How do I find the five-number summary if my dataset has an even number of values?

Sort the data, then split it into two equal halves. The median is the average of the two middle values. The first quartile is the median of the lower half, and the third quartile is the median of the upper half, using the same averaging rule if those halves have an even length.

Can the five-number summary contain decimal values?

Yes. If the sample size is even, the median and quartiles are often averages of two data points, producing decimals. This is correct and does not mean your dataset contains decimals.

What is the five-number summary used for besides box plots?

The five-number summary is a quick way to communicate the center and spread of data in reports or comparisons without displaying the raw dataset. It also forms the basis for identifying outliers using the Outlier Rule: flag any value outside Q_1 - 1.5 \cdot \text{IQR} or Q_3 + 1.5 \cdot \text{IQR}.

Should I use the five-number summary or the mean and standard deviation?

Use the five-number summary if your data is skewed, has outliers, or if you need to understand the shape of the distribution. Use mean and standard deviation when data is normally distributed and you need to make predictions or perform hypothesis tests based on the normal model.

Five Number Summary

The Five Number Summary divides a dataset into quartiles to reveal its center, spread, and range—essential for exploratory data analysis and box plots.

\{x_{\min}, Q_1, \tilde{x}, Q_3, x_{\max}\}

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What each symbol means

What Five Number Summary takes

Symbol	Meaning
$Q_1$	The first quartile (25th percentile): the value below which one quarter of the data falls; if misread as the first data point, you will miss how the bulk of the data clusters.
$Q_3$	The third quartile (75th percentile): the value below which three quarters of the data falls; confusing it with the third point in your list ignores the actual distribution.
$\tilde{x}$	The median (50th percentile): the middle value that splits the dataset in half; treating it as the mean instead can mislead you about typical values when data is skewed.

When to use it

Use the Five Number Summary when you need a quick, robust snapshot of how your data is distributed, especially before creating a box plot.

Level

Usually taught in: Statistics

Worked examples

1. Five-number summary of test scores

Problem

A class of 7 students scored

2, 5, 7, 8, 9, 12, 15

points on a quiz. Find the five-number summary.

$\text{Ordered data: } 2, 5, 7, 8, 9, 12, 15$
The data is already in order, which we need before finding quartiles.
$x_{\min} = 2, \quad x_{\max} = 15$
The minimum is the smallest value and the maximum is the largest value.
$\tilde{x} = 8$
With 7 values, the median is the 4th value (the middle position).
$Q_1 = 5$
The first quartile is the median of the lower half ( $2, 5, 7$ ), which is the middle value 5.
$Q_3 = 12$
The third quartile is the median of the upper half ( $9, 12, 15$ ), which is the middle value 12.

Answer: $\{2, 5, 8, 12, 15\}$

The five values divide the dataset into sections: the bottom 25% is 2–5, the bottom 50% is 2–8, the bottom 75% is 2–12, and the top 25% is 12–15. This summary shows the data is fairly symmetric around 8.

2. Five-number summary with negative values and even count

Problem

The temperatures (in °C) over 8 days were

-3, -1, 0, 2, 5, 6, 8, 10

. Find the five-number summary.

$\text{Ordered data: } -3, -1, 0, 2, 5, 6, 8, 10$
The data is already ordered from smallest (most negative) to largest.
$x_{\min} = -3, \quad x_{\max} = 10$
Identify the extreme values; note that the minimum is negative.
$\tilde{x} = \frac{2 + 5}{2} = 3.5$
With 8 values (even count), the median is the average of the 4th and 5th values ( $2$ and $5$ ).
$\text{Lower half: } -3, -1, 0, 2$
To find $Q_1$ , first identify the lower half (the 4 smallest values).
$Q_1 = \frac{-1 + 0}{2} = -0.5$
The first quartile is the median of the lower half, the average of the 2nd and 3rd values ( $-1$ and $0$ ).
$\text{Upper half: } 5, 6, 8, 10$
To find $Q_3$ , identify the upper half (the 4 largest values).
$Q_3 = \frac{6 + 8}{2} = 7$
The third quartile is the median of the upper half, the average of the 2nd and 3rd values of that subset ( $6$ and $8$ ).

Answer: $\{-3, -0.5, 3.5, 7, 10\}$

This dataset has negative values and an even number of points, requiring averages for all quartiles. The five-number summary reveals that temperatures ranged widely and were not centered around zero.

3. Five-number summary of crew work hours

Problem

A construction crew logged daily work hours over 8 days:

4, 5, 6, 5, 7, 8, 6, 9

hours. Find the five-number summary to understand typical workload.

$\text{Ordered data: } 4, 5, 5, 6, 6, 7, 8, 9$
Sort the hours from least to most; notice that 5 and 6 each appear twice.
$x_{\min} = 4, \quad x_{\max} = 9$
The crew worked as few as 4 hours on the lightest day and as many as 9 on the busiest.
$\tilde{x} = \frac{6 + 6}{2} = 6$
With 8 values, the median is the average of the 4th and 5th values, which are both $6$ .
$\text{Lower half: } 4, 5, 5, 6$
The lower half consists of the 4 smallest values.
$Q_1 = \frac{5 + 5}{2} = 5$
The first quartile is the average of the 2nd and 3rd values of the lower half ( $5$ and $5$ ).
$\text{Upper half: } 6, 7, 8, 9$
The upper half consists of the 4 largest values.
$Q_3 = \frac{7 + 8}{2} = 7.5$
The third quartile is the average of the 2nd and 3rd values of the upper half ( $7$ and $8$ ).

Answer: $\{4, 5, 6, 7.5, 9\}$

The five-number summary shows that the crew typically works between 5 and 7.5 hours (the middle 50%), with a median of 6 hours. The range from 4 to 9 hours suggests variable workload across the week.

Common mistakes

Where Five Number Summary usually goes wrong

The mistake	Why it is wrong	The fix
Confusing $Q_1$ with the 1st data point or $Q_3$ with the 3rd data point in the original list.	Quartiles are positions based on the sorted data's distribution, not the data's index.	Always sort the data first, then find which actual values mark the 25th and 75th percentiles by averaging positions if needed.
Using only half the data to calculate quartiles, or including the median in both the upper and lower halves.	Splitting the data inconsistently produces incorrect quartile values that misrepresent the data's spread.	For odd sample sizes, exclude the median from both halves; for even sample sizes, split into first and second halves without overlap.
Treating the five values as equally spaced or assuming equal spacing means symmetric data.	Quartiles reveal clustering only by comparing the gaps between consecutive values; unequal gaps indicate skew.	Compare the spacings: if $Q_1 - x_{\min}$ is much larger than $x_{\max} - Q_3$ , the lower tail is more spread out (left-skewed).

Tips and when to use something else

The five-number summary is the foundation for a box plot; use it whenever you sketch a box plot by hand.
If your data has outliers, the five-number summary is more robust than the mean and standard deviation because it does not depend on extreme values.
When summarizing a dataset to understand its shape, the five-number summary is essential; to make predictions with normally distributed data, use the mean, standard deviation, and Empirical Rule instead.
Always sort your data first; unsorted data leads to wrong quartile positions.

Related formulas

MedianThe median is the middle value of ordered data, most useful when your data contains outliers that would skew the mean value.Interquartile RangeThe interquartile range measures the spread of the middle 50% of your data, telling you how compressed or spread out typical values are.Range of a Data SetRange measures the spread of a data set by finding the difference between the maximum and minimum values, telling you how dispersed your data is.Outlier RuleThe Outlier Rule identifies data values farther than 1.5 times the interquartile range from the quartiles—use it to flag unusual observations.

Frequently asked questions

How do I find the five-number summary if my dataset has an even number of values?: Sort the data, then split it into two equal halves. The median is the average of the two middle values. The first quartile is the median of the lower half, and the third quartile is the median of the upper half, using the same averaging rule if those halves have an even length.
Can the five-number summary contain decimal values?: Yes. If the sample size is even, the median and quartiles are often averages of two data points, producing decimals. This is correct and does not mean your dataset contains decimals.
What is the five-number summary used for besides box plots?: The five-number summary is a quick way to communicate the center and spread of data in reports or comparisons without displaying the raw dataset. It also forms the basis for identifying outliers using the Outlier Rule: flag any value outside $Q_1 - 1.5 \cdot \text{IQR}$ or $Q_3 + 1.5 \cdot \text{IQR}$ .
Should I use the five-number summary or the mean and standard deviation?: Use the five-number summary if your data is skewed, has outliers, or if you need to understand the shape of the distribution. Use mean and standard deviation when data is normally distributed and you need to make predictions or perform hypothesis tests based on the normal model.

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Reviewed 2026-09-18