Five Number Summary — Definition, Examples & Table

Five Number Summary

For a set of data, the minimum, first quartile, median, third quartile, and maximum. Note: A boxplot is a visual display of the five-number summary.

Example: For data 2,5,6,9,12: minimum=2, Q1=3.5, median=6, Q3=10.5, maximum=12

Key Formula

\text{Five Number Summary} = \{\, \min,\; Q_1,\; Q_2,\; Q_3,\; \max \,\}

Where:

$\min$ = The smallest value in the data set
$Q_1$ = First quartile — the median of the lower half of the data (25th percentile)
$Q_2$ = Median — the middle value of the entire data set (50th percentile)
$Q_3$ = Third quartile — the median of the upper half of the data (75th percentile)
$\max$ = The largest value in the data set

Worked Example

Problem: Find the five number summary for the data set: 3, 7, 8, 5, 12, 14, 21, 15, 18, 14.

Step 1: Order the data from least to greatest.

3,\; 5,\; 7,\; 8,\; 12,\; 14,\; 14,\; 15,\; 18,\; 21

Step 2: Identify the minimum and maximum.

\min = 3, \quad \max = 21

Step 3: Find the median (Q₂). There are 10 values, so the median is the average of the 5th and 6th values.

Q_2 = \frac{12 + 14}{2} = 13

Step 4: Find Q₁, the median of the lower half (the first five values: 3, 5, 7, 8, 12). The middle value is the 3rd value.

Q_1 = 7

Step 5: Find Q₃, the median of the upper half (the last five values: 14, 14, 15, 18, 21). The middle value is the 3rd value.

Q_3 = 15

Answer: The five number summary is {3, 7, 13, 15, 21}.

Another Example

This example uses an odd number of data points, so the median is an actual data value rather than an average of two middle values. The lower and upper halves each have the same odd count, making Q₁ and Q₃ straightforward to find.

Problem: Find the five number summary for the data set: 10, 20, 30, 40, 50, 60, 70.

Step 1: The data is already in order and has 7 values (an odd count).

10,\; 20,\; 30,\; 40,\; 50,\; 60,\; 70

Step 2: Identify the minimum and maximum.

\min = 10, \quad \max = 70

Step 3: The median is the 4th value (the middle of 7 values).

Q_2 = 40

Step 4: The lower half consists of the values before the median: 10, 20, 30. The median of these three values is the middle one.

Q_1 = 20

Step 5: The upper half consists of the values after the median: 50, 60, 70. The median of these three values is the middle one.

Q_3 = 60

Answer: The five number summary is {10, 20, 40, 60, 70}.

Frequently Asked Questions

How do you find the five number summary?

First, sort the data from least to greatest. Identify the minimum and maximum. Find the median of the full data set. Then split the data into a lower half and an upper half (excluding the median if the count is odd). Q₁ is the median of the lower half, and Q₃ is the median of the upper half.

What is the difference between a five number summary and a boxplot?

A five number summary is a list of five values (min, Q₁, median, Q₃, max). A boxplot is a graphical representation of those same five values. The box spans from Q₁ to Q₃, a line inside the box marks the median, and whiskers extend to the minimum and maximum. Some boxplots also mark outliers separately.

When do you use the five number summary instead of the mean and standard deviation?

The five number summary is preferred when data is skewed or contains outliers, because the median and quartiles are resistant to extreme values. The mean and standard deviation work best for roughly symmetric distributions. If you are unsure about the shape of the data, the five number summary is a safer choice for a quick overview.

Five Number Summary vs. Mean & Standard Deviation

	Five Number Summary	Mean & Standard Deviation
What it reports	Min, Q₁, Median, Q₃, Max	Mean (center) and standard deviation (spread)
Sensitivity to outliers	Resistant — median and quartiles change little with extreme values	Sensitive — one outlier can shift the mean and inflate the standard deviation
Best used when	Data is skewed or has outliers	Data is roughly symmetric with no extreme outliers
Visual display	Boxplot	Often paired with a histogram or normal curve

Why It Matters

The five number summary is one of the first tools taught in statistics courses for describing a data set quickly. You will use it whenever you draw or interpret a boxplot, compare distributions side by side, or check for outliers using the interquartile range. It also appears frequently on standardized tests such as the AP Statistics exam.

Common Mistakes

Mistake: Forgetting to sort the data before finding quartiles and the median.

Correction: Always arrange the data in ascending order first. Quartiles and the median are defined by position in the ordered list, so unsorted data will give wrong results.

Mistake: Including the median in both the lower and upper halves when the data set has an odd number of values.

Correction: When the data count is odd, the median is an actual data point. Exclude it from both halves before computing Q₁ and Q₃. For example, in a set of 7 values, the lower half has the first 3 values and the upper half has the last 3.

Related Terms

Boxplot — Visual display of the five number summary
First Quartile — Q₁, the 25th percentile of the data
Median of a Set of Numbers — The middle value, Q₂ in the summary
Third Quartile — Q₃, the 75th percentile of the data
Set — A collection of data values being summarized
Interquartile Range — Q₃ − Q₁, measures the spread of the middle 50%
Outlier — Extreme values identified using the IQR from the summary
Range — Max − Min, the total spread of the data