Quartile Calculator – IQR and Five-Number Summary

How to use the calculator

1. Type or paste your numbers into the Data values field. Separate them with commas, spaces or new lines; empty entries are ignored and up to 500 values are accepted.
2. Choose a Quartile method from the dropdown: Exclusive — median of halves (Tukey), Inclusive — median included in both halves, or Linear interpolation — Excel QUARTILE.INC (R type 7).
3. Read the results: the sorted data, the count, the full five-number summary, the range, the IQR and the outlier fences.
4. Check the Outliers line to see which values (if any) fall outside the fences, and look at the box plot for a visual picture of the distribution.
5. Open How the quartiles were found to follow the exact steps the calculator used for your chosen method — handy for homework or verification.

What quartiles and the IQR tell you

Quartiles split a sorted data set into four roughly equal parts. The first quartile \(Q_1\) marks the point below which 25% of the values fall, the median \(Q_2\) marks 50%, and the third quartile \(Q_3\) marks 75%. Together with the minimum and maximum they form the five-number summary, a compact description of a distribution that works even when you only have a rough idea of the data.

The interquartile range, \( \text{IQR} = Q_3 - Q_1 \), measures the spread of the middle 50% of the data. Because it ignores the lowest and highest quarters, it is a robust measure of spread: a few extreme values barely move it, whereas they can inflate the range or the standard deviation dramatically. This makes the IQR the preferred spread measure for skewed data such as incomes, house prices or response times.

The IQR also powers the standard outlier detection rule. Values outside \( [Q_1 - 1.5\,\text{IQR},\; Q_3 + 1.5\,\text{IQR}] \) are considered unusually far from the bulk of the data. On a box plot these points appear as individual dots beyond the whiskers, and you should investigate them before deciding whether they are errors, rare events, or legitimate extremes that belong in the analysis.

A quick look at a box plot built from the five numbers tells you a lot: if the median sits near the middle of the box the distribution is roughly symmetric in its centre; if it hugs one end the middle of the data is skewed in that direction. To quantify asymmetry more precisely, see the Skewness and Kurtosis Calculator.

Quartile methods, assumptions and practical tips

Unlike the median, quartiles have no single universally agreed definition, which is why the calculator offers three methods:

• Exclusive (Tukey): the median is excluded from both halves when the count is odd. This is the method behind classic box plots and many textbooks.
• Inclusive: the median is included in both halves when the count is odd. It gives slightly narrower quartiles for odd-sized samples.
• Linear interpolation (R type 7): quartiles are interpolated between data points at fixed positions. This is what Excel's QUARTILE.INC, R's quantile() and NumPy's default percentile() produce, so use it when you need to match those programs.

The differences between the methods shrink as the sample grows, but with small data sets they can shift \(Q_1\), \(Q_3\), the IQR and the fences enough to change which points count as outliers. If you are comparing your results against a spreadsheet or a statistics package, first check that you have selected the same method.

A few things to keep in mind when interpreting the results:

• The 1.5·IQR rule is a convention, not a law. For very large or heavy-tailed data sets, flagging values as "outliers" is normal and does not automatically mean anything is wrong.
• Quartiles say nothing about the shape between the quarter marks; two very different distributions can share the same five-number summary.
• The calculator assumes plain numeric data. If your values come in groups or with frequencies, expand them into individual entries first.
• If you need an arbitrary percentile rather than specifically Q1/Q3, the Percentile Calculator supports nine interpolation methods; for other summary statistics such as mean, median and mode, see the Median and Mode Calculator.

Once you have identified outliers and a robust spread, you may want to describe the data further — for example with the Coefficient of Variation Calculator to compare variability between data sets measured on the same scale.