Five-number summary: formula breakdown, examples, and how it works

The five-number summary is one of the fastest ways to “see” a dataset without drawing a full chart. Instead of staring at a long list of values, you compress the information into five landmarks: the smallest value (minimum), the largest value (maximum), and three values that split the data into quarters (Q1, median, and Q3).

Why is this so popular? Because those five numbers tell you three big stories at once: center (where the middle is), spread (how wide the data is), and shape (whether the lower half and upper half are balanced). They also power the classic box-and-whisker plot, which is a favorite in stats classes, lab reports, and business dashboards.

• Minimum (Min): the smallest number in the dataset.
• First quartile (Q1): the value where about 25% of data points are at or below it.
• Median (Q2): the “middle” value (50th percentile).
• Third quartile (Q3): the value where about 75% of data points are at or below it.
• Maximum (Max): the largest number in the dataset.

Once you have those, two extra numbers become almost automatic: Range = Max − Min tells you the total spread, and IQR = Q3 − Q1 tells you the spread of the middle 50% of the data. The IQR is especially useful because it is resistant to extreme values. That’s why it shows up in outlier rules and boxplots.

Every five-number summary starts the same way: sort the data from smallest to largest. If your dataset is [12, 18, 22, 25, 31], it’s already sorted. If it’s messy like [25, 12, 31, 18, 22], sorting turns it into the first list. Sorting is the “unlock” that makes percentiles and medians meaningful.

Next, the calculator finds the median. There are two cases: if you have an odd number of data points, the median is the middle value. If you have an even number, the median is the average of the two middle values.

• Odd n: median is the value in position (n + 1) / 2.
• Even n: median is the average of positions n/2 and n/2 + 1.

Then come the quartiles, and this is where people get confused — not because the idea is hard, but because there are multiple accepted definitions. Two textbooks can both be “right” while giving different Q1 and Q3 values for the same dataset. That’s why this calculator lets you choose the quartile method.

Tukey quartiles are common in many intro statistics classes and in “hand calculation” problems. The rule is: split the sorted data into a lower half and an upper half, then take the median of each half. If the dataset has an odd number of points, the median itself is typically excluded from both halves.

Example (odd n): data = [1, 2, 3, 4, 5, 6, 7]. The median is 4. Lower half is [1, 2, 3] → Q1 = 2. Upper half is [5, 6, 7] → Q3 = 6.

Example (even n): data = [1, 2, 3, 4, 5, 6, 7, 8]. Median is the average of 4 and 5 → 4.5. Lower half is [1, 2, 3, 4] → Q1 = (2+3)/2 = 2.5. Upper half is [5, 6, 7, 8] → Q3 = (6+7)/2 = 6.5.

Many spreadsheets use a percentile-style approach. The inclusive method (similar to Excel’s QUARTILE.INC) treats the dataset as if the minimum is the 0th percentile and the maximum is the 100th percentile, then interpolates between points. A quick way to describe it is: the position for percentile p is 1 + (n−1)p (using p = 0.25 for Q1 and p = 0.75 for Q3). If the position is not an integer, you linearly interpolate between the two surrounding data points.

This approach has a nice property: it behaves smoothly as you add data points, and it aligns with many software tools. But because it “spreads” percentiles across the full range, it can produce quartiles that are not actual data points (especially with small datasets).

The exclusive method (similar to Excel’s QUARTILE.EXC) uses a slightly different position: (n + 1)p. It tends to push quartiles a bit inward (away from the minimum and maximum). It’s often used when you want to avoid giving too much weight to the endpoints in small samples. Like the inclusive method, it interpolates when the position lands between two ranks.

After computing Q1 and Q3, the calculator computes the interquartile range (IQR): IQR = Q3 − Q1. Then it builds “fences” using the classic rule: Lower fence = Q1 − 1.5×IQR and Upper fence = Q3 + 1.5×IQR. Any value outside those fences is flagged as a potential outlier.

This is popular because it is simple and robust. But remember: an outlier flag is a signal, not a judgment. Sometimes the outlier is a typo. Sometimes it’s a rare but real event (like a huge sale day, a special discount, or a measurement taken during a storm). In real analysis, you investigate why it’s extreme before you decide what to do.

Let’s use a dataset that looks like a typical homework list: 8, 9, 10, 12, 13, 13, 14, 18, 30. First, sort it (already sorted). Min = 8 and Max = 30. There are 9 values (odd), so the median is the 5th value → 13.

Using Tukey quartiles, exclude the median and split: lower half = [8, 9, 10, 12] → Q1 = (9+10)/2 = 9.5. upper half = [13, 14, 18, 30] → Q3 = (14+18)/2 = 16. Five-number summary = (8, 9.5, 13, 16, 30).

Now compute IQR = 16 − 9.5 = 6.5. Lower fence = 9.5 − 1.5×6.5 = 9.5 − 9.75 = −0.25. Upper fence = 16 + 9.75 = 25.75. So 30 is above 25.75 and gets flagged as an outlier. That doesn’t mean 30 is “wrong”, but it is far above the middle half of the data.

• Center: the median tells you what a “typical” value looks like.
• Spread: the range tells you how wide the full dataset is; the IQR tells you how wide the middle is.
• Skew: if (Median − Q1) is much smaller than (Q3 − Median), the data may skew upward, and vice versa.
• Outliers: points outside the fences are candidates for investigation.

In business terms: if your dataset is weekly sales, the median is your “typical week,” Q1 and Q3 define the usual band, and outliers might be holiday spikes or a sudden outage. In science, the summary helps you quickly compare two experimental conditions: which one has a higher median, and which one has a wider IQR?

Bottom line: the five-number summary is a small tool with a big impact. It’s fast, it’s visual (especially with a boxplot), and it’s one of the best first steps before you jump into more advanced stats.

Frequently Asked Questions

• Why do my Q1 and Q3 differ from my textbook?

  Quartiles have multiple valid definitions. Some books use Tukey (median of halves), while many spreadsheets use percentile interpolation (inclusive or exclusive). Use the dropdown to match the method your course expects.

• Do I have to remove outliers before calculating?

  No. Calculate first, then investigate. Outliers can be errors, or they can be real rare events. If you remove them, document why.

• Does the five-number summary work with negative numbers and decimals?

  Yes. The calculator supports negatives, decimals, and repeated values. Just make sure your separators are commas, spaces, or line breaks.

• What if my dataset has only 1–3 numbers?

  The calculator will still compute Min/Median/Max. Quartiles for tiny datasets can be ambiguous; in that case, we fall back to a consistent interpolation approach and show a note so you’re not surprised.

• Is this the same thing as a box plot?

  The five-number summary is the input for a box plot. A box plot is a visual chart drawn from these values. This page also shows a mini boxplot bar you can screenshot.

• How should I cite this calculator in a report?

  You can reference the page name and URL (five-number-summary.html). For academic work, also include the quartile method you used (Tukey / Inclusive / Exclusive) since that affects Q1 and Q3.