Stem & Leaf Diagrams (Edexcel GCSE Statistics)

Revision Note

Stem & Leaf Diagrams

What is a stem-and-leaf diagram?

  • A stem-and-leaf diagram is a simple but effective way of showing data

    • the raw data is still available as the numbers themselves create the diagram

  • A stem-and-leaf diagram

    • puts data into order

    • puts data into classes (groups)

  • A stem-and-leaf diagram makes patterns in the data easy to see

    • as the data is in order it is useful for finding the median and quartiles

Example of stem and leaf diagram
  • Stem-and-leaf diagrams are particularly useful for two-digit data but can be used for longer numbers

    • two-digit data could be something like 23 but could also be 2.3

    • due to this, it is essential a stem-and-leaf diagram has a key

Stems and Leaves, IGCSE & GCSE Maths revision notes

How do I draw a stem-and-leaf diagram?

  • The digits from each value in the data are split into two – stems and leaves

    • e.g. the data value 36 would be split into a stem of 3 and a leaf of 6

  • A stem can have more than one leaf

    • So the stems become our classes in our data

      • e.g. the stem of 3 becomes a class interval – covering values from 30 to 39

      • Any other values in the 30s would join the same stem/class as additional leaves

To draw a stem-and-leaf diagram

  • STEP 1
    Draw a rough version first

    • Always draw a rough stem-and-leaf diagram first

      • Though you can skip this step if the data is given to you in order

    • Work through the data one value at a time,

      • Split each value into a stem and a leaf

      • Lightly cross out each data value as you use it (to ensure none are missed out or used twice)

    • This gets the data into the correct basic form

      • grouped into its stems

      • with the correct number of leaves

  • STEP 2
    Draw a final diagram

    • Put the stems in ascending order

    • For each stem, rearrange the leaves into ascending order

    • Ensure your leaves are lined up in neat columns

      • so the size of each stem/class can be easily seen

  • STEP 3
    Add a key to your diagram

    • The key should explain how the values have been split into stems and leaves

      • e.g. does 3|6 mean 36 or 3.6?  

      • The key will let us know!

How do I find the median and quartiles from a stem-and-leaf diagram?

  • For the median ...

    • Lightly cross out numbers from the beginning and end

      • i.e. cross out the lowest number and the highest number

      • the highest number will be at the end of the last stem

    • Continue crossing out the next lowest/highest numbers until you meet in the middle

      • if one number remains in the middle, then it is the median

      • if two numbers remain find the midpoint between them

      • (if the midpoint isn't obvious then add them together and divide by 2)

  • For the lower quartile ...

    • Find the median first

      • Repeat the process for the median but on the lower half of the data

      • (i.e. up to but not including the median)

  • For the upper quartile ...

    • Find the median first

      • Repeat the process for the median but on the upper half of the data

      • (i.e. from but not including the median)

  • The interquartile range is the difference between the lower and upper quartiles

    • IQR = UQ - LQ

  • Remember to put the number back into its original format

    • i.e. don't only use the leaf

    • This is a common mistake when finding the median and quartiles

      • e.g. writing Median = 6 instead of Median = 36

What else can I do with a stem-and-leaf diagram?

  • Many things are easy to see in a stem-and-leaf diagram

    • The data is arranged into classes so it is easy to see the modal class

      • i.e. the stem with the greatest number of leaves

    • The data is arranged in order, so the maximum and minimum can be identified easily

      • This can be used to find the range (maximum value - minimum value)

    • Outliers can be easily identified (and removed if necessary)

  • You can compare data sets

    • Remember to comment on a measure of central tendency ('average')

      • (use the median)

    • and a measure of dispersion ('spread')

      • (use the interquartile range)

  • A box plot ('box and whisker diagram') can be drawn for the data

    • For this you need the median, upper and lower quartiles, and maximum and minimum values

What are back-to-back stem-and-leaf diagrams?

  • These are used when it is helpful for the data to be split into two comparable categories

    • e.g. boy/girl, child/adult, UK/non-UK, etc.

  • This makes it easier to compare two data sets

    • i.e. instead of having two separate stem-and-leaf diagrams

Example of a back-to-back stem and leaf diagram
  • Note that the leaves on the left-hand side of the stems (Boys) increase from the centre outwards

    • i.e. from the centre to the left

  • In all other respects back-to-back stem and leaf diagrams are the same as regular stem and leaf diagrams

Examiner Tips and Tricks

  • Accuracy is important

    • (Lightly) tick off values as you add them to a stem-and-leaf diagram

    • Check you have the right number of data values in total on your diagram

      • Other checks can include ensuring the median has the same number of values above and below it

Worked Example

A hospital is investigating a new drug that claims to reduce blood pressure.  They give a set of patients the new drug and three hours later record the amount the blood pressure of every patient has decreased by.  The results, measured in mmHG (millimetres of mercury), are given below.

12        31        24        18        21        34        40        19        23        17        16

(a) Draw a stem-and-leaf diagram to show these results.

The data is not in order so the first step is to draw a rough diagram

All values are two digit, so split each so that the first (tens) digit is a stem and the second (units) digit is a leaf

  Blood pressure reduction

1

2

8

9

7

6

3

1

4

2

4

1

3

4

0


For the final diagram put stems and leaves in order and add a key

  Blood pressure reduction

1

2

6

7

8

9

2

1

3

4

3

1

4

4

0

Key: 1|2 means a blood pressure reduction of 12 mmHG

(b) Use your stem-and-leaf diagram to find the median blood pressure reduction and the interquartile range.

For the median cross off highest and lowest numbers until we meet in the middle

 

  Blood pressure reduction

1

2

6

7

8

9

2

1

3

4

3

1

4

4

0

 
The median is a leaf of 1 in the stem of 2

Median = 21

Repeat for the lower half and upper half of the data to find the lower and upper quartile

Blood pressure reduction

1

2

6

7

8

9

2

1

3

4

3

1

4

4

0


The LQ is a leaf of 7 in the stem of 1; LQ = 17  
The UQ is a leaf of 1 in the stem of 3; UQ = 31
The question asks for the interquartile range

IQR = UQ - LQ = 31 - 17 = 14

Don't forget units in the final answer!

Median = 21 mmHG
Interquartile range = 14 mmHG

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Roger B

Author: Roger B

Expertise: Maths

Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.

Dan Finlay

Author: Dan Finlay

Expertise: Maths Lead

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.