1. “Teach A Level Maths” Statistics 1
Stem and Leaf Diagrams
© Christine Crisp

2. Statistics 1 AQA EDEXCEL MEI/OCR OCR
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3. You met some statistical diagrams when you did GCSE.
The next three presentations and this one remind you of them and point out some details that you may not have met before.
We will start with stem and leaf diagrams
( including back-to-back ).
Stem and leaf diagrams are sometimes called stem plots.

4. and the leaves the units
e.g. The table below gives the number of hours worked in a particular week by a sample of 30 men 35 41 33 31 30 45 36 51 32 28 34 42 21 46
I’ll use intervals of 5 hours to draw the diagram i.e , etc.
Weekly hours of 30 men 5 1 4 6 2 3 8 5 4 3 2
The stem shows the tens . . .
and the leaves the units
e.g. 46 is 4 tens and 6 units

5. and the leaves the units
e.g. The table below gives the number of hours worked in a particular week by a sample of 30 men 35 41 33 31 30 45 36 51 32 28 34 42 21 46
I’ll use intervals of 5 hours to draw the diagram i.e , etc. 5 1 4 6 2 3 8
Weekly hours of 30 men
Weekly hours of 30 men
The stem shows the tens . . .
and the leaves the units
e.g. 46 is 4 tens and 6 units

6. and the leaves the units
e.g. The table below gives the number of hours worked in a particular week by a sample of 30 men 35 41 33 31 30 45 36 51 32 28 34 42 21 46
I’ll use intervals of 5 hours to draw the diagram i.e , etc. 5 1 4 6 2 3 8
Weekly hours of 30 men
Weekly hours of 30 men
The stem shows the tens . . .
and the leaves the units
e.g. 46 is 4 tens and 6 units
N.B. 35 goes here . . .
not in the line below.

7. and the leaves the units
e.g. The table below gives the number of hours worked in a particular week by a sample of 30 men 35 41 33 31 30 45 36 51 32 28 34 42 21 46
I’ll use intervals of 5 hours to draw the diagram i.e , etc. 5 1 4 6 2 3 8
Weekly hours of 30 men
The stem shows the tens . . .
and the leaves the units
e.g. 46 is 4 tens and 6 units
We must show a key.
Key: 3 5 means 35 hours

8. 5 1 4 6 2 3 8
Weekly hours of 30 men
Key: 3 5 means 35 hours
If you tip your head to the right and look at the diagram you can see it is just a bar chart with more detail.
Points to notice:
The leaves are in numerical order
The diagram uses raw ( not grouped ) data

9. Finding the median and quartiles
Finding these is easy because the data are in order. 1 2 8 4 3 6 5
Weekly hours of 30 men
Key: 3 5 means 35 hours
Median: The median is the middle item, . . .
so with 30 observations we need the th item, the average of 15 and 16.
Tip: To find the middle use where n is the number of items of data.

10. Finding the median and quartiles
Finding these is easy because the data are in order. 1 2 8 4 3 6 5
Weekly hours of 30 men
15th
Key: 3 5 means 35 hours
Median: The median is the middle item, . . .
so with 30 observations we need the th item, the average of 15 and 16.

11. Finding the median and quartiles
Finding these is easy because the data are in order. 1 2 8 4 3 6 5
Weekly hours of 30 men
15th
16th
Key: 3 5 means 35 hours
Median: The median is the middle item, . . .
so with 30 observations we need the th item, the average of 15 and 16.

12. Finding the median and quartiles
Finding these is easy because the data are in order. 1 2 8 4 3 6 5
Weekly hours of 30 men
15th
16th
Key: 3 5 means 35 hours
Median: The median is the middle item, . . .
so with 30 observations we need the th item, the average of 15 and 16.
Since the 15th and 16th items are both 34, the median is 34.
( If the values are not the same we average them. )

13. Finding the median and quartiles
Finding these is easy because the data are in order. 1 2 8 4 3 6 5
Weekly hours of 30 men
7th
Key: 3 5 means 35 hours
For the lower quartile (LQ) we first need

14. Finding the median and quartiles
Finding these is easy because the data are in order. 1 2 8 4 3 6 5
Weekly hours of 30 men
7th
8th
Key: 3 5 means 35 hours
For the lower quartile (LQ) we first need

15. Finding the median and quartiles
Finding these is easy because the data are in order. 1 2 8 4 3 6 5
Weekly hours of 30 men
7th
8th
Key: 3 5 means 35 hours
For the lower quartile (LQ) we first need
The lower quartile is 32.

16. 7th value: 32 8th value: 36 0.75 of 4 is 3.
Finding the median and quartiles
If the values of the 7th and 8th observation are not the same, we interpolate to find the LQ.
e.g. If we had
7th value: 32
8th value: 36
and we want the 7·75th value, we need to add 0·75 of the gap between the 7th and 8th to the 7th value.
So,
The gap is 36 – 32 = 4.
0.75 of 4 is 3.
LQ = = 35

17. Finding the median and quartiles1 2 8 4 3 6 5
Weekly hours of 30 men
Key: 3 5 means 35 hours
For the upper quartile (UQ), we first need
( or from the top end )

18. Finding the median and quartiles1 2 8 4 3 6 5
Weekly hours of 30 men
23rd
Key: 3 5 means 35 hours
For the upper quartile (UQ), we first need
( or from the top end )

19. Finding the median and quartiles1 2 8 4 3 6 5
Weekly hours of 30 men
23rd
24th
Key: 3 5 means 35 hours
For the upper quartile (UQ), we first need
( or from the top end )

20. = 36 – 32 = 4 Finding the median and quartiles 1 2 8 4 3 6 56 5
Weekly hours of 30 men
23rd
24th
Key: 3 5 means 35 hours
For the upper quartile (UQ), we first need
( or from the top end )
The upper quartile is 36.
The interquartile range (IQR) = UQ - LQ
= 36 – 32 = 4

21. SUMMARY
Stem and Leaf diagrams are used for small, raw data sets (not grouped data).
e.g. 1 2 8 4 3 6 5
Weekly hours of 30 men
Key: 3 5 means 35 hours
leaves
stem
The diagram must have a title and a key.
The leaves are in numerical order ( away from the stem )
continued

22. The median is the th piece of data.
( If necessary, average 2 data items )
The quartiles are found at the th position and the th position.
The interquartile range (IQR) = UQ - LQ
( upper quartile – lower quartile )
If necessary, we can interpolate to find the LQ and UQ.

23. Back-to-back stem and leaf diagrams can be used to compare 2 sets of data relating to the same subject.
In our example we could add the data for 30 women. 1 2 8 4 3 6 5
Weekly hours
Men
Key: 3 5 means 35 hrs
Women 4 3 8 6 5 2 1
Notice how easily we can compare the variability of the 2 data sets
Key: 5 3 means 35 hrs

24. Exercise January Max Temperatures 1985 to 2005 Braemar Sheffield 9 2 82 8 1 2 4 5 9 7 1 2 2 4 8 8 5 2 1 6 4 5 9 8 6 1 5 2 3 4 5 9 8 4 3 4 4 9 7 5 1 3 4 7 5 2 6 1 1
Key: 3 4 means 4·3C
Key: 3 4 means 3·4C
Find the medians and quartiles. What can you say about the temperatures of the 2 places?

25. Answer: 4 5 6 7 8 9
January Max Temperatures 1985 to 2005
Sheffield 1 2 3
Braemar
Key: 3 4 means 3·4C
Key: 3 4 means 4·3C
Median LQ UQ
Braemar
4·4
3·6
5·95
Sheffield
7·1
5·35
8·15
The places have similar variability in temperature but Sheffield is about 2 ·7C warmer than Braemar.

27. The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied.
For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

28. SUMMARY
This is used for small, raw data sets (not grouped data). 5 5 1
Weekly hours of 30 men
e.g. 4 4 5 6
leaves 4 4 1 1 2
stem 3 3 5 5 5 5 5 5 6 6 3 3 1 1 2 2 2 2 3 3 3 4 4 4 2 2 8
Key: 3 5 means 35 hours 2 2 1
The diagram must have a title and a key.
The leaves are in numerical order ( away from the stem )

29. The median is the th piece of data.
The quartiles are found at the th position and the th position.
( If necessary, average 2 data items )
If necessary, we can interpolate to find the LQ and UQ.
The interquartile range (IQR) = UQ - LQ
( upper quartile – lower quartile )

30. 1 2 8 4 3 6 5
Key: 3 5 means 35 hours
Weekly hours
Men
Back-to-back stem and leaf diagrams can be used to compare 2 sets of data relating to the same subject.
In our example we could add the data for 30 women.
Women
Key: 5 3 means 35 hours
Notice how easily we can compare the variability of the 2 data sets