Five-number summary and boxplot calculator

Q: Does the five-number summary include the mean?

No. The five-number summary is the minimum, Q₁, median, Q₃, and maximum, all of which are positions in the sorted data. The mean is a separate measure of center, and you can compute it with the standard deviation calculator.

Q: Can I find outliers from just the five-number summary?

Partly. You can build the fences from Q₁ and Q₃ at Q₁ - 1.5 × IQR and Q₃ + 1.5 × IQR, then check whether the minimum or the maximum falls beyond them. If both ends sit inside the fences, there are no outliers, but pinpointing or counting every outlier in between takes the full data set.

By Jude Wallis · Published July 28, 2026

A five-number summary calculator sorts your data and returns the minimum, first quartile Q1, median, third quartile Q3, and maximum. It also finds the IQR = Q3 - Q1, builds 1.5 x IQR fences, flags outliers, and draws a boxplot. For 4, 8, 11, 12, 14, 15, 16, 18, 20, 42 you get 4, 11, 14.5, 18, 42.

Data valuesSeparate with commas, spaces, or new lines. Quartiles use the median-excluded (TI-84) convention.

min | Q1 | median | Q3 | max

1 | 3 | 5.5 | 8 | 40

Outliers by the 1.5 x IQR rule: 40. On a modified boxplot these plot as separate points.

Steps

1.sort the data; median = 5.5
2.Q1 = median of the lower half = 3; Q3 = median of the upper half = 8
3.IQR = Q3 - Q1 = 8 - 3 = 5
4.fences: Q1 - 1.5(IQR) = -4.5; Q3 + 1.5(IQR) = 15.5
5.values outside the fences: 40

AP Statistics: Unit 1 (topics 1.7 Summary Statistics for One Quantitative Variable, 1.8 Graphical Representations of Summary Statistics for One Quantitative Variable). In the Fall 2026 AP Statistics course, the five-number summary, the IQR, the 1.5 x IQR outlier rule, and the boxplot fall under Unit 1 topics 1.7 and 1.8.

What a five-number summary is

A five-number summary describes a dataset with five values: the minimum, the first quartile, the median, the third quartile, and the maximum. Together they show the extremes and how the middle of the data is spread.

The first quartile, written $Q_1$ , is the value with about a quarter of the data at or below it. The median is the middle value. The third quartile, written $Q_3$ , is the value with about three quarters of the data at or below it. The minimum and maximum are the smallest and largest values.

This five number summary calculator takes a list of numbers, sorts them, and returns all five values along with the range, the interquartile range, and any outliers. You type or paste your data and read the results without sorting by hand.

The quartiles, the IQR, and the convention this calculator uses

The interquartile range, written IQR, is the distance between the two quartiles:

IQR = Q_3 - Q_1

It measures the spread of the middle 50% of the data and ignores the two ends, so a single large value cannot inflate it. That makes this IQR calculator more reliable than the range on skewed data.

To locate the quartiles, the calculator uses the median-excluded method, the convention taught in AP Statistics and used by TI graphing calculators. First find the median of the whole sorted list, then split the data into a lower half and an upper half. $Q_1$ is the median of the lower half and $Q_3$ is the median of the upper half.

When the number of values $n$ is odd, the single middle value (the median) is left out of both halves. When $n$ is even, the median falls between two values, and every value goes into one half or the other.

One honest caveat: software differs. Spreadsheet functions and some statistics packages use interpolation and can report slightly different quartiles on the same data. For AP work, stay with the median-excluded method this calculator uses.

How the calculator flags outliers with 1.5 x IQR fences

To flag outliers, the calculator builds two cutoff values called fences from the IQR:

Lower fence $= Q_1 - 1.5 \times IQR$
Upper fence $= Q_3 + 1.5 \times IQR$

Any value below the lower fence or above the upper fence is flagged as an outlier, and values between the fences are treated as ordinary. This is the same 1.5 x IQR rule covered in the outliers guide.

The multiplier 1.5 is a standard convention. It catches genuinely unusual points without labeling too many ordinary values, and it holds up on skewed data because the quartiles it depends on are resistant to extreme values.

The boxplot this calculator draws

A boxplot is a picture of the five-number summary, and this boxplot calculator draws a modified one that shows outliers separately.

The box spans from $Q_1$ to $Q_3$ , so it covers the middle 50% of the data, with a line drawn at the median. The whiskers reach out from the box toward the smallest and largest values, but they stop at the most extreme values that are still inside the fences. Any outliers are drawn as isolated points beyond the whiskers.

Reading the plot from left to right tells you the shape. If the median line sits near the center of the box and the whiskers are about equal, the data is roughly symmetric. If the box or a whisker is stretched to one side, the data is skewed that way. To draw one by hand, see how to make a boxplot.

How to use the calculator

Enter your data as a list of numbers separated by commas or spaces. The calculator sorts the values and returns:

the five-number summary: minimum, $Q_1$ , median, $Q_3$ , maximum
the range (maximum minus minimum) and the IQR
the lower and upper fences
a list of any outliers
a boxplot drawn from the summary

Check the sorted list the tool shows against your input so you can catch a mistyped value. Then read the five-number summary and the outlier list together, because an outlier flagged here is exactly the point that sits beyond a whisker on the boxplot.

Five-number summary and outliers for 10 reading counts

Ten students read these numbers of books over the summer: 15, 4, 20, 12, 42, 8, 16, 11, 18, 14. Find the five-number summary, the range, the IQR, the 1.5 x IQR fences, and any outliers.

Sort the data: 4, 8, 11, 12, 14, 15, 16, 18, 20, 42. There are $n = 10$ values.
Read the ends: minimum $= 4$ and maximum $= 42$ .
Find the median. With 10 values it lies between the 5th value (14) and the 6th value (15), so the median $= (14 + 15)/2 = 14.5$ .
Split the sorted data at the median. Lower half: 4, 8, 11, 12, 14. Upper half: 15, 16, 18, 20, 42.
Find $Q_1$ , the median of the lower half. The middle of 4, 8, 11, 12, 14 is 11, so $Q_1 = 11$ .
Find $Q_3$ , the median of the upper half. The middle of 15, 16, 18, 20, 42 is 18, so $Q_3 = 18$ .
Write the five-number summary: minimum 4, $Q_1$ 11, median 14.5, $Q_3$ 18, maximum 42.
Compute the range: $42 - 4 = 38$ .
Compute the IQR: $IQR = Q_3 - Q_1 = 18 - 11 = 7$ .
Compute $1.5 \times IQR = 1.5 \times 7 = 10.5$ .
Lower fence $= Q_1 - 1.5 \times IQR = 11 - 10.5 = 0.5$ . Upper fence $= Q_3 + 1.5 \times IQR = 18 + 10.5 = 28.5$ .
Compare each value to the fences. The minimum 4 is above 0.5, so there is no low outlier. The value 42 is above the upper fence of 28.5, so 42 is an outlier.

Five-number summary: minimum 4, Q1 11, median 14.5, Q3 18, maximum 42. Range 38, IQR 7. The fences are 0.5 and 28.5, and 42 is the only outlier.

Boxplot for the same data

Using the summary from the previous example (minimum 4, Q1 11, median 14.5, Q3 18, maximum 42, with 42 flagged as an outlier), describe where each part of the modified boxplot goes.

Draw the box from $Q_1 = 11$ to $Q_3 = 18$ , with a line at the median 14.5.
Plot the outliers first. The value 42 is beyond the upper fence of 28.5, so it is drawn as a separate point, not inside a whisker.
The upper whisker stops at the largest value that is not an outlier. With 42 removed, the largest remaining value is 20, so the upper whisker ends at 20.
The lower whisker stops at the smallest value that is not an outlier. The minimum, 4, is above the lower fence of 0.5, so the lower whisker ends at 4.

Box from 11 to 18 with a median line at 14.5; lower whisker to 4; upper whisker to 20; the outlier 42 drawn as an isolated point beyond the upper whisker.

Frequently asked questions

What is the difference between the range and the IQR?

The range is the maximum minus the minimum, so it depends entirely on the two most extreme values. The IQR is $Q_3 - Q_1$ , the spread of the middle 50% of the data, so a single outlier barely moves it. That resistance is why the IQR is the spread reported alongside the median.

Why does my calculator give different quartiles than a spreadsheet?

Because they use different quartile conventions. This calculator and TI graphing calculators use the median-excluded method, while some spreadsheet functions interpolate between values. Both are valid, but on AP problems use the median-excluded quartiles this tool reports.

Does the five-number summary include the mean?

No. The five-number summary is the minimum, $Q_1$ , median, $Q_3$ , and maximum, all of which are positions in the sorted data. The mean is a separate measure of center, and you can compute it with the standard deviation calculator.

Can I find outliers from just the five-number summary?

Partly. You can build the fences from $Q_1$ and $Q_3$ at $Q_1 - 1.5 \times IQR$ and $Q_3 + 1.5 \times IQR$ , then check whether the minimum or the maximum falls beyond them. If both ends sit inside the fences, there are no outliers, but pinpointing or counting every outlier in between takes the full data set.