What is a Box Plot

Welcome to our What is a Box Plot page.

This page will help you understand all about box plots and how they work.

There is also help on how to create a box plot from a set of data and lots of worked examples.

What is a Box Plot

A box plot is a visual way of recording a set of data values.

The data from a box plot can show us 5 facts:

  • the minimum data value;
  • the 1st quartile value (the lower quartile);
  • the median value.
  • the 3rd quartile value (the upper quartile);
  • the maximum data value;

box plot diagram labelled image

Using these facts, we can also quickly use the box plot work out:

  • the range of the data;
  • the interquartile range of the data set;
  • whether the data is skewed to the left or right.

What is the Median?

The median is the midpoint (or middle value) of a set of numbers.

It is found by ordering the set of numbers and then finding the middle value in the set.

The median can also be thought of as the 2nd quartile or Q2.

About ½ (50%) of the data values are lower than the median and ½ (50%) of the data values are higher than the median.

how to find median example 1

How to Make a Box Plot (or Box and Whisker Plot) from a set of data

How to Create a Box Plot

  1. Put the data in order, smallest to largest.
  2. Find the middle (median) value - also called Q2.
  3. Find the lower and upper quartiles by finding the middle of the first half of the data and the middle of the second half of the data.
  4. Draw a number line using a suitable scale, from the minimum to maximum data value.
  5. Draw a rectangle above the number line from the Q1 (first quartile) value to Q3 (third quartile) value.
  6. Draw a vertical line inside the rectangle to show the median value.
  7. Draw horizontal lines (these are the whiskers) from the either side of the rectangle to the minimum and maximum data points.

Creating a Box Plot - Examples

Example 1) Create a box plot of these test scores:

box plots example 1a

Step 1) Put the scores in order, smallest first.

box plots example 1b

Step 2) Find the median (middle) score and use the median to split the scores into two halves.

box plots example 1c

Step 3) Find the 1st and 3rd quartiles by finding the middle of the first half of the data and the middle of the second half of the data.

box plots example 1d

Step 4) Draw a number line from the minimum to maximum values.

Step 5) Draw a rectangle about the number line from the Q1 (first quartile) value to the Q3 (3rd quartile) value.

Step 6) Draw a vertical line inside the rectangle to show the median value.

box plots example 1e

Step 7) Draw the whiskers from either side of the rectangle to the minimum and maximum data points.

box plots example 1f


Example 2) Create a box plot of this data which shows the number of visitors each hour at a museum:

box plots example 2a

Step 1) Put the scores in order, smallest first.

box plots example 2b

Step 2) Find the median (middle) score and use the median to split the scores into two halves.

box plots example 2c

The median value is (55 + 59) ÷ 2 = 57

Step 3) Find the 1st and 3rd quartiles by finding the middle of the first half of the data and the middle of the second half of the data.

box plots example 2d

The 1st quartile is halfway between 47 and 53 = (47 + 53) ÷ 2 = 50

The 3rd quartile is halfway between 64 and 71 = (64 + 71) ÷ 2 = 67 ½

Step 4) Draw a number line from the minimum to maximum values.

Step 5) Draw a rectangle about the number line from the Q1 (first quartile) value to the Q3 (3rd quartile) value.

Step 6) Draw a vertical line inside the rectangle to show the median value.

box plots example 2e

Step 7) Draw the whiskers from either side of the rectangle to the minimum and maximum data points.

box plots example 2f

What is a Box Plot - Interpreting a Box Plot

As well as creating a box plot, another important skill is to be able to interpret a box plot from a given set of data.

We already know that the box plot directly shows us the 5 facts:

  • the minimum value
  • the maximum value
  • the median value (Q2)
  • the 1st quartile value (Q1)
  • the 3rd quartile value (Q3)

box plot diagram labelled 2 image

Using this information we can also find the following information:

  • the range of data values (by subtracting the minimum value from the maximum value)
  • the interquartile range (by subtracting the 1st quartile value (Q1) from the 3rd quartile value (Q3)

We also know that about 25% of the data set is below the 1st quartile (Q1), and about 75% of the data set is above Q1.

We also know that about 75% of the data set is below the 3rd quartile (Q3) and about 25% of the data set is above Q3.

What is a Box Plot - Data Distribution

By looking at the box plot, we can also tell how the data has been distributed.

We can see if the data is skewed to the right (positively skewed) or skewed to the left (negatively skewed), or if the data has no skew (symmetrically distributed.

We can tell the skew of the data by looking at the position of the median in the box.

Skewed Right or Positively Skewed

If the median is towards the left of the center of the box, and the whisker on the left is shorter, then we say that the data is skewed right, or positively skewed.

box plot skewed right image

Skewed Left or Negatively Skewed

If the median is on the right hand side of the box, and the whisker on the right is shorter, then we say that the dats is skewed left, or negatively skewed.

box plots skewed left image

No Skew or Symmetric Distribution

If the median is in the center of the box, and the whiskers are about the same length, then we say that there is no skew and that the data has a symmetric distribution.

box plots symmetric distribution image

We are now going to put all this into practice by looking at a ready-made box plot and identifying all the information it shows.

Interpreting a Box Plot - Examples

This box plot shows the number of books read by a group of students during a semester.

box plot example 3

So what information can this box plot tell us?

box plot example 3a

  • The minimum value is 11 books.
  • The 1st quartile (Q1) is 14 books.
  • The median value is 16 books.
  • The 3rd quartile (Q3) is 21 books.
  • The maximum value is 24 books.

The range of books read is maximum value - minimum value = 24 - 11 = 13 bookx

The interquartile range is the 3rd quartile (Q3) - the 1st quartile (Q1) = 21 - 14 = 7 books.

The median is to the left of the center line, and the whisker is slightly shorter on the left hand side, so the data is skewed right (or positively skewed).

About ¼ (25%) of the students read fewer than 14 books and 75% read more than 14 books.

About ¾ (75%) of the students read fewer than 21 books and 25% read more than 21 books.

About half the students read fewer than 16 books, and half read more than 16 books.

More Recommended Math Worksheets

Take a look at some more of our worksheets similar to these.

Lower Quartile and Upper Quartile

If you want some more help identifying the lower and upper quartiles, then take a look at this page.

Looking for some Box Plot Worksheets?

We also have a selection of box plot worksheets.

The worksheets will help you to practice creating and interpreting box plots.

 

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