## What you need to know

A stem-and-leaf diagram is a way of displaying a collection of numbers. The ‘stem’ consists of the first part of every number (usually the first digit/digits) and the ‘leaf’ consists of the latter part of every number (usually the last digit). In any stem-and-leaf diagram, there must be a key that explains how the diagram should be read and understood.

Let’s see an example of what a stem and leaf diagram looks like and how we can use it.

Example: Ramona wanted to learn more about the ages of her colleagues. She collected the ages of all of them and recorded her results using a stem-and-leaf diagram.

a) Write down the mode.

b) Write down the median.

Calculate the range.

Before answering the question, let’s refer to the key shown above and to the right of the diagram. It tells us that the numbers in the ‘stem’ are tens, whilst the numbers in the ‘leaf’ section are ones (this makes sense since we’re talking about ages).

This means that where there is a ‘7’ in the ‘2’ row (or a ‘7’ leaf with a ‘2’ stem) it’s referring to the number 27.

a) The mode means the number that appears most often. Here we can see that the number 2 appears three times in the ‘3’ row – more times than any other value appears. Each of these 2s refer to the number 32, therefore the mode is 32.

b) The median is the middle value. A common method for finding the median is to cross out the smallest value, then the largest, then the 2nd smallest, then the 2nd largest, and so on until you only have 1 value left.

Note: do this in pencil in case you have to read off the numbers to answer later questions!

In this case, doing so we’re left with a ‘2’ leaf and a ‘3’ stem, therefore the median is 32.

c) The range is the largest value takeaway the smallest. Reading of the diagram, we see the smallest value is 19 (a ‘9’ leaf with a ‘1’ stem), whilst the largest is 55 (a ‘5’ leaf with a ‘5’ stem). So, we get

$\text{range }=55-19=36$

Now we’ve seen how to read a stem-and-lead diagram, we’ll see about how we can draw one.

Example: Scott measured the height of all the people in his choir. The values, in cm, are given below.

$154,\,\,180,\,\,176,\,\,153,\,\,162,\,\,165,\,\,154,\,\,186,$

$190,\,\,187,\,\,176,\,\,176,\,\,172,\,\,182,\,\,177,\,\,169$

Draw a stem-and-leaf diagram of Scott’s data.

To draw a stem-and-leaf we must first figure out what our key will be. The data is all 3-digit numbers, so we can’t follow the exact structure of the last example. Instead, the stem will be the first two digits, whilst the leaf will be the last digit. This means our stem will have to include: 15, 16, 17, 18, and 19. Without filling in the data, the bare bones of our stem-and-leaf diagram will look like the figure on the right.

At this point, it’s probably a good idea to rewrite your data to be in numerical order before committing it to the diagram.

$153,\,\,154,\,\,154,\,\,162,\,\,165,\,\,169,\,\,172,\,\,176,$

$176,\,\,176,\,\,177,\,\,180,\,\,182,\,\,186,\,\,187,\,\,190$

Now, we can fill in the ‘leaf’ values (the last digit of every value) alongside the correct ‘stem’ value, to complete the stem-and-leaf diagram. The result is shown below.

Note: one extra benefit of stem-and-lead diagrams is that if you tilt your head to the right, they seem to look a little like a bar chart, with each row of numbers forming a bar. This gives you a good visualisation of the data without having to draw any additional graphs.

## Example Questions

All of these values are 2-digit numbers, so the first digits will be in the ‘stem’ section, and the second digits will be in the ‘leaf’ section. This means our set-up, including the key, should look like

Note: if you chose a different value to use in the key then that is equally correct.

Now, we’ll rewrite the values in order:

$14,\,\,19,\,\,22,\,\,24,\,\,29,\,\,32,\,\,35,\,\,35,\,\,36,\,\,38,\,\,41,\,\,47$

Then, filling in the ‘leaf’ section with the last digit of each value, the completed diagram should look like

a) The only number to appear more than once is the ‘2’ leaf on the ‘1’ part of the stem. Therefore, the mode is 12.

b) The largest value is 51 (‘1’ leaf on the ‘5’ stem) and the smallest is 9 (‘9’ leaf on the ‘0’ stem). So, we get

$\text{range }=51-9=42$

c) All the journeys under 20 minutes are the ones that appear in the ‘0’ stem or the ‘1’. Reading off the diagram, we see that there are 8 of those, and in total there are 15 times recorded. So, we get the percentage of Wallace’s journeys that were under 20 minutes to be

$\dfrac{8}{15}\times 100=53.3\%\text{ (1dp)}$