A five number summary calculator condenses a data set into five values: minimum, first quartile, median, third quartile, and maximum. Those five numbers describe the centre, spread, and range of the data without listing every value. They are also the five values drawn in a box plot, which is why this summary is a standard first step in descriptive statistics.
The summary is useful because a mean alone can hide shape. Two classes can have the same mean but very different spreads, and a few extreme values can pull a mean away from most of the data. Quartiles divide the sorted data into quarters, so the summary shows where the middle half lies and how far the extremes stretch. That makes it easier to compare groups and to spot outliers.
What this calculator does
Start by sorting the data from smallest to largest. The minimum is the first value and the maximum is the last. The median is the middle value, or the average of the two middle values when the count is even. The first quartile, Q1, is the median of the lower half, and the third quartile, Q3, is the median of the upper half. The calculator below uses that median-of-halves method, which is the version most commonly taught for a five number summary.
The interquartile range, Q3 – Q1, is the spread of the middle half of the data. It is less sensitive to extremes than the full range, maximum minus minimum, because it ignores the outer quarters. Both spreads are worth reporting: the range shows the full extent, and the interquartile range shows where most values actually sit.
Formula
Sorted data: minimum = smallest value, Q1 = median of lower half, median = middle value, Q3 = median of upper half, maximum = largest value. Interquartile range = Q3 – Q1.
How to use it
- Enter the data values separated by commas, in any order. The calculator sorts them for you.
- Read the minimum and maximum from the ends of the sorted list.
- Find the median from the centre of the sorted list.
- Find Q1 from the lower half and Q3 from the upper half, using the median method.
- Check that minimum, Q1, median, Q3, and maximum are in non-decreasing order.
Try the calculator
Worked example
Find the five number summary for 3, 7, 8, 12, 15, 18, 21. The data are already sorted. The minimum is 3 and the maximum is 21. There are 7 values, so the median is the 4th value, 12. The lower half is 3, 7, 8, whose median is 7, so Q1 = 7. The upper half is 15, 18, 21, whose median is 18, so Q3 = 18. The five number summary is therefore 3, 7, 12, 18, 21, and the interquartile range is 18 – 7 = 11.
For an even count, the median is the average of the two middle values. For example, the data 2, 4, 6, 8 have median (4 + 6) / 2 = 5. The same averaging rule applies when finding quartiles from a half with an even count.
How to check your answer
Sort the data first and recount it. An unsorted list or a missing value changes every quartile.
Verify the order minimum, Q1, median, Q3, maximum. If any value is out of order, a quartile was taken from the wrong half.
Compare the interquartile range with the full range. A much larger full range than interquartile range suggests outliers in the tails.
Common mistakes
- Forgetting to sort the data before finding quartiles.
- Including the median in both halves when the method being used excludes it, or excluding it when the method includes it, which shifts Q1 and Q3.
- Reporting only the mean and ignoring the summary, which hides spread and skew.
- Confusing Q1 with the first value in the list and Q3 with the last value.
Frequently asked questions
What is the difference between median and Q1?
The median is the middle of all the data. Q1 is the middle of the lower half, so about 25 percent of values fall at or below Q1 and 75 percent fall at or above it.
Why use quartiles instead of just minimum and maximum?
Minimum and maximum show the full range but are sensitive to one extreme value. Quartiles show where the bulk of the data lies and are more stable.
How is a five number summary related to a box plot?
A box plot draws exactly these five values: the box spans Q1 to Q3 with a line at the median, and the whiskers extend to the minimum and maximum, sometimes with outliers marked separately.
