Measuring Spread of Data – work with a partner
(1)
Read the following scenario and answer the question as best you can:
Suppose you have two types of biscuit packing machines in your factory, A and B. Each type of machine is supposed
to insert 200g of biscuits into a packet. If packets are overweight or underweight you could be losing profits or
running the risk of prosecution from the government trading standards office. In order to guide your future
purchasing decisions, you decide to test which type of machine, A or B, is more reliable. As part of your testing
procedure, you take a random sample of ten packets of biscuits from each machine and record the mass of each
packet, to the nearest gram. Here are the results:
Machine A (mass in grams): 196, 198, 198, 199, 200, 200, 201, 201, 202, 205
Machine B (mass in grams): 195, 195, 195, 198, 200, 201, 203, 204, 204, 205
Which type of machine, A or B, is the best one to buy in the future? Explain your reasoning,
showing any diagrams, graphs or calculations you used to help you make your decision.
(2)
Please discuss in your group these questions.

Give examples of situations where it is important to measure the spread of a data set

What methods do you already know for measuring the spread of a data set? Which are the most
sophisticated? Which are the most problematic and why?
What was your conclusion to the question in the above scenario and why?
Come see Ms. Makunja and be ready for a chat about your understanding.
Cumulative Frequency – group task
Take 10 minutes to read and digest the following scenario and questions and discuss it with a partner.
Cumulative frequency graphs answer questions such as
“What proportion of the data has values less than …?”
Cumulative frequency on the y-axis is plotted against the observed value on the x-axis. Describe the shape the graph
would take.
With grouped data the first step is to produce a table of cumulative frequencies (the sum of the frequencies up to
each particular class). These are then plotted against the corresponding upper class boundaries (ucb). The
successive points are then connected by a curve.
Scenario
In studying bird migration a standard technique is to put coloured rings around the legs of the
young birds at their breeding colony. This means that the source of any bird (with a coloured ring)
later seen somewhere else can be identified. The following data, which refer to the recoveries of
razorbills, consist of the distances (measured in hundreds of miles) between the recovery point
and the breeding colony. We can illustrate the data using a cumulative frequency curve and
estimate the distance exceeded by 50% of the birds.
Distance (miles) x
x < 100
100  x < 200
200  x < 300
300  x < 400
400  x < 500
500  x < 600
600  x < 700
700  x < 800
800  x < 900
900  x < 1000
1000  x < 1500
1500  x < 2000
2000  x < 2500
Frequency
2
2
4
3
5
7
5
2
2
0
2
0
2
Cumulative Frequency
2
4
8
11
16
23
28
30
32
32
34
34
36
The final value in the cumulative
frequency column represents what?
Cumulative Frequency
Cumulative Frequency Graph
The graph (when drawn accurately)
tells us that 50% of the birds had
travelled for more than 520 miles. To
get this answer we look at 18 on the
vertical scale and read a value off the
horizontal scale. Why?
38
36
34
32
30
28
26
24
22
20
18
16
14
12
10
8
6
4
2
0
0
500
1000
1500
2000
2500
3000
Distance (miles)
Read and make notes on the information given on page 451 of your text.
What are percentiles?
What are quartiles?
What is the Interquartile Range? What does it show you about a set of data?
The graph below shows how the interquartile range is found and how quartiles are
related to a Box Plot.
Use the graph to calculate an approximation for the interquartile range for your data.
Show your work here
Cumulative Frequency
Cumulative Frequency Graph
38
36
34
32
30
28
26
24
22
20
18
16
14
12
10
8
6
4
2
0
0
500
1000
1500
2000
Distance (miles)
2500
3000
Example
Below are two sets of data. For each set find the upper and lower quartiles (Q1 and Q3) and the
median (Q2) and the interquartile range. For textbook help with this section, refer to page 446, and
read examples 13 and 14.
1)
2, 2, 3, 3, 4, 4, 4, 5
2)
2, 2, 3, 3, 3, 4, 4, 5, 6, 6, 7, 7
Practice:
Cumulative Frequency – Exercise 18C page 443 Q. 3, 4, 9
Measuring the Spread of Data – Exercise 18D.1 Q. 1d (by hand and with calculator), 2
Percentiles and Quartiles - Exercise 18D.2 Q. 2, 5, 7 (use technology), 8
4) Standard Deviation – teacher led discussion
The standard deviation is a way of measuring all the variation within a sample.
How do you think standard deviation is calculated from a frequency table?
x
Frequency
x
Frequency (f)
1
2
fx
4
1
5
1
(x- x )
6
1
9
2
(x- x )2
f(x- x )2
1
4
5
6
9
Standard Deviation for Grouped Data – partner activity – participation
recorded
Work with a partner on this example and be ready to take part in a whole class discussion. Try to explain the steps
being followed. Why do these steps lead to an estimate for the standard deviation? The question is: Estimate the
standard deviation of the weights of 40 year 11 students whose weights were measured to the nearest kg:
Weight (kg)
50-54
55-59
60-64
65-69
70-74
75-79
Midpoint
(x)
52
57
62
67
72
77
Frequency
(f)
2
1
7
18
11
1
Total = 40
104
57
434
1206
792
77
2670
435.1250
95.0625
157.9375
1.1250
303.1875
105.0625
1097.5000
Explain what these process are showing us:
(x- x )
(x- x )2
f(x- x )2
 fx  2670  66.75
x
 f 40
s
 f (x  x)
f
2

1097.5
 5.24
40
Standard deviation with the graphic calculator - work through these steps
Raw data set:
5, 17, 15, 3, 9, 11
To find the standard deviation of this data set, follow these steps:
STAT EDIT 1:Edit ENTER Highlight heading L1
CLEAR ENTER
then type the data into list L1
STAT CALC 1:1-Var Stats ENTER 2nd 1 ENTER
To see  x  5 (the standard deviation is 5, the variance is 52 = 25)
Frequency Table:
Here are the scores achieved by 20 students in a test:
3, 8, 9, 1, 4, 2, 7, 6, 5, 9, 10, 3, 4, 6, 2, 8, 7, 6, 3, 7
Here is the data organised into a frequency table:
Score out of ten (x)
1
2
3
4
5
6
7
8
Frequency (f)
1
2
3
2
1
3
3
2
Follow these steps to find the standard deviation of the scores of the group of students:
STAT EDIT 1:Edit ENTER Highlight heading L1 CLEAR ENTER
then type the integers from 1 to 10 into list L1
Highlight heading L2
CLEAR ENTER
Then type the frequencies shown in the table (1, 2, 3, 2, and so on) into list L2
STAT CALC 1:1-Var Stats ENTER 2nd 1 , (comma key is above the 7) 2nd 2 ENTER
To see  x  2.578759392 (so 2.58 to 3 s.f.)
9
2
10
1
To find the variance accurately now do: VARS 5 4 x2 ENTER to get variance = 6.65
graded on homework completion
Please show all steps of working – you may use a calculator to check your answers. Page 150 Ex. 5I.1 questions 1, 2,
3, 4a,b,c, 6.