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box plot calculator

Updated Sep 18, 2026 By Infinity Calculator
Rate Formulas
Input Mode
Raw Data parses your numbers; Summary Statistics plots a five-number summary directly.
Sample = exclusive interpolation, position p(n+1), variance ÷ (n−1). Population = inclusive linear interpolation, position p(n−1)+1, variance ÷ n.
Data Sets (up to 4 groups)
Group 1 *
Separate values with commas, spaces, tabs or new lines. Minimum 4 valid numbers.
Whisker Mode
Tukey flags points beyond Q1 − 1.5×IQR and Q3 + 1.5×IQR as outliers.
Chart Options
Maximum 120 characters.
Parsed Input
Step-by-Step Solution
Box Plot
Statistical Summary
Extended Dispersion Statistics

Introduction

A box plot is a simple picture that shows how your numbers spread out. This free box plot calculator draws that picture from your data.

Just type or paste your numbers. The tool sorts them, then finds the five-number summary: the minimum, first quartile (Q1), median, third quartile (Q3), and maximum. It also shows the interquartile range (IQR), mean, range, and standard deviation. If a number sits far from the rest, the calculator marks it as an outlier.

You can compare up to four groups on one chart. You can also skip the raw data and enter a five-number summary directly. Pick Tukey (1.5×IQR) whiskers, min/max whiskers, or your own percentile cutoffs. Change the colors, flip the plot sideways or upright, add a title, and turn the grid on or off.

Every answer comes with step-by-step math, so you can see how each quartile was found. When you are done, save your chart as a PNG or SVG image, or download the summary table as a CSV file. It works well for homework, class projects, lab reports, and quick data checks at work.

How to use our Box Plot Calculator

Type in your numbers or a five-number summary. The calculator draws the box plot and shows the median, quartiles, IQR, whiskers, and outliers with step-by-step math.

Input Mode: Pick "Raw Data" if you have a list of numbers. Pick "Summary Statistics" if you already know the min, quartiles, and max.

Data Type: Choose "Sample" if your numbers are part of a bigger group. Choose "Population" if they are the whole group. This changes how quartiles and variance are found.

Group Label: Name each data set, like "Class A." The name shows on the chart and in the tables.

Values: Paste or type at least 4 numbers. Split them with commas, spaces, tabs, or new lines.

Add another group: Click this to compare up to 4 data sets side by side. Use "Remove" to drop a group.

Minimum, Q1, Median, Q3, Maximum: In Summary Statistics mode, enter these five values. They must go from smallest to largest.

Mean (optional): Add the average if you know it. It shows as a diamond on the plot.

Whisker Mode: "Tukey" marks points past 1.5×IQR as outliers. "Min/Max" stretches whiskers to the smallest and largest values. "Custom Percentile" lets you set your own cutoffs.

Low and High percentile cutoff: Only used with Custom Percentile. Set the low end (0 to 49) and the high end (51 to 100) for the whiskers.

Plot Orientation: Show the boxes across the page (horizontal) or up and down (vertical).

Color Theme: Pick the color used for the boxes and whiskers.

Chart Title: Write a title for your graph, up to 120 letters.

Notched boxes: Check this to add a notch showing the median's confidence range.

Show grid: Check this to add light grid lines behind the plot.

Sort data automatically: Check this to list your numbers in order from low to high.

Highlight outliers: Check this to show far-off points as red triangles.

Show mean marker: Check this to mark the average with a diamond.

Include summary table in export: Check this to add the stats table to your saved PNG or SVG image.

Generate Plot: Click this to build the box plot, steps, and summary tables. Use "Clear All" to start over, "Reset Options to Defaults" to undo chart changes, and the export buttons to save your plot as PNG, SVG, or CSV.

What Is a Box Plot?

A box plot (also called a box-and-whisker plot) is a picture that shows how a set of numbers spreads out. Instead of showing every value, it shows five key points. That makes it easy to see the middle of your data, how wide the data is, and if any values look strange.

The Five-Number Summary

Every box plot is built from five numbers:

  • Minimum: the smallest value.
  • Q1 (first quartile): the value where 25% of the data falls below.
  • Median (Q2): the middle value; half the data is below it.
  • Q3 (third quartile): the value where 75% of the data falls below.
  • Maximum: the largest value.

Parts of the Plot

The box stretches from Q1 to Q3. The middle 50% of your data sits inside that box. The line inside the box is the median. The whiskers are the lines that stick out from each side of the box and show the rest of the data. Dots or triangles past the whiskers are outliers. A diamond marker shows the mean (the average).

IQR and Outliers

The interquartile range (IQR) is the width of the box: IQR = Q3 − Q1. It tells you how spread out the middle half of the data is. The most common rule for spotting outliers is Tukey's 1.5×IQR rule:

  • Lower fence = Q1 − 1.5 × IQR
  • Upper fence = Q3 + 1.5 × IQR

Any value below the lower fence or above the upper fence is marked as an outlier. Whiskers then stop at the last real data value inside the fences.

Whisker Choices

  • Tukey (1.5×IQR): the standard method, which flags outliers.
  • Min / Max: whiskers reach the smallest and largest values, so nothing is an outlier.
  • Custom percentile: you pick the cutoffs, like the 5th and 95th percentile.

Sample vs. Population Quartiles

There is more than one way to find quartiles. The sample (exclusive) method puts a quartile at position p(n+1) and divides the variance by n − 1. The population (inclusive) method uses position p(n−1)+1 and divides the variance by n. Use sample when your numbers are part of a bigger group. Use population when your numbers are the whole group. The two methods can give slightly different Q1 and Q3 values, and that is normal.

How to Read the Shape

  • If the median sits near the center of the box, the data is fairly even (symmetric).
  • If the median is closer to Q1 and the right whisker is long, the data is skewed right.
  • If the median is closer to Q3 and the left whisker is long, the data is skewed left.
  • A short box means values are close together. A long box means they are spread out.

Notched Boxes

A notch is a pinch in the box around the median. It shows a rough 95% confidence range for the median, found with median ± 1.57 × IQR ÷ √n. If the notches of two boxes do not overlap, their medians are likely truly different. When the notch would be wider than the box, it is skipped because it would not mean much.

Why Box Plots Are Useful

Box plots are great for comparing groups side by side, like test scores from different classes or wait times at different stores. They pack a lot of information into a small space, show skew and spread at a glance, and make unusual values easy to spot. They are used in school statistics, science labs, business reports, and quality checks.

Formulas used

Quartile position (sample / exclusive method)
\text{pos}(Q_p) = p\,(n+1)
Quartile position (population / inclusive method)
\text{pos}(Q_p) = p\,(n-1) + 1
Quartile value by linear interpolation
Q_p = x_{(i)} + f\left(x_{(i+1)} - x_{(i)}\right), \quad i = \lfloor \text{pos} \rfloor,\; f = \text{pos} - i
Interquartile range and range
IQR = Q_3 - Q_1, \qquad R = x_{\max} - x_{\min}
Tukey fences for outlier detection
\text{Lower fence} = Q_1 - 1.5\,IQR, \qquad \text{Upper fence} = Q_3 + 1.5\,IQR
Mean, variance and standard deviation
\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}, \qquad s = \sqrt{\frac{\sum_{i=1}^{n}(x_i-\bar{x})^2}{n-1}}, \qquad \sigma = \sqrt{\frac{\sum_{i=1}^{n}(x_i-\bar{x})^2}{n}}
Standard error of the mean and coefficient of variation
SEM = \frac{s}{\sqrt{n}}, \qquad CV = \frac{s}{|\bar{x}|} \times 100\%
Skewness (adjusted Fisher-Pearson for samples)
g_1 = \frac{m_3}{m_2^{3/2}}, \quad m_k = \frac{1}{n}\sum_{i=1}^{n}(x_i-\bar{x})^k; \qquad G_1 = g_1\,\frac{\sqrt{n(n-1)}}{n-2}

Frequently asked questions

What percent of the data is inside the box on a box plot?

About 50%. The box runs from Q1 to Q3, so it holds the middle half of your numbers.

The rest breaks down like this:

  • About 25% sits below the box (left or lower whisker).
  • About 25% sits inside the box below the median.
  • About 25% sits inside the box above the median.
  • About 25% sits above the box (right or upper whisker).

Each of the four parts holds roughly a quarter of the data, even when one part looks much longer than another.

Does a box plot show the mean or the median?

The line inside the box is always the median, not the mean. A plain box plot does not show the mean at all.

Some plots add the mean as a diamond or a small dot. Comparing the two tells you about shape:

  • Mean close to the median means the data is fairly even.
  • Mean much higher than the median means the data is skewed right.
  • Mean much lower than the median means the data is skewed left.

Why do Excel, my calculator, and my textbook give different Q1 and Q3 values?

Because there is more than one accepted way to find quartiles. Nobody is wrong.

The two common ways are:

  • Exclusive (sample): leaves the median out when splitting the data. Position = p(n+1).
  • Inclusive (population): includes the median in each half. Position = p(n−1)+1.

They match when the data size fits neatly, and differ by a small amount when it does not. Just say which method you used when you report your answer.

Why does my box plot have a missing whisker?

A whisker disappears when it has zero length. That happens when Q1 equals the smallest value, or Q3 equals the largest value.

Common causes:

  • Many repeated numbers, like lots of zeros.
  • A very small data set.
  • Data that is bunched hard against one end, like scores capped at 100.

It is not an error. It just means at least a quarter of your values are all the same.

What is a modified box plot?

A modified box plot draws outliers as separate dots or stars, and stops each whisker at the last real value inside the 1.5×IQR fences.

A plain (simple) box plot stretches the whiskers all the way to the smallest and largest values, so nothing gets flagged.

Most box plots you see in class, in software, and in reports today are modified box plots.

Should I remove outliers from my data?

No, not automatically. An outlier is a signal, not always a mistake.

Check it first:

  • Typo or bad reading? Fix it or drop it, and say so in your notes.
  • Real but unusual value? Keep it. It may be the most important number in your set.

Never delete a point just because it makes a chart look messy. If you do remove any value, report the result both with and without it.

How many numbers do you need to make a box plot?

You need at least 5 numbers to get a real five-number summary, but that is the bare minimum.

With fewer than about 8 to 10 values, the quartiles jump around a lot and the box can be misleading. For a plot people can trust, aim for 15 or more values per group.

For very small sets, a simple dot plot showing every value is often clearer.

Box plot vs histogram: which one should I use?

Use a histogram when you want to see the shape of one data set, including peaks, gaps, and bumps.

Use a box plot when you want to compare several groups side by side, or when space is tight.

The big trade-off: a histogram can show two peaks, but a box plot cannot. Two very different data sets can produce the exact same box.

What does it mean when two box plots overlap?

Heavy overlap means the two groups share many of the same values, so any difference between them may just be chance.

  • Boxes barely touching or not touching at all: the groups are likely truly different.
  • One median falling inside the other box: the difference is weak.
  • Notched boxes that do not overlap: strong sign the medians really differ.

Box plots hint at a difference. To prove it, run a test like a t-test or Mann-Whitney test.

Can the mean fall outside the box on a box plot?

Yes, though it is uncommon. The mean gets pulled toward extreme values, while the box only depends on rank order.

If one huge value sits far above the rest, the mean can be dragged past Q3. That is a strong sign of heavy skew, and it means the median describes your typical value better than the mean does.

What can a box plot not tell you?

Box plots hide a lot on purpose. They do not show:

  • How many values you have.
  • Whether the data has two peaks (bimodal).
  • Gaps or clusters inside the box.
  • The exact individual values.

Two data sets that look nothing alike can make the same box plot. When shape matters, pair the box plot with a histogram or dot plot.

How do you find the median when there is an even number of values?

Sort the numbers, find the two in the middle, and average them.

Example: 4, 7, 9, 12. The middle two are 7 and 9, so the median is (7 + 9) ÷ 2 = 8.

Notice the median does not have to be one of your actual data values. With an odd count, the median is simply the single middle number.

Are a box plot and a box-and-whisker plot the same thing?

Yes. They are two names for the same chart. "Box-and-whisker plot" is the longer name used in most school textbooks, while "box plot" is the short name used in software and research papers. Statistician John Tukey created it in the 1970s.

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