1. Year 8: Charts & Quartiles
Dr J Frost
Learning Outcomes: To understand stem and leaf diagrams, frequency polygons, box plots and cumulative frequency graphs.
Last modified: 19th November 2015

2. Lower and Upper Quartile
Suppose that we line up everyone in the school according to height.
50%
The height of the person 25% along the line is known as the:
lower quartile
We already know that the median would be the middle person’s height.
50% of the people in the school would have a height less than them.
The upper quartile is the height of the person 75% along the data. ?

3. Check your understanding
50% of the data has a value more than the median.
75% of the data has a value less than the upper quartile.
25% of the data has a value more than the upper quartile.
75% of the data has a value more than the lower quartile. ? ? ? ? 0%
25%
50%
75%
100% LQ
Median UQ

4. 12 Median/Quartile Example 12, 13, 14, 14, 15, 16, 16, 17, 19, 24 ? ?
LQ: Find the median of the first half.
UQ: Find the median of the second half.
Here are the ages of 10 people at Pablo’s party. Choose the correct value.
12, 13, 14, 14, 15, 16, 16, 17, 19, 24
(Click to vote) 16
15.5 16
Median: 13
13.5 14
LQ: 17 18 19
UQ:
Interquartile
Range:
𝑈𝑄−𝐿𝑄 =17−14 =3 ? ? 12
Range:

5. Another Example
Rule for lower quartile:
Even num of items: find median of bottom half.
Odd num of items: throw away middle item, find medium of remaining half.
-2, 4, 6, 7, 9, 10, 13, 17, 20, 24, 25, 30, 50
(Click to vote) 10 13 15
Median: 6
6.5 7
LQ: 24
24.5 25
UQ:
Interquartile
Range:
𝑈𝑄−𝐿𝑄 =24.5−6.5 =18 ? ? 52
Range:

6. Quickfire Quartiles LQ Median UQ 1, 2, 3 1 2 3 ? ? ? 1, 2, 3, 4 1.5
2.5
3.5 ? ? ?
1, 2, 3, 4, 5
1.5 3
4.5 ? ? ? 2
3.5 5
1, 2, 3, 4, 5, 6
Rule for lower quartile:
Even num of items: find median of bottom half.
Odd num of items: throw away middle item, find medium of remaining half.

7. What if there’s lots of items?
There are 31 items, in order of value. What items should we use for the median and lower/upper quartiles? LQ
Use the 8 th item ?
Use 𝑛+1 𝑡ℎ item
Median ?
Use 𝑛+1 𝑡ℎ item
Use the 16th item ?
Use 𝑛+1 𝑡ℎ item UQ
Use the 24th item

8. What if there’s lots of items?
Num items LQ
Median UQ ? ? ? 15
4th
8th
12th ? ? ? 23
6th
12th
18th ? ? ? 39
10th
20th
30th
Emphasise that GCSE exams always seem to use a number of items 1 less than a multiple of 4. ? ? ? 47
12th
24th
36th

9. Check Your Understanding
Here are the ages of 15 different cats:
3, 4, 8, 9, 10, 11, 11, 13, 14, 15, 15, 16, 17, 18, 20 ?
Lower Quartile = 9
Median = 13
Upper Quartile = 16
Interquartile Range = 7 ? ? ?

10. Stem and Leaf Diagram - What is it?
Suppose this “stem and leaf diagram” represents the lengths of beetles.
The key tells us how two digits combine. 1 2 3 4 5 4
Key:
2 | 1 means 2.1cm
Value represented = 4.5cm
The numbers must be in order.
These numbers represent the second.
These numbers represent the first digit of the number. ?

11. Example
Here are the weights of a group of cats. Draw a stem-and-leaf diagram to represent this data.
36kg kg kg 50kg 11kg 36kg 38kg 47kg 12kg 30kg 18kg 57kg 1 2 3 4 5 ? ? 7
0 7
Key:
3 | 8 means 38kg ? ? ?
What do you think are the advantages of displaying data in a stem-and-leaf diagram?
Shows how the data is spread out.
Identifies gaps in the values.
All the original data is preserved (i.e. we don’t ‘summarise’ in any way). ? ? ?

12. Your turn ? ? ? ? ? ? ? Median width = 2.1cm ? Lower Quartile = 1.7cm
Here is the brain diameter of a number of members of 8IW. Draw a stem and leaf diagram representing this data.
1.3cm cm cm cm cm cm cm cm 1.3cm cm 1.9cm 1 2 3 4 5 ?
0 1
2 3
2 6 3 ?
Key:
3 | 8 means 3.8cm ? ? ? ? ?
Median width = 2.1cm ?
Lower Quartile = 1.7cm ?
Upper Quartile = 4.2cm ?

13. Quick Exercises ? ? 15 Q1 and Q2 on your provided worksheet. 0 5 7 8 8
(Ref: Yr8-ChartsAndQuartilesWorksheet.doc) ?
3 3 5
3 | 5 = 35 mins ? 15

14. Quick Exercises
Q1 and Q2 on your provided worksheet.
(Ref: Yr8-ChartsAndQuartilesWorksheet.doc) ? 32 ?
45 – 21 = 24

15. Frequency Diagram
Suppose we wanted to plot the following data, where each value has a frequency.
A suitable representation of this data would be a bar chart. ?
100 80 60 40 20
Shoe size
Frequency
Shoe Size
Frequency 8 26 9 42 10
103 11 34 12 5
When bar charts have frequency on the y-axis, they’re known as frequency diagrams.

16. Frequency Polygons
But suppose that we had data grouped into ranges.
What would be a sensible value to represent each range?
IQ (x)
Frequency
90 ≤ x < 100 2
100 ≤ x < 110 15
110 ≤ x < 120 8
120 ≤ x < 130
130 ≤ x < 140 4 16 14 12 10 8 6 4 2
Join the points up with straight lines.
Modal class interval:
100 ≤ x < 110 ?
This is known as a frequency polygon.

17. Frequency Polygons – Exercises on sheet? ?
b) 30 < x ≤ 40
c) 16% ? ?
b) 20 < x ≤ 30
c) 16% ? ?

18. The Whole Picture Widths (cm): Determine Median/LQ/UQ
Frequency Polygon
Histogram
Grouped Frequency Table
Cumulative Frequency Table
Widths (cm):
4, 4, 7, 9, 11, 12, 14, 15, 15, 18, 28, 42
Width (cm)
Frequency
0 < w < 10 4
10 < w < 25 6
25 < w < 60 2
Width (cm)
Cum Freq
0 < w < 10 4
0 < w < 25 10
0 < w < 60 12
Determine Median/LQ/UQ
Median/LQ/UQ class interval
Cumulative Frequency Graph
Estimate of Median/LQ/UQ/num values in range
Box Plots

19. Box Plots
Box Plots allow us to visually represent the distribution of the data.
Minimum
Maximum
Median
Lower Quartile
Upper Quartile 3 27 17 15 22
Sketch
Sketch
Sketch
Sketch
Sketch
range
IQR
How is the IQR represented in this diagram?
How is the range represented in this diagram?
Sketch
Sketch

20. Box Plots ? ? ? ? ? 5lb, 6lb, 7.5lb, 8lb, 8lb, 9lb, 12lb, 14lb, 20lb
Sketch a box plot to represent the given weights of cats:
5lb, 6lb, 7.5lb, 8lb, 8lb, 9lb, 12lb, 14lb, 20lb
Minimum
Maximum
Median
Lower Quartile
Upper Quartile 5 20 8
6.75 13 ? ? ? ? ?
Sketch

21. Box Plots
Sketch a box plot to represent the given ages of people at Dhruv’s party:
5, 12, 13, 13, 14, 16, 22
Minimum
Maximum
Median
Lower Quartile
Upper Quartile 5 22 13 12 16 ? ? ? ? ?
Sketch

22. Comparing Box Plots
Box Plot comparing house prices of Croydon and Kingston-upon-Thames.
Croydon
Kingston
£100k £150k £200k £250k £300k £350k £400k £450k
“Compare the prices of houses in Croydon with those in Kingston”. (2 marks)
For 1 mark, one of:
In interquartile range of house prices in Kingston is greater than Croydon.
The range of house prices in Kingston is greater than Croydon.
For 1 mark:
The median house price in Kingston was greater than that in Croydon.
(Note that in old mark schemes, comparing the minimum/maximum/quartiles would have been acceptable, but currently, you MUST compare the median) ? ?

23. Test Your Understanding

24. 100m times at the 2012 London Olympics
Modal class interval
10.05 < t ≤ 10.2 ? ?
Median class interval
10.05 < t ≤ 10.2 ?
Estimate of mean
10.02 ?
Time (s)
Frequency
Cum Freq
9.6 < t ≤ 9.7 1
9.7 < t ≤ 9.9 4 5
9.9 < t ≤ 10.05 10 15
10.05 < t ≤ 10.2 17 32
TOTAL ? ? ? ?

25. ? ? ? ? ? Median = 10.07s Lower Quartile = 9.95s Upper Quartile
Cumulative Frequency Graphs
Time (s)
Frequency
Cum Freq
9.6 < t ≤ 9.7 1
9.7 < t ≤ 9.9 4 5
9.9 < t ≤ 10.05 10 15
10.05 < t ≤ 10.2 17 32
Plot
‘Cumulative’ means
‘running total’.
This graph tells us how many people had “up to this value”.
Plot ?
Plot 32 28 24 20 16 12 8 4
Plot
Median = s ?
Lower Quartile
= 9.95s
Cumulative Frequency ?
Upper Quartile
= s ?
Interquartile Range
= 0.18s ?
Time (s)

26. ? ? ? Cumulative Frequency Graphs
A Cumulative Frequency Graph is very useful for finding the number of values greater/smaller than some value, or within a range.
Cumulative Frequency Graphs
Estimate how many runners had a time less than 10.15s.
26 runners
Estimate how many runners had a time more than 9.95
32 – 8 = 24 runners
Estimate how many runners had a time between 9.8s and 10s
11 – 3 = 8 runners 32 28 24 20 16 12 8 4 ?
Cumulative Frequency ? ?
Time (s)

27. Cumulative Frequency Graph Frequency Polygon
Cumulative Frequency Graph vs Frequency Polygon
(students always get them mixed up!)
Time (s)
Frequency
Cum Freq
9.6 < t ≤ 9.7 1
9.7 < t ≤ 9.9 4 5
9.9 < t ≤ 10.05 17 22
10.05 < t ≤ 10.2 10 32
Plot
Plot
Sketch Line
Plot
Plot
Cumulative Frequency Graph
Frequency Polygon 32 28 24 20 16 12 8 4 18 16 14 12 10 8 4 2
Cumulative Frequency
Frequency
Time (s)
Time (s)

28. Worksheet Cumulative Frequency Graphs
Printed handout. Q5, 6, 7, 8, 9, 10
Solutions on next slides.

29. 5 23 35 39 40 ? ? ? ? ? ?
179 ?

30. ? 34
Lower Quartile = 16
Upper Quartile = 44.5 ? ?

31. We previously found:
Minimum = 9, Maximum = 57, LQ = 16, Median = 34, UQ = 44.5 ?
1 mark: Range/interquartile range of boys’ times is greater.
1 mark: Median of boys’ times is greater. ?

32. ? 44
100
134
153
160
25<𝐴≤35 ? ? ? 30
40.9−24.1=16.8 ?

33. C ? D ? B ? A ?

34. ? 73 ?
80 – 65 = 15