Five-Number Summary: Find the Minimum, Quartiles and Maximum

A five-number summary describes a data set with its minimum, first quartile, median, third quartile, and maximum. Enter or write down a list of numerical values, then sort them; the result gives a quick view of the data’s center and spread, though quartile conventions can vary.

Formula

The five-number summary is (minimum, Q1, median, Q3, maximum). The minimum and maximum are the smallest and largest observations. The median is the middle observation, or the average of the two middle observations when the number of values is even. Q1 is the median of the lower half, and Q3 is the median of the upper half.

• x means an observed data value, with the same unit as the original measurements.
• n is the number of observations, a count.
• Q1 and Q3 are the first and third quartiles, in the original measurement unit.

This guide uses the median-of-halves convention: for an odd number of observations, leave the overall median out when splitting the data into lower and upper halves. Some calculators and textbooks use other percentile conventions, so quartiles may differ slightly between sources.

Worked Example

Use the exact default list in the calculator: 2, 4, 6, 8, 10, 12, 14. It is already in ascending order, and it contains seven values.

1. Minimum: the first value is 2.
2. Median: the fourth value is 8.
3. Lower half, excluding the median: 2, 4, 6. Its middle value is Q1 = 4.
4. Upper half, excluding the median: 10, 12, 14. Its middle value is Q3 = 12.
5. Maximum: the last value is 14.

The five-number summary is (2, 4, 8, 12, 14), rounded to two decimal places as displayed by the calculator. The units are the same as the data values; the example uses plain numbers.

What You’ll Need

• A complete list of numerical observations, such as measurements, scores, or counts.
• Paper and a pencil, a spreadsheet, or a calculator that accepts a list of values.
• The quartile convention required for your class, report, or analysis, if one has been specified.

Keep each value in its original unit. If the data include measurements in different units, convert them to one consistent unit before calculating.

Step-by-Step

1. Check the data. Make sure every entry is a number that belongs in the data set. Decide how to handle missing values, and do not silently include headings or labels as observations.
2. Sort the values. Arrange them from smallest to largest. Sorting is essential because the summary is based on positions in the ordered list, not on the order in which values were collected.
3. Find the minimum and maximum. Record the first and last values in the sorted list. These are the smallest and largest observations, respectively.
4. Find the overall median. Count the observations. If the count is odd, select the single middle value. If it is even, average the two middle values: add them and divide by 2.
5. Split the data into halves. If there is an odd number of observations, omit the overall median from both halves under the median-of-halves convention. If the count is even, divide the list into equal lower and upper halves without omitting a value.
6. Find Q1. Calculate the median of the lower half. For an odd number of values in that half, use its middle value; for an even number, average its two middle values.
7. Find Q3. Calculate the median of the upper half using the same rule.
8. Write the results in order. Record minimum, Q1, median, Q3, and maximum. Check that they are nondecreasing: each number should be less than or equal to the one after it.

Common Mistakes

• Forgetting to sort: A value that appears first in the raw data is not necessarily the minimum. Always order the list first.
• Including the median in both halves: With an odd-sized data set, this changes Q1 and Q3 under the median-of-halves method. Leave the central value out of both halves.
• Using a different quartile rule without noticing: Percentile algorithms do not all define quartiles in the same way. State the method or follow the convention requested.
• Taking the wrong middle value: For an even number of values, the median is the average of the two central values, not either one alone.
• Rounding too early: Keep full precision while averaging middle values. Round only the final reported results, and use a consistent number of decimal places.
• Treating repeated values as one observation: Keep duplicates if they are genuine measurements. Each observation counts toward the positions used to find the median and quartiles.

Tips

• Write the sorted values with their positions numbered; this makes middle positions easier to verify.
• For a quick plausibility check, confirm that Q1 is between the minimum and median, and Q3 is between the median and maximum.
• The difference Q3 − Q1 is the interquartile range (IQR). It describes the spread of the middle half of the data, but is not one of the five summary values.
• A box plot often uses the five-number summary, but some box plots show whiskers based on an outlier rule rather than extending to the minimum and maximum. Check the plot’s definition.
• For large lists, use a spreadsheet or calculator to sort and calculate, then verify which quartile convention it applies.

Troubleshooting

• Your quartiles differ from a calculator: Check whether it includes the overall median in each half or uses an interpolated percentile method. Recalculate using the required convention and label it clearly.
• You cannot identify a middle value: Recount the sorted observations. An even count has two central values; average them to find the median of that list.
• The summary is out of order: Recheck the sorting and the values copied into each half. A five-number summary cannot decrease as you move from minimum to maximum.
• Results look implausible: Look for misplaced decimal points, data entered as text, an accidental extra value, or a missed duplicate. Confirm that all entries use the same unit.
• There is only one observation: The minimum, Q1, median, Q3, and maximum are all that observation. With very small data sets, quartiles may be less informative, and conventions can have a noticeable effect.

Frequently Asked Questions

What are the five numbers in a five-number summary?
They are the minimum, first quartile (Q1), median, third quartile (Q3), and maximum, listed in that order.

How do I find Q1 and Q3 when there is an odd number of values?
Using the median-of-halves method, find the overall median, leave it out, and take the median of each remaining half.

Do different methods give different five-number summaries?
They can give different Q1 and Q3 values because quartile and percentile conventions vary. The minimum, maximum, and usual median do not depend on the quartile convention.

Can values repeat?
Yes. Keep each valid occurrence in the list. Repeated observations affect the count and may determine the median or quartiles.

What does a five-number summary tell me?
It shows the data’s endpoints, center, and the boundaries of its lower and upper quarters. It does not reveal every value or show the full shape of the distribution.

Is the five-number summary the same as the IQR?
No. The summary contains five values. The IQR is one additional measure, calculated as Q3 minus Q1.

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list data | Data values | 2, 4, 6, 8, 10, 12, 14
out summary | Five-number summary (min, Q1, median, Q3, max) | - | minl(data), q1(data), median(data), q3(data), maxl(data) | 2