Stem and leaf diagrams Revision Notes for OCR GCSE Maths

Stem and leaf diagrams

1. What is a Stem and Leaf Diagram?

A stem and leaf diagram is simply a method of presenting data in a way that breaks each number into two parts:

The stem (the leading digit or digits of the number).

The leaf (the final digit of the number).

It's a form of data presentation that groups similar numbers together, making it easier to summarise data compared to a long, unorganised list.

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Example: Stem and Leaf Diagram

Suppose you have the following set of numbers representing the times (in minutes) that students took to complete a task:

Data: $31, 33, 35, 36, 38, 42, 43, 45, 48$

You can organise these into a stem and leaf diagram like this:

Stem	Leaves
$3$	$1\ 3\ 5\ 6\ 8$
$4$	$2\ 3\ 5\ 8$

The stem represents the tens digit (e.g., $30s, 40s$ ).
The leaves represent the ones digits of the numbers. So, for the number $31$ , the stem is $3$ (for the tens place), and the leaf is $1$ (for the ones place). This process is repeated for all the other numbers.

Key: The key at the bottom explains how to read the diagram. In this case:

Key:3∣1=31 minutes

This means that $"3 | 1"$ represents the number 31.

2. Constructing a Stem and Leaf Diagram

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Example: Mr Barton's Wake-up Times

Here are the times (in minutes) that it takes Mr Barton to get out of bed after his alarm has sounded on a Monday morning:

Data:

$12, 6, 20, 24, 52, 41, 35, 55, 12, 11, 3, 25, 39, 41, 52, 13, 59, 18, 22, 29, 35$

Step 1: Identify the Stems

The stems are the first digit (or digits) of each number. For this example, where most values range between $0$ and $59$ , the stems will be the tens digits:

For numbers in the $0s$ (e.g., $6$ ), the stem is $0$ .
For numbers in the $10s$ (e.g., $12$ ), the stem is $1$ .
Continue this for the other tens values ( $2, 3, 4, 5$ ). Our stems will be: $0, 1, 2, 3, 4$ , and $5.$

Step 2: Add the Leaves

The leaves are the units digits of the numbers. We will now place each leaf (the last digit of each number) next to its respective stem. First, we create an unordered stem and leaf diagram, simply adding the numbers in the order they appear:

Stem	Leaves
$0$	$6\ 3\ 2$
$1$	$2\ 1\ 3\ 3 \ 8\ 2$
$2$	$0\ 4\ 5\ 2\ 9$
$3$	$5\ 5\ 8\ 9$
$4$	$1\ 1$
$5$	$2\ 5\ 2\ 9$

Step 3: Organise the Leaves in Order

Now we order the leaves for each stem, so they appear in increasing order:

Stem	Leaves
$0$	$2\ 3\ 6$
$1$	$1\ 2\ 2\ 3\ 3\ 8$
$2$	$0\ 2\ 4\ 5\ 9$
$3$	$5\ 5\ 8\ 9$
$4$	$1\ 1$
$5$	$2\ 2\ 5\ 9$

Step 4: Add a Key

Finally, include a key to help others interpret the diagram. The key explains how to read the numbers in the diagram. In this case:

Key:2∣0=20 minutes

This tells us that the stem " $2$ " combined with the leaf " $0$ " represents the number 20.

3. Finding the Median and Interquartile Range from a Stem and Leaf Diagram

(a) Finding the Median

The median is the middle number in an ordered list of data. If there is an even number of data points, the median will be the average of the two middle numbers.

Step-by-Step Process to Find the Median:

Count the total number of data points
Find the middle position
Calculate the median

Count the total number of data points: In this example, we have 24 data points.

Find the middle position: Since 24 is an even number, the median will be the average of the 12th and 13th numbers.

In the stem and leaf diagram, the 12th and 13th numbers are 26 and 27 (the leaves are $5$ and $9$ from the $20$ stem).

Calculate the median:

Median=\frac{26+27}2=26.5

Thus, the median time it takes Mr Barton to wake up is 26.5 minutes.

(b) Finding the Interquartile Range ( $IQR$ )

The interquartile range ( $IQR$ ) is a measure of the spread of the middle 50% of the data. The $IQR$ is calculated as the difference between the upper quartile ( $Q3$ ) and the lower quartile ( $Q1$ ).

Step-by-Step Process to Find the $IQR$ :

Find the Lower Quartile ( $Q1$ )
Find the Upper Quartile ( $Q3$ )
Calculate the $IQR$

Find the Lower Quartile ( $Q1$ ): $Q1$ is the median of the lower half of the data (the first 12 values).

The middle of the lower half (6th and 7th numbers) is 13 and 13 (from the $10$ stem).

Q1=\frac{13+13}2=13

Find the Upper Quartile ( $Q3$ ): Q3 is the median of the upper half of the data (the last 12 values).

The middle of the upper half (18th and 19th numbers) is 40 and 40 (from the $30$ and $40$ stems).

Q3=\frac{40+40}2=40

Calculate the $IQR$ : The interquartile range is the difference between the upper quartile and the lower quartile:

IQR=Q3−Q1=40−13=27

Thus, the $IQR$ for Mr Barton's wake-up times is 27 minutes.

4. Advantages of Stem and Leaf Diagrams

Keeps original data: Unlike bar charts and histograms, no data is lost. Every piece of data is shown in its original form.
Easily ordered: It provides an effective way of ordering and displaying relatively small sets of data, which helps in identifying patterns or calculating statistics like the median or mode.

5. Disadvantages of Stem and Leaf Diagrams

Time-consuming for large data sets: While stem and leaf diagrams are practical for small sets, they become impractical for very large data sets (e.g., hundreds of data points). Sorting and organising large amounts of data in this format can be complex and time-consuming.
Difficult to interpret with many data points: If you have too many leaves, the diagram can become cluttered and hard to interpret.

Stem and leaf diagrams (OCR GCSE Maths): Revision Notes