Five Number Summary Calculator

A five-number summary is a concise summary of the properties of a data set. It consists of five values that divide sorted data into four equal parts and provide insight about the spread and central tendency of the data.

The five values are the minimum, first quartile, median, third quartile, and maximum. These form the basis of box plots which allow you to easily see where data is concentrated, how spread out it is, and whether there are any skewnesses in the distribution.

This calculator will also calculate the interquartile range and range in addition to the five values. Outlier limits, variance and standard deviation can be shown if required. Enter your numbers into the text box to find out the summarized results at a glance.

The values that make up a five-number summary are:

A five-number summary of a sorted data set consists of the following values:

How to calculate a five-number summary.

If you arrange your data in a particular order, then it's simple.

1. Sort the data.The values are sorted in ascending order. This calculator will automatically sort the values for you.

2. Check the minimum and maximum value.The values at the beginning and end of a sorted list are directly the smallest or largest value respectively.

3. Calculate the median.If there are an odd number of data points, the middle value is taken; if even, then the average of the two middle values.

4. Calculate Q1 and Q3.Q1 is the median of the lower half of data values, and Q3 is the median of the upper half of data values.

There are two methods for calculating quartiles and there are reasons why results may differ.

If the number of data points is even, it's easy to split them into two groups with the median as a dividing point. If the number of data points is odd, then the median itself becomes an actual data point, and there are two generally accepted ways to handle this. This is often the most confusing part, since different textbooks and calculation tools use different rules.

Exclusion method (default)

In this method the overall data median is not included in either group and medians are calculated for each group. This method is used by many textbooks and TI graphing calculators and is also the default setting of this calculator.

Inclusion method

This method includes the overall median in both the upper and lower groups, and calculates medians for each group. This method tends to make Q1 and Q3 closer to the center than the exclusive method does.

It is important to note that when the number of data points is even, both methods will always produce the same quartiles. The difference only arises when the number of data points is odd. As neither method is wrong, it is advisable to specify which method was used. By switching the method in the settings of the calculation tool, both results can be displayed.

Example calculation:

We will take the nine values of 3, 7, 8, 5, 12, 14, 21, 13, 18 which are shown at the start of this calculator as an example. In ascending order they are: 3, 5, 7, 8, 12, 13, 14, 18, 21.

Since the number of data points is odd, the median is the middle value, 12. For the exclusion method, the lower halves are 3, 5, 7, 8 and 13, 14, 18, 21. The median of the first half is (5 + 7) / 2 = 6, and the median of the second half is (14 + 18) / 2 = 16.

If we switch to the inclusive method, then both groups will include the median of 12. So the lower halves are 3, 5, 7, 8, 12, and the median is 7. The upper halves are 12, 13, 14, 18, 21, and the median is 14. In this case, the five-number summary is 3, 7, 12, 14, 21. The smallest value, the median, and the largest value are all the same, but the quartiles differ.

Range, interquartile range and box plot:

Two indicators for dispersion can be directly calculated from these five measures.

The range covers the entire data set while the interquartile range (IQR) only considers the middle half and is therefore less sensitive to extreme values.

A box plot is a graphical representation of these values. The rectangle represents the range from Q1 to Q3, and a line indicates the position of the median value. The "whiskers" extend from the minimum to the maximum or, if outliers are present, to the limits. In other words: A five-number summary and a box plot show the same information in different forms.

Outlier limits and 1.5 x IQR

The IQR is also often used to determine outliers. According to the Tukey method, values that are below Q1 and exceed 1.5 IQR or above Q3 and exceed the same amount are considered potential outliers.

Values that exceed the limits are candidates for outliers and should be investigated further. When outlier checking is enabled you can display the limit values of the entered data as well as the number of values identified as outliers.

Applications of five-number summary

In education this method is often used as an introduction to statistics to summarize data sets before creating a box plot.

In business and science it allows the properties of a distribution to be captured quickly in a way that is less sensitive to outliers. Quartiles are useful because they are not directly affected by extreme values, so can show typical trends without being distorted by individual outliers. They are also helpful when comparing several data sets side-by-side.

In sports and business analytics it is used to compare players, seasons or processes at a glance and identify outliers that require further investigation.

Common mistakes.

The data is not sorted first.Each of the following steps assumes that your data is sorted in ascending order. If it's not, then the quartiles won't make sense.

I'm confused about how to calculate quartiles.If the number of data points is odd and your result differs from that in your textbook or a calculator, you are probably using a different method. Check which method is required.

Sometimes, interquartile range is confused with range.The range takes into account all values while the IQR only considers the middle half. Both are indicators of how spread out the data is but they reflect different aspects.

Symbols used:

The table below summarizes the symbols used in previous formulas.

This calculator is for general learning and analysis purposes. If certain rules must apply to quartiles, check whether the exclusive method or inclusive method is required.