Transcript KS3: Stem and Leaf Diagrams

Year 8: Data Handling 2
Dr J Frost ([email protected])
Learning Outcomes: To understand stem and leaf
diagrams, frequency polygons, box plots and
cumulative frequency graphs.
Last modified: 28th November 2013
Stem and Leaf Diagram - What is it?
Suppose this “stem and leaf diagram” represents the lengths of beetles.
1
2
3
4
5
4
1
2
0
0
The key tells us how two
digits combine.
2
5
1
1
4
6
2
1
5
6 6 7 7 8 8
2 4 5 6 7 7 7 7 8
2
Value represented = 4.5cm
?
These numbers represent the
first digit of the number.
Key:
2 | 1 means 2.1cm
The numbers must be in
order.
These numbers represent the second.
Example
Here are the weights of a group of cats. Draw a stem-and-leaf diagram to represent this
data.
36kg
15kg
1
35kg 50kg 11kg 36kg 38kg 47kg 12kg 30kg 18kg 57kg
1 2 5 ?8
2
?
Key:
3 | 8 means
? 38kg
5 6? 6 8
?3
0
4
7
?
5
0 7
?
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).
Your turn
Here is the brain diameter of a number of members of 8IW. Draw a stem and leaf diagram
representing this data.
1.3cm 2.1cm
5.3cm 2.0cm
1
3 3 7 ?9
2
0 1
?
?3
2 3
?
4
2 6
?
5
3
?
1.7cm
?
Median width = 2.1cm
?
Lower Quartile = 1.7cm
?
Upper Quartile = 4.2cm
4.2cm
3.3cm 3.2cm 1.3cm 4.6cm 1.9cm
Key:
3 | 8 means
? 3.8cm
Exercises
Q1 on your provided worksheet.
(Ref: Yr8-DataHandlingWorksheet.doc)
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.
?
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.
100
Frequency
Shoe Size
80
60
40
20
8
9
10
Shoe size
11
12
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
14
120 ≤ x < 130
0
12
130 ≤ x < 140
4
10
Modal class interval:
100 ≤ x < ?
110
16
Join the points
up with straight
lines.
8
6
4
This is known as
a frequency
polygon.
2
90
100
110
120
130
140
Frequency Polygons – Exercises on sheet
Q1
Q2
?
?
b) 30 < x ≤?40
c) 16%
?
b) 20 < x ≤?30
c) 16%
?
The Whole Picture
Histogram
Widths (cm):
4, 4, 7, 9, 11, 12, 14, 15,
15, 18, 28, 42
Determine
Median/LQ/UQ
Frequency
Polygon
Grouped
Frequency Table
Cumulative
Frequency Table
Width (cm)
Frequency
Width (cm)
Cum Freq
0 < w < 10
4
0 < w < 10
4
10 < w < 25
6
0 < w < 25
10
25 < w < 60
2
0 < w < 60
12
Median/LQ/UQ
class interval
Box Plots
Cumulative
Frequency
Graph
Estimate of
Median/LQ/UQ/num
values in range
Recap: 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.
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%
LQ
50%
Median
75%
UQ
100%
Median/Quartile Revision
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
Median:
15

15.5

16

Lower:
13

13.5

14

UQ:
17

18

19

Interquartile
Range:
3?
Range:
?
12
(Click to vote)
Quickfire Quartiles
LQ
Median
UQ
1?
2?
3?
1, 2, 3, 4
?
1.5
?
2.5
?
3.5
1, 2, 3, 4, 5
1.5
?
2?
4.5
?
2?
3.5
?
5?
1, 2, 3
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.
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?
0 1 1 2 4 5 5 6 7 8 10 10 14 14 14 14 15 16 17
29 31 31 37 37 38 39 40 40 41 43 44
LQ
Use the 8? th item
Use
1
4
𝑛+1
𝑡ℎ
item
Median
? th item
Use the 16
Use
1
2
𝑛+1
𝑡ℎ
item
Use
3
4
𝑛+1
𝑡ℎ
item
UQ
Use the
? th
24
item
What if there’s lots of items?
Num items
LQ
Median
UQ
15
4th?
8?th
? th
12
23
6th?
12? th
18? th
39
10?th
20? th
30? th
47
12?th
24? th
36? th
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
Sketch
Sketch
Sketch
22
Sketch
Sketch
range
IQR
0
5
How is the IQR represented
in this diagram?
10
Sketch
15
20
25
30
How is the range
Sketch
represented in this diagram?
Box Plots
Sketch a box plot to represent the given weights of cats:
5lb, 6lb, 7.5lb, 8lb, 8lb, 9lb, 12lb, 14lb, 20lb
Minimum
5
?
Maximum
Median
?
20
0
?
8
4
Lower Quartile Upper Quartile
8
6.75
12
Sketch
?
16
13
20
?
24
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
5
?
Maximum
Median
?
22
0
?
13
4
Lower Quartile Upper Quartile
8
12
12
Sketch
?
16
16
20
?
24
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)
?
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
1?
9.7 < t ≤ 9.9
4
5?
9.9 < t ≤ 10.05
10
15?
10.05 < t ≤ 10.2
17
32?
TOTAL
?
32
Time (s)
Frequency
Cum Freq
9.6 < t ≤ 9.7
1
1
Plot
9.7 < t ≤ 9.9
4
5
Plot
9.9 < t ≤ 10.05
10
15
Plot
10.05 < t ≤ 10.2
17
32
Plot
Cumulative Frequency Graphs
This graph tells us how
many people had “this
value or less”.
Median = 10.07s
?
Cumulative Frequency
28
24
Lower Quartile
?
= 9.95s
20
16
Upper Quartile
?
= 10.13s
12
8
Interquartile Range
?
= 0.18s
4
0
9.5
9.6
9.7
9.8
9.9
10.0
Time (s)
10.1
10.2
10.3
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.
32
Cumulative Frequency
28
26
? runners
24
Estimate how many
runners had a time more
than 9.95
20
16
32 – ?
8 = 24 runners
12
Estimate how many
runners had a time
between 9.8s and 10s
8
4
11 –?3 = 8 runners
0
9.5
9.6
9.7
9.8
9.9
10.0
Time (s)
10.1
10.2
10.3
Time (s)
Frequency
Cum Freq
9.6 < t ≤ 9.7
1
1
Plot
9.7 < t ≤ 9.9
4
5
Plot
9.9 < t ≤ 10.05
17
22
Plot
10.05 < t ≤ 10.2
10
32
Plot
Sketch Line
Cumulative Frequency
Cumulative Frequency Graph
Frequency Polygon
18
28
16
Frequency
32
24
20
16
14
12
10
12
8
8
4
4
2
0
9.5
9.6
9.7
9.8
9.9
Time (s)
10.0
10.1
10.2
0
10.3 9.5
9.6
9.7
9.8
9.9
Time (s)
10.0
10.1
10.2
Worksheet
Cumulative Frequency Graphs
Printed handout. Q5, 6, 7, 8, 9, 10
Reference: GCSE-GroupedDataCumFreq
5?
23
?
35
?
39
?
40
?
?
179
?
34
?
Lower Quartile = 16
?
Upper Quartile = 44.5
?
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.
?
25 < 𝐴 ≤ 35
44
100
134
153
160
?
30
?
?
40.9 − 24.1
? = 16.8
C?
D?
B?
A?