Calculators

Statistics

What is the Five Number Summary?

The five number summary is a set of numeric values that describe the center, spread and distribution of an ordered data set. The five number summary values are:

1. Minimum — the smallest number in the data set
2. First Quartile (Q₁) — the median of the lower half of the data
3. Median (Q₂ or x tilde) — the middle number in the data set
4. Third Quartile (Q₃) — the median of the upper half of the data
5. Maximum — the largest number in the data set

The five number summary is the foundation of the box and whisker plot (box plot) and is useful for quickly understanding the distribution, spread, and skewness of a data set. This box and whisker plot is an example of a uniform distribution of data points.

Using the Five Number Summary Calculator

Enter a data set with values separated by commas, spaces, or line breaks. You can copy and paste data from spreadsheets or text documents. See examples of data formats in the Data Formats table below.

If you'd like to try out the calculator and need a data set, use the CalculatorSoup Random Number Generator. Using this link you'll get a set of 100 numbers between 1 and 100, but you can adjust the settings to customize your random number set. Copy the numbers in the answer box and return to this page to paste the sample data set.

Find the Five Number Summary

To find the five number summary, follow these steps:

1. Sort the data from smallest to largest values
2. Count the number data points to get the size
3. Find the minimum which is the smallest value, and maximum which is the largest value
4. Find the median which is the middle number or average of two middle numbers
5. Find Q₁, the median of the lower half, not including the median data point
6. Find Q₃, the median of the upper half, not including the median data point

Five Number Summary Formulas

For an ordered data set x₁ ≤ x₂ ≤ x₃ ≤ ... ≤ x_n, the following formulas define each value in the five number summary.

Minimum

The minimum is the smallest value in the ordered data set.

\[ \text{Min} = x_1 = \text{min}(x_i)_{i=1}^{n} \]

Maximum

The maximum is the largest value in the ordered data set.

\[ \text{Max} = x_n = \text{max}(x_i)_{i=1}^{n} \]

Median (Q₂ or x tilde)

The median is the middle value of the ordered data set. It separates the data into an upper half and a lower half. The median is also called the second quartile, Q₂.

If n is odd the median is the middle value at position p where

\[ p = \dfrac{n + 1}{2} \] \[ \widetilde{x} = x_p \]

If n is even the median is the average of the two middle values at positions p and p + 1 where

\[ p = \dfrac{n}{2} \] \[ \widetilde{x} = \dfrac{x_{p} + x_{p+1}}{2} \]

First Quartile (Q₁)

The first quartile Q₁ is the median of the lower half of the data set. The lower half is the set of values below the median, not including the median value itself when n is odd.

To find Q₁, take all values below the median and find their median using the same median formula above.

Third Quartile (Q₃)

The third quartile Q₃ is the median of the upper half of the data set. The upper half is the set of values above the median, not including the median value itself when n is odd.

To find Q₃, take all values above the median and find their median using the same median formula above.

This is one of several methods for calculating quartiles.^[1]

Other Supporting Statistics Formulas

Range

The range is the difference between the maximum and minimum values.

\[ \text{Range} = x_n - x_1 \]

Interquartile Range (IQR)

The interquartile range is the difference between the third and first quartiles. The IQR describes the spread of the middle 50% of the data.

\[ IQR = Q_3 - Q_1 \]

Outliers

Outliers are data values that fall unusually far from the rest of the data. Using the five number summary, potential outliers are values that lie above the upper fence or below the lower fence.

\[ \text{Upper fence} = Q_3 + 1.5 \times IQR \] \[ \text{Lower fence} = Q_1 - 1.5 \times IQR \]

Five Number Summary and Box Plots

A box and whisker plot (box plot) is a graph of the five number summary. The box is drawn from Q₁ to Q₃ with a vertical line at the median. Whiskers extend from the box to the minimum and maximum values, or to the fences when outliers are present. Outliers are plotted as individual points beyond the whiskers.

Box plots make it easy to compare distributions between data sets and to quickly identify the center, spread, and skewness of a distribution.

Five Number Summary Example

Find the five number summary for the data set:

3, 7, 8, 5, 12, 14, 21, 13, 18

Step 1: Sort the Data

Order the values from smallest to largest:

3, 5, 7, 8, 12, 13, 14, 18, 21

The data set has n = 9 values.

Step 2: Find the Minimum and Maximum

\[ \text{Min} = 3 \] \[ \text{Max} = 21 \]

Step 3: Find the Median

Since n = 9 is odd, the median is the middle value at position:

\[ p = \dfrac{9 + 1}{2} = 5 \] \[ \widetilde{x} = x_5 = 12 \]

Step 4: Find Q₁

The lower half consists of all values below the median: 3, 5, 7, 8 (4 values).

Since the lower half has an even count, Q₁ is the average of the two middle values:

\[ Q_1 = \dfrac{5 + 7}{2} = 6 \]

Step 5: Find Q₃

The upper half consists of all values above the median: 13, 14, 18, 21 (4 values).

Since the upper half has an even count, Q₃ is the average of the two middle values:

\[ Q_3 = \dfrac{14 + 18}{2} = 16 \]

Result

The five number summary is:

3, 6, 12, 16, 21

From the five number summary you can also find:

\[ \text{Range} = 21 - 3 = 18 \] \[ IQR = Q_3 - Q_1 = 16 - 6 = 10 \]

Checking for outliers:

\[ \text{Lower Fence} = 6 - 1.5 \times 10 = -9 \] \[ \text{Upper Fence} = 16 + 1.5 \times 10 = 31 \]

All data values fall between −9 and 31, so there are no outliers.

 

References

[1] Wikipedia contributors. "Quartile." Wikipedia, The Free Encyclopedia. Last visited 21 May, 2026.