Five Number Summary: What is it and how to calculate it

A five number summary is a simple way to describe a dataset using just five key values: the minimum, first quartile (Q1), median, third quartile (Q3), and maximum. These numbers give you a quick snapshot of how data is spread, helping you understand patterns, detect unusual values, and summarize large datasets efficiently. Knowing the five number summary is useful for students, analysts, and anyone working with data, as it simplifies complex information and supports decision-making. In this article, we will explain what each number represents, show how to calculate them by hand, and explore how you can visualize the results in a box plot.

What is a Five Number Summary?

A five number summary is a simple way to describe a dataset. It gives you a quick snapshot of the data’s distribution. The summary consists of five key numbers, which include:

1. Minimum – the smallest value in the dataset.
2. First Quartile (Q1) – the value that separates the lowest 25% of the data.
3. Median (Q2) – the middle value that divides the dataset into two equal halves.
4. Third Quartile (Q3) – the value that separates the highest 25% of the data.
5. Maximum – the largest value in the dataset.

The five number summary is useful for understanding how data is spread. It helps you quickly spot outliers, trends, and the overall range of values.

Unlike the mean or standard deviation, which give a single measure of central tendency or spread, the five number summary shows both the middle values and the extremes. This makes it easier to understand the full picture of your dataset.

How to Calculate the Five Number Summary by Hand

Calculating a five number summary manually (by Hand) is simple and straightforward. Thus, given a sample data, you can easily find the find number summary by following these steps:

1. Arrange all your data points in ascending order from smallest to largest.
2. Identify the minimum and maximum. The first number is the minimum, and the last number is the maximum.
3. Find the median (Q2) – This is the middle value of the dataset. If there is an even number of values, take the average of the two middle numbers.
4. Find Q1 and Q3 – The first quartile (Q1) is the median of the lower half of the data (below the median). The third quartile (Q3) is the median of the upper half of the data (above the median).

While manual calculation is useful for small datasets, it can be time-consuming and error-prone for large datasets. That’s why using an online five number summary calculator is often faster and more accurate.

Example

Suppose you have the following dataset. 3, 7, 8, 5, 12, 14, 21, 13, 18. To calculate the five number summary by hand, follow these steps:

• Step 1: Sort the data in ascending order. The sorted data will be:
  • 3, 5, 7, 8, 12, 13, 14, 18, 21
• Step 2: Identify the minimum and maximum value from the sorted data.
  • Minimum = 3 and Maximum = 21
• Step 3: Find the median (Q2) of the dataset. Since the number of observations is odd (n = 9), then the median valus is the one at the middle.
  • Thus, median = 12 (middle value)
• Step 4: Find the lower quartile (Q1) and Upper quartile (Q3) from the sorted data.
  • Q1 is the median of the lower half values (sorted values). Thus, you need to find the median of 3, 5, 7, 8. This gives (5+7)/2 = 6. Hence, Q1 = 6
  • Q3 is the median of the upper half of the sorted values. In this case, we find the median of these values, 13, 14, 18, 21. Thus, median of the upper half = (14+18)/2 = 16. Hence Q3 = 16

Therefore, the five number summary for the sample data is:

• Minimum = 3
• Q1 = 6
• Median = 12
• Q3 = 16
• Maximum = 21

Want a quick way to compute the 5 number summary? Our online five number summary calculator allows you to compute these 5 key numbers with a click of a button. It also provides a step-by-step explanation of how to compute the numbers by hand.

Visualizing the 5 Number Summary

In some AP Statistics exams, students are actually required to visualize the five number summary, not just calculate it. In these cases, the best and most common way to display it is by using a box plot. A box plot takes the five key values and turns them into a simple visual that shows the spread and shape of your data at a glance.

Here’s a quick breakdown of what each part of a box plot represents:

• The left/lower whisker points to the minimum value in the dataset.
• The start of the box on the left represents Q1, where the lowest 25% of the data ends.
• The line inside the box is the median (Q2), the middle of the dataset.
• The end of the box on the right marks Q3, where 75% of the data lies below it.
• The right/upper whisker extends out to the maximum value.

This visual makes it much easier to understand how the data is spread out, whether it’s skewed, and whether any potential outliers exist.

To make things even clearer, we’ve included an annotated box plot (Figure 2) based on the same example dataset used earlier in this article. You can see exactly where the Min, Q1, Median, Q3, and Max appear on the plot and how they shape the overall distribution.

Ready to explore more insights from your data? Try our Descriptive Statistics Calculator to instantly get the mean, median, mode, variance, standard deviation, and more — all in one place.

Frequently Asked Questions