How to Calculate a Five-Number Summary

Learning how to calculate a five-number summary by hand is straightforward when you follow the values in the correct order. A five-number summary describes a dataset using its minimum, Q1, median, Q3, and maximum.

Before calculating the quartiles, always arrange the data from smallest to largest. Then work from the outside values toward the center.

1. Sort the Data

Start by arranging every value in the dataset from smallest to largest.

For example, consider this dataset:

18, 7, 12, 25, 10, 15, 21

After sorting:

7, 10, 12, 15, 18, 21, 25

Sorting is important because the median and quartiles depend on the positions of the values in the ordered dataset.

2. Find the Minimum and Maximum

Once the data is sorted, finding the minimum and maximum is simple.

• Minimum: The first value
• Maximum: The last value

For the example above:

• Minimum = 7
• Maximum = 25

These values define the overall range of the dataset.

3. Find the Median

The median is the middle value of the ordered dataset.

If there is an odd number of values, the median is the single value in the center.

Example:

7, 10, 12, 15, 18, 21, 25

There are 7 values, so 15 is the middle value.

Median = 15

If there is an even number of values, there are two middle values. Add them together and divide by 2.

Example:

7, 10, 12, 15, 18, 21

The two middle values are 12 and 15.

Median = (12 + 15) ÷ 2 = 13.5

4. Find Q1

Q1, or the first quartile, is the median of the lower half of the ordered dataset.

Using:

7, 10, 12, 15, 18, 21, 25

The overall median is 15. Because there is an odd number of values, leave the median out when dividing the data into lower and upper halves.

Lower half:

7, 10, 12

The middle value is 10, so:

Q1 = 10

5. Find Q3

Q3, or the third quartile, is the median of the upper half of the ordered dataset.

For the same example, the upper half is:

18, 21, 25

The middle value is 21, so:

Q3 = 21

For an even-sized dataset, divide the ordered data into two equal halves and find the median of each half.

Note: Different statistics courses or software may use different quartile conventions. If your class specifies a particular quartile method, use that method consistently.

6. Write the Five-Number Summary

After finding all five values, write them in this standard order:

Minimum, Q1, Median, Q3, Maximum

For our example:

• Minimum = 7
• Q1 = 10
• Median = 15
• Q3 = 21
• Maximum = 25

Therefore, the five-number summary is:

(7, 10, 15, 21, 25)

This format gives you a quick picture of the dataset’s center and spread. You can also use Q1 and Q3 to calculate the interquartile range:

IQR = Q3 − Q1

For this dataset:

IQR = 21 − 10 = 11

If you want to check your work, the 5 Number Summary Calculator can calculate the values and show the sorting and calculation steps automatically.

Five-Number Summary Example

Let’s work through a complete five-number summary example step by step. The key is to sort the data first, then find the median and quartiles from the appropriate halves.

Consider this dataset:

4, 7, 8, 10, 12, 15, 18, 20, 24

Step 1: Sort the Dataset

The values are already arranged from smallest to largest:

4, 7, 8, 10, 12, 15, 18, 20, 24

There are 9 values, so the dataset has an odd number of observations.

Step 2: Find the Minimum

The smallest value is:

Minimum = 4

Step 3: Find the Median

Because there are 9 values, the median is the value in the exact middle.

4, 7, 8, 10, 12, 15, 18, 20, 24

The fifth value is 12.

Median = 12

Step 4: Find the Lower Half

Because the dataset has an odd number of values, leave the median out when dividing the data into two halves.

Lower half:

4, 7, 8, 10

The median of these four values is the average of the two middle values:

Q1 = (7 + 8) ÷ 2 = 7.5

Therefore:

Q1 = 7.5

Step 5: Find the Upper Half

The upper half is:

15, 18, 20, 24

Again, there are four values, so average the two middle values:

Q3 = (18 + 20) ÷ 2 = 19

Therefore:

Q3 = 19

Step 6: Find the Maximum

The largest value in the dataset is:

Maximum = 24

Final Five-Number Summary

The five values are always written in this order:

Minimum, Q1, Median, Q3, Maximum

For this dataset:

5-Number Summary: 4, 7.5, 12, 19, 24

This summary gives a compact view of the dataset’s center and spread. From Q1 and Q3, you can also calculate the interquartile range:

IQR = Q3 − Q1 = 19 − 7.5 = 11.5

Five-Number Summary for Even Number of Values

Finding a five-number summary for an even number of values follows the same basic process, but the median and quartiles may require averaging two middle values.

Consider this dataset:

6, 8, 11, 14, 17, 19, 22, 26

Step 1: Sort the Dataset

The values are already sorted:

6, 8, 11, 14, 17, 19, 22, 26

There are 8 observations, so the dataset has an even number of values.

Step 2: Find the Minimum and Maximum

The first value is the minimum:

Minimum = 6

The last value is the maximum:

Maximum = 26

Step 3: Find the Median

With 8 values, there is no single middle value. Instead, use the two middle observations.

The middle values are 14 and 17.

Calculate their average:

Median = (14 + 17) ÷ 2 = 15.5

Therefore:

Median = 15.5

Step 4: Determine the Lower Half

For an even-sized dataset, split the ordered data into two equal groups.

Lower half:

6, 8, 11, 14

The middle values are 8 and 11.

Calculate their average:

Q1 = (8 + 11) ÷ 2 = 9.5

Therefore:

Q1 = 9.5

Step 5: Determine the Upper Half

The upper half is:

17, 19, 22, 26

The middle values are 19 and 22.

Calculate their average:

Q3 = (19 + 22) ÷ 2 = 20.5

Therefore:

Q3 = 20.5

Final Five-Number Summary

Now combine the five values in the standard order:

Minimum, Q1, Median, Q3, Maximum

5-Number Summary: 6, 9.5, 15.5, 20.5, 26

This example shows why it is important to check whether your dataset contains an odd or even number of observations. For even-sized datasets, the median and quartiles can involve averaging two middle values.

Note: Quartile conventions can vary between textbooks, calculators, and statistical software. If your course or instructor specifies a particular quartile method, use that method consistently.

Five-Number Summary Formula

The five-number summary formula is not one single mathematical equation. Instead, it is a set of five descriptive statistics that summarize the key positions and spread of a dataset.

The standard format is:

Minimum, Q1, Median, Q3, Maximum

The five-number summary also makes it easy to calculate the interquartile range (IQR). The IQR measures the spread of the middle 50% of the data.

IQR = Q3 − Q1

For example, if Q1 = 10 and Q3 = 24:

IQR = 24 − 10 = 14

A larger IQR indicates that the middle 50% of the observations covers a wider range of values.

Five-Number Summary vs. Mean and Standard Deviation

A five-number summary and measures such as the mean and standard deviation describe data in different ways.

The five-number summary is especially useful when you want to see the minimum, maximum, median, and quartile positions of a dataset. Mean and standard deviation instead focus on the average and how observations vary around that average.

Five-Number Summary and Box Plot

A five-number summary provides the key values used to construct a box-and-whisker plot. The box plot turns these numerical values into a visual representation of the dataset’s center and spread.

The five positions are:

Minimum → Q1 → Median → Q3 → Maximum

In a standard box plot:

• Box: Extends from Q1 to Q3

• Line inside the box: Represents the median

• Whiskers: Extend toward the lower and upper ends of the data

The exact whisker endpoints can vary by box-plot convention. In some methods, whiskers extend to the minimum and maximum. In other methods, they extend to the most extreme values within a specified range, while potential outliers are plotted separately.

This connection makes a five-number summary useful beyond a list of calculations. Once you know the minimum, Q1, median, Q3, and maximum, you have the essential numerical information needed to understand or construct the corresponding box plot.

How to Find the IQR From a Five-Number Summary

You can find the interquartile range (IQR) directly from a five-number summary using Q1 and Q3. The IQR measures the spread of the middle 50% of the dataset.

The formula is:

IQR = Q3 − Q1

Simple Example

Suppose a dataset has this five-number summary:

Minimum = 4, Q1 = 8, Median = 12, Q3 = 20, Maximum = 27

To find the IQR, subtract Q1 from Q3:

IQR = 20 − 8

IQR = 12

Therefore, the IQR is 12.

Notice that the minimum, median, and maximum are not needed for this calculation. Only Q1 and Q3 are used.

A larger IQR means the middle 50% of the data covers a wider interval, while a smaller IQR means those values are more closely grouped.

Finding a five-number summary by hand can take a few steps, especially when a dataset contains many values. Our 5 Number Summary Calculator makes the process faster while showing the key calculations clearly.

Enter your dataset to find the minimum, Q1, median, Q3, and maximum. You can also review the sorted data, IQR, calculation details, and box-and-whisker plot to better understand your results.

Whether you are checking statistics homework, completing an assignment, or learning how to calculate a five-number summary, the calculator gives you a convenient way to verify your work.