Stem and Leaf Diagrams

A stem and leaf diagram shows a list of numbers in rows. Each number is split into two parts: the stem, written to the left of the line, and the leaf, the last digit, written to the right. For 23 the stem is 2 and the leaf is 3

Each leaf is one number. The row 2 \mid 0\ 3\ 3\ 7 shows the four numbers 20, 23, 23 and 27. The key tells you how to read the diagram, so a stem and leaf diagram always needs one.

In the exam you are usually asked to show a list of numbers in an ordered stem and leaf diagram with a key, for 3 marks, or to find the median, mode or range from a diagram, for 1 or 2 marks. Harder questions ask for a probability, or ask you to compare the numbers with a second group.

In an ordered stem and leaf diagram, the leaves in each row go from smallest to largest. So the smallest value is the first leaf in the top row, and the largest value is the last leaf in the bottom row.

To find how many values there are, count the leaves. Don't count the stems.

Read the words of the question carefully. More than 45 does not include 45 itself, but 45 or more does.

To find a probability from the diagram, count the leaves you want and put that number over the total number of leaves:

\text{probability} = \frac{\text{number of values you want}}{\text{total number of values}}

• Find the smallest and largest numbers. Use the tens digits as the stems, and write them down a column with a line to their right.
• Go along the list. Write each units digit as a leaf on its row, and cross the number off as you go.
• Draw the diagram again with the leaves in each row in order, smallest first.
• Add a key with units, such as 2 \mid 3 represents 23 minutes.
• Count the leaves. There must be as many leaves as there are numbers in the list.

Write every stem in order, with no gaps, even if a row has no leaves. Keep the leaves in neat, evenly spaced columns, so that a row with more leaves looks longer.

For bigger numbers, the leaf is still the last digit and the stem is everything before it. For heights from 126 cm to 158 cm, use the stems 12, 13, 14 and 15, with a key such as 14 \mid 3 represents 143 cm. A number such as 107 has stem 10 and leaf 7

Don't use 1 as the only stem for numbers such as 126 and 158. Every number would then go on the same row, and the diagram would show nothing.

The mode is the value that comes up most often. Look for a leaf repeated in the same row, then use the key to write the whole value. In the row 2 \mid 4\ 4\ 4 the mode is 24, not 4

The median is the middle value when the values are in order. An ordered diagram has already put them in order. If there are n values, the median is the value in position

\frac{n+1}{2}

Count the leaves along the rows, starting from the smallest, until you reach that position.

If n is even, the median is halfway between the two middle values. With 12 values, \frac{12+1}{2} = 6.5, so the median is halfway between the 6th and 7th values.

The range is the largest value subtract the smallest value. It is one number, not two.

\text{range} = \text{largest value} - \text{smallest value}

For the mean, add up all the values, the whole numbers and not just the leaves, then divide by how many values there are.

Some questions give you a diagram for one group and the median and range of a second group, and ask you to compare them. Work out the median and the range for the diagram first.

• Compare the medians: the group with the higher median has higher values on average.
• Compare the ranges: the group with the bigger range is more spread out, so its values are less consistent.

Writing down the numbers is not enough. Say which is bigger, and what that means for the people or things in the question. For example: the median time for Team A (24 minutes) is lower than the median time for Team B (31 minutes), so on average Team A were quicker.

• Leaving the leaves out of order. In an ordered diagram each row goes from smallest to largest, so draw a rough diagram first, then a neat ordered one.
• Missing out a number. Cross off each number as you place it, then check that the number of leaves matches the number of values.
• Leaving out the key or its units. Write a key with an example and units, such as 2 \mid 3 represents 23 minutes.
• Giving a leaf as the answer. Use the key to write the whole value: 5 \mid 7 is 57, not 7, and the mode is the most common value, not the most common digit.
• Writing the range as two numbers. The range is one number, the largest value subtract the smallest, such as 72 - 38 = 34
• Writing down the values without comparing them. Say which median and which range is bigger, and what that means in the question.