Answers to activity questions
Check your skills
Mean
1.
For the data (right) about the Average Daily
Hours of Sunshine, calculate the mean, mode
and median.
2.
For the same data, calculate the range,
interquartile range and standard deviation.
3.
Median
There is
no mode
Is the 5th
value.
Median =
6.9
Range
IQR
SD
8.8 - 2.3 = 6.5
Q1=5.95
Q3=7.3
IQR=1.35
1.78
sample
assumed
Σx 58.3
=
= 6.48
n
9
For the Latitude Data, use a 5 number summary to draw a box and whisker plot.
L=2.3 Q1=5.95 M=6.9 Q3=7.3 H=8.8
2.0
4.
=
x
Mode
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
Use Excel to do a scatterplot of Hours of Sunshine vs Latitude. Latitude is the independent
variable.
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Av Daily Hours of Sunshine
Daily Hours of Sunshine for 9 Locations
10
9
8
7
6
5
4
3
2
1
0
0
10
20
30
40
50
60
Latitude (°S)
5.
Use Excel to determine the correlation and the regression equation.
Correlation = - 0.69 Moderate negative correlation
Regression line:
H=
−0.11L + 10.2
where L is the latitude
H is the av. daily hours of sunshine
Organising Data – Tables
1.
(a)
The speed (in km/hr) of 40 cars passing a school at 9am on a school day.
Speed
Tally
Frequency
Cumulative
Frequency
10 to but less than 20 km/hr
ΙΙΙ
3
3
20 to but less than 30 km/hr
ΙΙΙΙ ΙΙΙ
8
11
30 to but less than 40 km/hr
ΙΙΙΙ ΙΙΙ
8
19
40 to but less than 50 km/hr
ΙΙΙΙ ΙΙΙΙ
10
29
50 to but less than 60 km/hr
ΙΙΙΙ ΙΙ
7
36
60 to but less than 70 km/hr
ΙΙΙΙ
4
40
40
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(b)
Enter the data into Stem and Leaf Plot.
Stem
1
2
3
4
5
6
(c)
2.
Leaf
2 4 9
0 2 5 5 7 7 7 8
1 1 2 2 3 5 8 9
0 0 1 2 4 5 6 6 8 9
0 0 2 2 2 7 8
1 2 4 9
4|5 means 45 km/hr
What percentage of cars were doing 40km/hr or more?
Using the Cumulative Frequency Column, we know that 19 motorists were doing less
than 40 km/hr. Therefore the number doing 40km/hr or more is 40 (Total) – 19 =
21motorists.
The number doing 40km/hr or more = 21
The fraction doing 40km/hr or more = 21/40
Make this a percentage: 21/40 x 100 = 52.3 or approximately 53%
The number of students attending a class (maximum 25) for 30 lessons is given in the table
below:
(a)
Is the data discrete or continuous? The data is discrete because the number of student
can only be a whole number.
(b)
Enter the data into a frequency distribution table (groups not required). Include a
relative frequency and % relative frequency column.
Number attending
Tally
Frequency
Relative
Frequency
% Relative
Frequency
20
Ι
1
1
= 0.0333
30
3.3%
21
Ι
1
1
= 0.0333
30
3.3%
22
ΙΙ
2
2
= 0.0666
30
6.7%
23
ΙΙΙΙ
5
5
= 0.166
30
16.7%
24
ΙΙΙΙ ΙΙΙΙ Ι
11
11
= 0.366
30
36.7%
25
ΙΙΙΙ ΙΙΙΙ
10
10
= 0.3333
30
33.3%
30
1.00
100%
Total
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3.
(c)
What proportion of lessons contained 22 students? The relative frequency for 22
students is 0.067 (to 3 decimal places)
(d)
What percentage of lessons were fully attended? From the table - 33.3%
Systolic Blood Pressures of 35 patients at a Cardiac Clinic
(a)
Enter the data into a Stem and Leaf Plot. Use the key: 11|4 means 114.
Stem
8
9
10
11
12
13
14
15
16
17
18
(b)
4.
Leaf
8
2 5
2 7 9
0 2 4 5 8
0 1 2 2 4 6 7 8
1 3 4 4 7 8
1 3 4 5 6 9
5
1
5
8
11|4 means 114
If hypertension (high blood pressure) is defined by a systolic blood pressure 140 or
above, what percentage of this group are suffering hypertension? With a Stem and
Leaf, the original values are retained, so counting is required. There are 10 out of 35
which is 28.5% (to 1 d.p.)
The Shot Put distances thrown by 27 world champion shot putters are given in the table
below. The unit is metres (m).
22.25
21.72
21.07
20.22
22.67
(a)
20.19
20.37
21.55
20.38
21.58
21.39
20.45
23.12
22.37
21.72
21.25
23.09
20.91
22.19
21.19
21.19
22.58
21.70
22.07
21.22
21.97
20.54
Enter the data into a Frequency Distribution Table. Your FDT must have at least 5
groups.
Distance
Cumulative
Frequency
Tally
Frequency
20 to but less than 20.5
ΙΙΙΙ
5
5
20.5 to but less than 21
ΙΙ
2
7
21 to but less than 21.5
ΙΙΙΙ Ι
6
13
21.5 to but less than 22
ΙΙΙΙ Ι
6
19
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% Relative
Frequency
5
× 100 =
18.5%
27
2
7.4%
× 100 =
27
6
× 100 =
22.2%
27
6
× 100 =
22.2%
27
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22 to but less than 22.5
ΙΙΙΙ
4
23
22.5 to but less than 23
ΙΙ
2
25
23 to but less than 23.5
ΙΙ
2
27
Total
27
4
× 100 =
14.8%
27
2
7.4%
× 100 =
27
2
7.4%
× 100 =
27
100%
(b)
How many have thrown less than 22m? (Use cumulative frequency to answer this)
The cumulative frequency of the group before 19, this is the sum of the frequencies
less than 22.
(c)
What percentage threw 21 to but less than 22m? (Use a % Relative Frequency
Column) This is two groups of the table. Each group has 22.2%, so 44.4% threw 21 to
but less than 22m.
Graphs
1.
The height of a plant is measured every Friday morning for twelve weeks. Which type of
graph would be best to show the growth of the plant? Line
2.
A ‘sound and vision’ shop sells CDs, DVDs, console games and computer games. Which
type of graph would best display the relative sales of the different items sold? Pie
3.
A student wishes to compare the number of motorcycle fatalities between the states. To get
a more accurate picture, the data is collected for the years 2007, 2008 and 2009. Which type
of graph allows all this data to be presented in one graph? Composite Column
4.
The number of students attending a class (maximum 25) for 30 lessons is given in the table
below:
Frequency
Column Graph of Student Numbers in Class
12
10
8
6
4
2
0
20
21
22
23
24
25
Number Attending Class
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5.
The Shot Put distances thrown by 27 world champion shot putters are given in the
Frequency Distribution Table below. The unit is metres (m).
(a) Construct a frequency histogram for this information. As a second step, put a frequency
polygon on the histogram.
Shot Put Records
7
6
Frequency
5
4
3
2
1
0
20
20.5
21
21.5
22
22.5
23
23.5
Distance Thrown (m)
(b)
Construct a cumulative frequency histogram and then add an Ogive to the histogram.
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Shot Put Records
30
Cumulative Frequency
25
20
15
10
5
0
20
20.5
21
21.5
22
Distance Thrown (m)
22.5
23
23.5
Measures of Central Tendency
1.
The lifetime, in hours, of a sample of 15 light bulbs is: 351, 429, 885, 509, 317, 753, 827,
737, 487, 726, 395, 773, 926, 688, 485. Calculate the mean, mode and median of the values.
Do not organise into a table.
In order to calculate the median, the values must be ordered.
317, 351, 395, 429, 485, 487, 509, 688, 726, 737, 753, 773, 827, 885, 926
Mean
∑X
X=
n
317 + 351 + .... + 885 + 926
X=
15
X = 619.2
Mode
No value is repeated – there
is no mode.
The mode would not
significant in this example.
As there are 15 values, the
position of the median is:
n + 1 15 + 1 16
=
= = 8
2
2
2
The 8th value is the median.
Counting through the ordered
th
list, the 8 value is 688. The
median is also a good indicator
or typical value in this question.
The mean is approx. 619 hours.
The mean is a good indicator of a
central or typical value in this
question.
2.
Median
The number of children in a 10 families is: 1, 5, 2, 2, 2, 3, 1, 4, 3, 2. Calculate the mean,
mode and median of the values. Do not organise into a table.
In order to calculate the median, the values must be ordered.
1, 1, 2, 2, 2, 2, 3, 3, 4, 5
Mean
Mode
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Median
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X=
∑X
n
1 + 1 + .... + 4 + 5
X=
10
X = 2.5
The mode is 2, it occurs 4
times, or has a frequency of 4.
The mode is a good indicator
of a typical value in this
example.
The mean is 2.5 children. The
mean is a good indicator of a
central or typical value in this
question. However it is not
possible to have 2.5 children
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As there are 15 values, the
position of the median is:
n + 1 10 + 1 11
=
= = 5.5
2
2
2
The 5.5th value is the median.
th
th
The 5 value is 2 and the 6
value is 2 also, so the median is
2. The median is also a good
indicator or typical value in this
question.
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3.
For the Car Speed Data, calculate the mean, mode and median of the values after
organising the data in Frequency Distribution Table (This can be found in the answers to the
Topic ‘Graphs’).
Car Speed Data (in km/hr)
Speed(x)
Group Midpoint
Frequency(f)
Cumulative
Frequency
f ×x
10 to but less than 20
km/hr
15
3
3
45
20 to but less than 30
km/hr
25
8
11
200
30 to but less than 40
km/hr
35
8
19
280
40 to but less than 50
km/hr
45
10
29
450
50 to but less than 60
km/hr
55
7
36
385
60 to but less than 70
km/hr
65
4
40
260
n =Σf =40
Mean
ΣfX
Σf
1620
X=
40
X = 40.5
X=
The mean is approx. 40.5
km/hr. The mean is a good
indicator of a central or
typical value in this
question.
Mode
The modal group is 40 to
but less than 50 km/hr.
1620
Σfx =
Median
As there are 40 values, the position of the
median is:
n + 1 40 + 1 41
=
= = 20.5
2
2
2
The 20.5th value is the median.
th
st
The median is the average of the 20 and 21
values. The group ’40 to but less than 50 km/hr’
th
th
contains the 20 to the 29 value.
 n +1

 -cf m -1
2 

× Group width
Median =
Lm +
fm
20.5 - 19
× 10
10
Median = 41.5
Median=40+
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There is also a graphical method for working out the median.
C.F. Ogive for Car Speed Data
45
40
35
C.F.
30
25
20
15
Median is approx. 42
10
5
0
4.
10
20
30
40
50
Speed (km/hr)
60
70
Students Attending Class
Frequency(f)
f ×x
20
1
20
1
21
1
21
2
22
2
44
4
23
5
115
9
24
11
264
20
25
10
250
30
30
Σf =
714
Σfx =
Number attending
(x)
Total
(a)
Cumulative
Frequency
Calculate the mean, mode and median.
Mean
ΣfX
Σf
714
X=
30
X = 23.8
X=
Mode
The modal group is 24
students.
The mean is approx. 23.8
students. The mean is a good
indicator of a central or typical
value in this question.
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Median
As there are 30 values, the
position of the median is:
n + 1 30 + 1 31
=
= = 15.5
2
2
2
The 15.5th value is the median.
The median is the average of the
th
th
15 and 16 values. The group 24
th
th
contains the 10 to the 20
th
th
values, including the 15 and 16
values. The median is 24.
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(b)
Discuss the appropriateness of each measure of central tendency as a typical value.
All measures of centre are relevant. On average 24 students attend lectures.
5.
The Shot Put distances thrown by 27 world champion shot putters are given in the table
below. The unit is metres (m).
Distance (x)
Group
Centre
Frequency
(f)
20 to but less than 20.5
20.25
5
5
101.25
20.5 to but less than 21
20.75
2
7
41.5
21 to but less than 21.5
21.25
6
13
127.5
21.5 to but less than 22
21.75
6
19
130.5
22 to but less than 22.5
22.25
4
23
89
22.5 to but less than 23
22.75
2
25
45.5
23 to but less than 23.5
23.25
2
27
46.5
Total
27
Σf =
Cumulative
Frequency
f ×x
581.75
Σfx =
Calculate the mean, mode and median.
Mean
Mode
ΣfX
Σf
581.75
X=
27
X = 21.5
This data is bimodal.
The modal groups are
21 to but less than 21.5
and 21.5 to but less
than 22.
As there are 27 values, the position of
the median is:
The mode is of limited
use in this example.
The median is located in the group
’21.5 to but less than 22 m’.
 n +1

 -cf m -1
2 
Median =
Lm + 
× Group width
fm
X=
The mean is approx.
21.5 m. The mean is a
good indicator of a
central or typical value in
this question.
Median
n + 1 27 + 1 28
=
= = 14
2
2
2
The 14th value is the median.
14 - 13
× 0.5
6
Median = 21.6 (rounded to 1dp)
Median=21.5+
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There is also a graphical method for working out the median.
Shot Put Records
30
Cumulative Frequency
25
20
15
10
Median is approx 21.6
5
0
20
20.5
21
21.5
22
Distance Thrown (m)
22.5
23
23.5
Measures of Spread
1.
The lifetime, in hours, of a sample of 15 light bulbs is: 351, 429, 885, 509, 317, 753, 827,
737, 487, 726, 395, 773, 926, 688, 485.
In order to calculate the median, the values must be ordered.
317, 351, 395, 429, 485, 487, 509, 688, 726, 737, 753, 773, 827, 885, 926
Range
SD
Range = Highest Value - Lowest Value
Range = 926 - 317
Using a Scientific Calculator
s = 203 (rounded to nearest whole number)
Range = 609
IQR
The first quartile (Q1) is the
The third quartile (Q3) is the
3 ( n + 1)
n +1
4
15 + 1
=
4
16
=
4
= 4th value
Q1 is 429
4
3 × 16
=
4
48
=
4
= 12th value
Q3 is 773
The IQR is 773 – 429 = 344
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2.
The number of children in a 10 families is: 1, 5, 2, 2, 2, 3, 1, 4, 3, 2. Calculate the range,
inter-quartile range and standard deviation of the values. Do not organise into a table.
In order to calculate the median, the values must be ordered.
1, 1, 2, 2, 2, 2, 3, 3, 4, 5
Range
SD
Range = Highest Value - Lowest Value
Using a Scientific Calculator
Range = 5 - 1
s = 1.27 (rounded to 2dp)
(assuming sample)
Range = 4
IQR
The first quartile (Q1) is the
The third quartile (Q3) is the
3 ( n + 1)
n +1
4
10 + 1
=
4
11
=
4
= 2.75th value
4
3 × 11
=
4
33
=
4
= 8.25th value
Q1 is 1.75
Q3 is 3.25
The IQR is 3.25 – 1.75 = 1.5
3.
For the Car Speed Data, calculate the range, inter-quartile range and standard deviation of
the values after organising the data in Frequency Distribution Table. The assumption is that
the original values are lost and it is a sample.
Car Speed Data (in km/hr)
Speed(x)
Group
Midpoint
Frequency(f)
Cumulative
Frequency
f ×x
10 to but less than 20
km/hr
15
3
3
45
20 to but less than 30
km/hr
25
8
11
200
30 to but less than 40
km/hr
35
8
19
280
40 to but less than 50
km/hr
45
10
29
450
50 to but less than 60
km/hr
55
7
36
385
60 to but less than 70
km/hr
65
4
40
260
n =Σf =40
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1620
Σfx =
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Range
SD
Range = Highest Value - Lowest Value
Using a Scientific Calculator
Range = 70 - 10
s = 14.5 (rounded to 1dp)
(assuming sample)
Range = 60
IQR
The first quartile (Q1) is the
The third quartile (Q3) is the
3 ( n + 1)
n +1
4
40 + 1
=
4
41
=
4
= 10.25th value
4
3 × 41
=
4
123
=
4
= 30.75th value
using interpolation
10.25 − 3
×10
The first quartile (Q1 ) =
20 +
8
7.25
=
×10
20 +
8
= 29.1
using interpolation
30.75 − 29
The third quartile (Q3 ) =
50 +
×10
7
1.75
50 +
=
×10
7
= 52.5
The IQR is 52.5 – 29.1 = 23.4
The Ogive can also be used to calculate the quartiles and IQR
C.F. Ogive for Car Speed Data
45
40
35
C.F.
30
25
20
15
IQR = 52 – 29 = 23
10
5
0
10
20
30
40
50
Speed (km/hr)
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60
70
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4.
The number of students attending a class (maximum 25) for 30 lessons is given in the table
below. The assumption is that the original values are lost and it is a sample.
Frequency(f)
f ×x
20
1
20
1
21
1
21
2
22
2
44
4
23
5
115
9
24
11
264
20
25
10
250
30
30
Σf =
714
Σfx =
Number attending
(x)
Total
Cumulative
Frequency
Range
SD
Range = Highest Value - Lowest Value
Range = 25 - 20
Using a Scientific Calculator
Range = 5
s = 1.27 (rounded to 2dp)
(assuming sample)
IQR
The first quartile (Q1) is the
The third quartile (Q3) is the
n +1
4
30 + 1
=
4
31
=
4
= 7.75th value
4
3 × 31
=
4
93
=
4
= 23.25th value
The 7th and 8th values are both 23.
Q1 = 23
3 ( n + 1)
The 23rd and 24th values are both 25.
Q3 = 25
The IQR is 25 – 23 = 2
5.
The Shot Put distances thrown by 27 world champion shot putters are given in the table
below. The unit is metres (m). The assumption is that the original values are lost and it is a
sample.
Distance (x)
20 to but less than 20.5
20.5 to but less than 21
21 to but less than 21.5
21.5 to but less than 22
22 to but less than 22.5
22.5 to but less than 23
23 to but less than 23.5
Group
Centre
20.25
20.75
21.25
21.75
22.25
22.75
23.25
Total
Frequency
(f)
5
2
6
6
4
2
2
27
Σf =
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Cumulative
Frequency
5
7
13
19
23
25
27
f ×x
101.25
41.5
127.5
130.5
89
45.5
46.5
581.75
Σfx =
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Range
SD
Range = Highest Value - Lowest Value
Range = 23.5 - 20
Range = 3.5
Using a Scientific Calculator
s = 0.901 (rounded to 3dp)
(assuming sample)
IQR
The third quartile (Q3) is the
The first quartile (Q1) is the
3 ( n + 1)
n +1
4
27 + 1
=
4
28
=
4
= 7th value
Interpolation is not required here as the
7th value is 21.
4
3 × 28
=
4
84
=
4
= 21st value
By interpolation
21 − 19
×10
4
2
= 22 + × 0.5
4
= 22.25
The third quartile (Q3 ) =
22 +
The IQR is 22.25 – 21 = 1.25
The Ogive can also be used to calculate the quartiles and IQR
Shot Put Records
30
Cumulative Frequency
25
20
15
IQR = 22.2 – 21 = 1.2
10
5
0
20
20.5
21
21.5
22
Distance Thrown (m)
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22.5
23
23.5
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Comparing Data
1.
The lifetime, in hours, of a sample of 15 light bulbs is: 351, 429, 885, 509, 317, 753, 827,
737, 487, 726, 395, 773, 926, 688, 485.
In order to calculate the median, the values must be ordered.
317, 351, 395, 429, 485, 487, 509, 688, 726, 737, 753, 773, 827, 885, 926
From the Measures of Central Tendency section, the median is 688.
From the Measures of Spread section, the quartiles are 429 and 773.
The lowest and highest values are 317 and 926.
317
300 350
2.
429
400
688
450 500
550
600
650
700
773
750 800
926
850 900
950
The Car Speed Data was collected from a school zone just prior to an advertising
campaign. The data collected was:
12
31
22
40
41
35
50
52
Car Speed Data (in km/hr)
45
28
40
20
59
27
14
33
48
27
61
42
44
31
46
57
32
49
25
39
62
19
32
64
46
58
52
52
25
38
69
27
After an advertising campaign to make motorists more aware of speeding in school zones,
the following data was collected.
22
31
34
38
45
48
50
41
42
30
14
50
27
16
59
14
27
28
33
31
40
42
42
39
45
19
30
25
26
55
32
46
75
46
36
39
25
38
18
27
To organise the data, a back to back stem and leaf plot will be used.
After advertising
9 8 6 4
8 7 7 7 6 5 5
9 9 8 8 6 4 3 2 1 10
8 6 6 5 5 2 2 2 1
9 50
4
2
0
0
0
5
Stem
1
2
3
4
5
6
7
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2
0
1
0
0
1
Prior to Advertising
4 9
2 5 5 7 7 7 8
1 2 2 3 5 8 9
0 1 2 4 5 6 6 8 9
2 2 2 7 8 9
2 4 9
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For the Prior to Advertising Data
Lowest
Value = 12
First Quartile is the
n +1
4
40 + 1
=
4
41
=
4
= 10.25th value
This means that the first
Median is the
n +1
2
40 + 1
=
2
41
=
2
= 20.5th value
Median = 40
quartile is a quarter of the way
Third Quartile is the
3 ( n + 1)
4
3 × 41
=
4
= 30.75th value
Highest
Value = 69
This means that the third
quartile is three quarters of
the way between the 30th and
31th values.
between the 10th and 11th
Third Quartile
= 50 + 0.75 x (52-50)
=51.5
values.
Lowest
Value = 12
First Quartile
= 27 + 0.25 x (28-27)
= 27.25
First Quartile (Q1) =
27.25
Median =
40
Third Quartile (Q3) =
51.5
Highest
Value = 69
Third Quartile is the
Highest
Value = 75
For the After the Advertising Data
Lowest
Value = 14
First Quartile is the
n +1
4
40 + 1
=
4
41
=
4
= 10.25th value
This means that the first
quartile is a quarter of the way
Median is the
n +1
2
40 + 1
=
2
41
=
2
= 20.5th value
Median =
(34+36)/2=35
3 ( n + 1)
4
3 × 41
=
4
= 30.75th value
This means that the third
quartile is three quarters of
the way between the 30th and
31th values.
between the 10th and 11th
Third Quartile
= 42 + 0.75 x (45-42)
=44.25
values.
First Quartile
= 27
Lowest
Value = 14
First Quartile (Q1) =
27
Median =
35
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Third Quartile (Q3) =
44.25
Highest
Value = 75
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12
27.25
40
51.5
69
Prior
After
14
10
27
15
20
25
35
30
35
44.25
40
45
75
50
55
60
65
70
75
80
The quartiles and the median are lower after the campaign, based on this there is some evidence
of a drop in speed. The lowest and highest value have increased, these can be outliers and so
unreliable. The Stem and Leaf Plot suggests that the highest is an ‘odd’ value.
3.
The heart rate of a group of athletes was compared to the heart rate of a group of office
workers after climbing a set of stairs. The data is presented below:
Athletes
96
104
85
112
111
106
117
106
88
121
126
110
79
93
97
116
101
103
83
91
91
96
93
88
114
118
145
100
107
103
132
138
Office Workers
94
113
121
127
118
108
120
113
117
145
97
131
126
Office Workers
8 7
8 8 7 4
7 6
8
7
3
3
1
2
5
4
0
3
0
1
5
Stem
7
8
9
10
11
12
13
14
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Athletes
9
3
1
1
0
1
5
1
3
1
6
8
3
4
2
8
3 6 6 7
6 6
6 7
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For the Athletes
Lowest
Value = 79
First Quartile is the
Median is the
n +1
4
24 + 1
=
4
25
=
4
= 6.25th value
Third Quartile is the
3 ( n + 1)
4
3 × 25
=
4
= 18.75th value
n +1
2
24 + 1
=
2
25
=
2
= 12.5th value
This means that the first
quartile is a quarter of the way
Highest
Value = 126
This means that the third
quartile is three quarters of
Median =
(97+101)/2=99
the way between the 18th and
19th values.
between the 6th and 7th values.
First Quartile
= 91
First Quartile (Q1) =
91
Lowest
Value = 79
Third Quartile
= 110 + 0.75 x (111110)
=110.75
Third Quartile (Q3) =
110.75
Median =
99
Highest
Value = 126
For the Office Workers
Lowest
Value = 94
Lowest
Value = 94
First Quartile is the
Median is the
First Quartile
= 107 + 0.5 x (108107)
= 107.5
First Quartile (Q1) =
107.5
91
3 ( n + 1)
4
3 × 22
=
4
= 16.5th value
n +1
2
21 + 1
=
2
22
=
2
= 11th value
n +1
4
21 + 1
=
4
22
=
4
= 5.5th value
79
Third Quartile is the
99
Highest
Value = 145
Third Quartile
= 127 + 0.5 x (131127)
=129
Median = 114
Third Quartile (Q3) =
129
Median =
114
110.75
Highest
Value = 145
126
Athletes
Office
Workers
70
75
80
85
90
94
95
100
107.5
105 110
114
115 120
125 130
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129
135
140
145
145 150
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It is quite clear that there is a difference between the two groups.
Correlation and Regression
1.
(a)
Excel scatterplot. The scatterplot suggests Very Strong Correlation.
Energy Content of various Milks
1200
Energy (kJ)
1000
800
600
400
200
0
0
5
10
15
20
Percentage Level of Fat (%)
(b)
The correlation coefficient = 0.989; This also suggests very strong correlation.
(c)
The regression line;
=
y 44.4 x + 300 or
=
E 44.4 F + 300
where E - energy content of the milk
F - percentage Fat level of the milk
2.
A large company is making note of the amount spent on advertising each month and then
comparing this with the amount of monthly sales.
(a)
The scatterplot suggests reasonable to strong positive correlation.
Sales ($x1000)
Chart Title
180
160
140
120
100
80
60
40
20
0
1
2
3
4
5
6
7
Advertising Cost ($x1000)
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(b)
The correlation coefficient = 0.735 suggesting Quite Strong Positive Correlation
(c)
The regression line.
y 11.2 x + 70 or
=
S 11.2 A + 70
=
where S - Sales per month ($000)
A - Advertising Costs per month ($000)
3.
(a)
Distance Inland vs January Average Max Temp. The scatterplot suggests strong
positive correlation.
January Max Temp for Towns
40
Temperature (°C)
38
36
34
32
30
28
26
0
200
400
600
800
1000
1200
Inland Distance (km)
Correlation coefficient = 0.92; this suggests Strong Positive Correlation.
Regression line:
=
y 0.0071x + 31.5 or
TJan max 0.0071d + 31.5
=
where TJan max - Jan Average Max Temp ( °C)
d - Inland Distance (km)
(b)
Distance Inland vs July Average Minimum Temp. There is some (negative?) correlation
here. It appears that once you get 200km inland that the minimum temperature stops
decreasing and then increases at a lower rate.
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July Min Temp for Towns
13
12
Temperature (°C)
11
10
9
8
7
6
5
4
0
200
400
600
800
1000
Distance Inland (km)
1200
Correlation Coefficient = 0.55; this also suggests slight positive correlation. It is not
meaningful to proceed to regression.
(c)
Distance Inland vs Annual Rainfall. Scatterplot suggests strong negative correlation.
This means the greater the distance inland, the lower the annual rainfall.
Average Annual Rainfall (mm)
Annual Rainfall for Towns
1000
900
800
700
600
500
400
300
200
100
0
0
200
400
600
800
1000
1200
Distance Inland (km)
Correlation coefficient = -0.94; this suggests Strong Positive Correlation.
Regression line:
-0.505 x + 796 or
y=
-0.505d + 796
RAv =
where RAv - Average Annual Rainfall (mm)
d - Inland Distance (km)
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Comments:
Graph (a)
is not surprising. Maximum temperature can be influenced by other factors such as
altitude but distance from the sea is a significant one.
Graph (b)
is a very interesting graph. Perhaps altitude has an effect over 200 km inland. This
could lead to further investigations.
Graph (c)
is expected. However rainfall can be affected by topography such as mountain ranges.
This usually creates a rainy side or a rain shadow depending on the direction of the
prevailing wind. There is one location (Blackwater) that is lower than the regression
line suggests, perhaps this location is in a rain shadow.
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