresholds themselves.
Why does another calculator show different quartiles?
It may use exclusive interpolation, nearest ranks, or a median-of-halves convention. Compare the quartile method before treating the difference as an error.
What if the IQR is zero?
The interpolated first and third quartiles are equal. This can occur with many tied middle observations even if some outer observations differ.
Sources & further reading
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