Five Number Summary Calculator

1. Enter the data. Type or paste the values in any order. Commas, spaces and line breaks all work as separators.
2. Pick the quartile method. Keep the textbook default unless your course or software uses another rule. The tool lists the other three results for you.
3. Read the summary and the box plot. The five numbers appear in order, followed by the spread, the fences and any potential outliers.

A five number summary describes a data set with five values: the minimum, the first quartile, the median, the third quartile and the maximum. OpenStax Introductory Statistics builds every box plot from these five values.

The median splits the ordered data into two halves. The first quartile, Q1, is the middle of the lower half, and the third quartile, Q3, is the middle of the upper half. About one quarter of the data lies at or below Q1 and about three quarters lies at or below Q3.

The summary shows center and spread at once. The median gives the center. The range and the interquartile range give the spread. The gaps between the five numbers show whether the data lean to one side.

The summary does not use every value the way a mean does. One extreme value moves the maximum but leaves Q1, the median and Q3 almost unchanged.

The calculator sorts the data, finds the median, and then finds Q1 and Q3 with the method you choose.

1. Sort. Values are ordered from smallest to largest.
2. Median. With an odd count it is the middle value. With an even count it is the average of the two middle values.
3. Quartiles. The chosen rule gives Q1 and Q3. For the interpolated rules, p is 0.25 and 0.75.
4. Fences and outliers. Any value below Q1 - 1.5 x IQR or above Q3 + 1.5 x IQR is listed as a potential outlier.

The minimum, the median and the maximum are the same under every method. Only Q1 and Q3 change.

Textbook Example with 14 Values

OpenStax uses this data: 1, 11.5, 6, 7.2, 4, 8, 9, 10, 6.8, 8.3, 2, 2, 10, 1.

1. Sort: 1, 1, 2, 2, 4, 6, 6.8, 7.2, 8, 8.3, 9, 10, 10, 11.5.
2. There are 14 values, so the median is the average of the 7th and 8th: (6.8 + 7.2) / 2 = 7.
3. Lower half: 1, 1, 2, 2, 4, 6, 6.8. Its middle value is 2, so Q1 = 2.
4. Upper half: 7.2, 8, 8.3, 9, 10, 10, 11.5. Its middle value is 9, so Q3 = 9.
5. The summary is 1, 2, 7, 9, 11.5. The IQR is 9 - 2 = 7.

These are the values the textbook reports. The tool loads this set when the page opens.

Example Where the Methods Disagree

Nine quiz scores: 4, 7, 8, 10, 12, 13, 15, 18, 30. The median is the 5th value, 12.

The score of 30 sits exactly on the upper fence under two rules and beyond it under the other two. That is why the method matters on small data sets.

The data decide the minimum, median and maximum. The method and the sample size decide how far Q1 and Q3 can shift.

The Quartile Method

For the values 1 to 8, the two median-of-halves rules give Q1 = 2.5 and Q3 = 6.5. The (n - 1) rule gives 2.75 and 6.25. The (n + 1) rule gives 2.25 and 6.75.

Odd or Even Count

With an even count, the median-excluded and median-included rules always agree. They can differ only when the count is odd.

Sample Size

Differences between methods shrink as the data set grows. The NIST handbook notes that there is no single accepted way to interpolate, and that all the methods are used in practice.

Extreme Values

One extreme value changes the maximum and the range but barely moves the quartiles. In the 13 house prices of OpenStax Example 2.13, a 5,500,000 price leaves Q3 at 649,000.

Repeated Values

Ties are kept. Each repeated value holds its own place in the sorted list.

The two summaries answer the same question, center and spread, with different tools.

For the nine quiz scores, the mean is 13 and the median is 12. The single score of 30 pulls the mean up. The standard deviation calculator gives the other summary for the same data.

Drawing a Box Plot

The five values are the box, the median line and the two ends. The NIST handbook describes a version with fences, where whiskers stop at the last value inside each fence and outliers are drawn as points.

Comparing Two Groups

OpenStax compares a day class and a night class this way. The day class has an IQR of 26.5 and the night class has 11, so the day scores vary more.

Screening for Outliers

The night class has fences at 61.5 and 105.5, so the scores 25.5 and 45 are potential outliers. Load the third preset to see it.

Describing Skewed Data

Incomes, prices and waiting times often have a long tail. The median and quartiles describe them better than a mean. The percentile calculator finds any other cut point.

1. Not Sorting First

Quartiles are positions in ordered data. The middle of an unsorted list means nothing.

2. Mixing Methods

Comparing your hand answer with software that uses another rule gives a false mismatch. Check the other-methods line before you assume an error.

3. Including the Median in the Halves by Accident

With an odd count, the default textbook rule leaves the median out of both halves.

4. Calling Every Flagged Value an Error

OpenStax calls these potential outliers and says they need further investigation. A flagged value can be real.

5. Confusing the Fences with the Whiskers

Fences are computed limits. Whiskers end at actual data values inside those limits.

6. Typing Thousands Separators

The tool reads 1,200 as the two numbers 1 and 200. Type 1200.

7. Reading Q2 as the Mean

Q2 is the median. The median calculator shows that step alone.

The results are exact for the data and the method chosen, and they reproduce the OpenStax textbook answers. The limits come from the choice of method, not from the arithmetic.

They are the minimum, the first quartile (Q1), the median, the third quartile (Q3) and the maximum, listed in that order.

Sort the data and find the median. Q1 is the median of the values below it and Q3 is the median of the values above it. For 1 to 8, Q1 is 2.5 and Q3 is 6.5.

It uses a different quartile rule. For 4, 7, 8, 10, 12, 13, 15, 18, 30, one rule gives Q1 = 7.5 and another gives 8. Both are accepted methods.

Use the method your textbook teaches. The default here matches OpenStax Introductory Statistics, where the median is left out of both halves.

Compute the IQR, then the fences Q1 - 1.5 x IQR and Q3 + 1.5 x IQR. Values outside the fences are potential outliers.

Yes. The second quartile, the median and the 50th percentile are the same value.

Yes. In the data 5, 5, 5, 5, 9 the minimum, Q1 and median are all 5. Repeats in the summary show that many values are tied.

It is Q3 minus Q1. It measures the spread of the middle half of the data and ignores the extremes.

One value is enough to compute a result, but quartiles carry little meaning with fewer than 4 values. The tool shows a warning in that case.

No. The math runs in your browser. The Save button keeps results only in your own browser storage.

• Measures of the location of the data: quartiles, IQR and potential outliers (OpenStax Introductory Statistics 2e, section 2.3).
• Box plots and the five values they are built from (OpenStax Introductory Statistics 2e, section 2.4).
• Percentiles: the p(N + 1) method and its alternatives (NIST/SEMATECH e-Handbook of Statistical Methods, 7.2.6.2).
• Box plot with fences (NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.3.7).
• Sample quantiles: the nine quantile types (R stats package documentation).