Name of Lecturer: Mr. J.Agius
Course: HVAC1
Lesson 64
Chapter 13: Statistics
Finding a Central Value
When you have two or more numbers it is nice to find a value for the "center".
2 Numbers
With just 2 numbers the answer is easy: go half-way in-between.
Example: what is the central value for 3 and 7?
Answer: Half-way in-between, which is 5.
You can calculate it by adding 3 and 7 and then dividing the result by 2:
(3+7) / 2 = 10/2 = 5
3 or More Numbers
You can use the same idea when you have 3 or more numbers:
Example: what is the central value of 3, 7 and 8?
Answer: You calculate it by adding 3, 7 and 8 and then dividing the results by 3
(because there are 3 numbers):
(3+7+8) / 3 = 18/3 = 6
Notice that we divided by 3 because we had 3 numbers ... very important!
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Name of Lecturer: Mr. J.Agius
Course: HVAC1
The Mean
So far we have been calculating the Mean (or the Average):
Mean: Add up the numbers and divide by how many numbers.
But sometimes the Mean can let you down:
Example: Birthday Activities
Uncle Bob wants to know the average age at the party, to choose an activity.
There will be 6 kids aged 13, and also 5 babies aged 1.
Add up all the ages, and divide by 11 (because there are 11 numbers):
(13+13+13+13+13+13+1+1+1+1+1) / 11 = 7.5...
The mean age is about 7½, so he gets a Jumping Castle!
The 13 year olds are embarrassed,
and the 1-year olds can't jump!
The Mean was accurate, but in this case it was not useful.
The Median
But you could also use the Median: simply list all numbers in order and choose the
middle one:
Example: Birthday Activities (continued)
List the ages in order:
1, 1, 1, 1, 1, 13, 13, 13, 13, 13, 13
Choose the middle number:
1, 1, 1, 1, 1, 13 , 13, 13, 13, 13, 13
The Median age is 13 ... so let's have a Disco!
Sometimes there are two middle numbers. Just average them:
Example: What is the Median of 3, 4, 7, 9, 12, 15
There are two numbers in the middle:
3, 4, 7, 9 , 12, 15
So we average them:
(7+9) / 2 = 16/2 = 8
The Median is 8
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Page 2
Name of Lecturer: Mr. J.Agius
Course: HVAC1
The Mode
The Mode is the value that occurs most often:
Example: Birthday Activities (continued)
Group the numbers so we can count them:
1, 1, 1, 1, 1, 13, 13, 13, 13, 13, 13
"13" occurs 6 times, "1" occurs only 5 times, so the mode is 13.
But Mode can be tricky, there can sometimes be more than one Mode.
Example: What is the Mode of 3, 4, 4, 5, 6, 6, 7
Well ... 4 occurs twice but 6 also occurs twice.
So both 4 and 6 are modes.
When there are two modes it is called "bimodal", when there are three or more modes
we call it "multimodal".
Conclusion
There are other ways of measuring central values, but Mean, Median and Mode are
the most common.
Use the one that best suits your data. Or better still, use all three!
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Page 3
Name of Lecturer: Mr. J.Agius
Course: HVAC1
The Range (Statistics)
The Range is the difference between the lowest and highest values.
Example: In {4, 6, 9, 3, 7} the lowest value is 3, and the highest is 9.
So the range is 9-3 = 6.
It is that simple!
But perhaps too simple ...
The Range Can Be Misleading
The range can sometimes be misleading when there are extremely high or low values.
Example: In {8, 11, 5, 9, 7, 6, 3616}:


the lowest value is 5,
and the highest is 3616,
So the range is 3616-5 = 3611.
The single value of 3616 makes the range large, but most values are around 10.
Range of a Function
Range can also mean all the output values of a
function, seeDomain, Range and Codomain.
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Page 4
Name of Lecturer: Mr. J.Agius
Course: HVAC1
Question 1
In his end of year exams, Joe scored the following:
Science 34%
English 90%
History 87%
Math 34%
Geography 55%
What was the mean?
A
C
34%
60%
A
C
34%
60%
B
D
55%
87%
What was the median?
B
D
55%
87%
What was the mode?
A
C
34%
60%
B
D
55%
87%
Question 2
The table shows the average temperatures for London, England, for each month of
the year.
What is the mean of these values?
A
C
48.5oF
49.5oF
A
C
48.5oF
49.5oF
A
C
48.5oF
49.5oF
B
D
60oF
62oF
What is the median of these values?
B
D
60oF
62oF
What is the mode of these values?
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B
D
60oF
62oF
Page 5
Name of Lecturer: Mr. J.Agius
Course: HVAC1
Question 3
A booklet has 12 pages with the following numbers of words:
271, 354, 296, 301, 333, 326, 285, 298, 327, 316, 287 and 314
What is the mean number of words per page?
A
C
307
311
B
D
309
313
Question 4
The average mark scored by 29 students in a science test was 56%
John was sick, so sat the test late and scored 71%
Including John's, what was the new value of the mean mark?
A
C
56%
60%
B
D
56.5%
63.5%
Question 5
The mean of two numbers is 10, and one number is 6 more than the other.
What is the value of the smaller of the two numbers?
A
C
6
8
B
D
7
9
Question 6
What is the median of the numbers 3, 11, 6, 5, 4, 7, 12, 3 and 10 ?
A
C
4
6
B
D
5
7
Question 7
A booklet has 12 pages with the following numbers of words:
271, 354, 296, 301, 333, 326, 285, 298, 327, 316, 287 and 314
What is the median number of words per page?
A
C
301
307.5
B
D
305.5
314
Question 8
A
C
For the numbers 13, 16, 12, 11, 8, 14, 12 and 18
which of the following is true?
B
median > mean > mode
mean > median > mode
D
mean > mode > median
median > mode > mean
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Name of Lecturer: Mr. J.Agius
Course: HVAC1
Question 9
The numbers 7, 6, 10, 13, 7, 2, 5, 6 and x have only one mode
And the mean, median and mode are all equal.
What is the value of x?
A
C
x=5
x = 6.5
B
D
x=6
x=7
Question 10
What is the range for the following set of numbers?
15, 21, 57, 43, 11, 39, 56, 83, 77, 11, 64, 91, 18, 37
A
C
22
80
B
D
72
91
Question 11
Olivia wrote a 100 word story.
She then counted the number of letters in each word. Her results are shown in the
following bar graph:
What is the range?
A
C
7
29
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B
D
8
30
Page 7
Name of Lecturer: Mr. J.Agius
Course: HVAC1
Question 12
The population of a town was recorded every twenty years from 1900 to 2000. The
results are shown in the line graph.
What was the range over that period?
A
C
100
4600
B
D
4400
5400
Question 13
The frequency table shows the numbers of goals the Lakers scored in their last
twenty matches.
What was the range?
A
C
4
6
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B
D
5
7
Page 8
Name of Lecturer: Mr. J.Agius
Course: HVAC1
Exercise 30A
1)
2)
3)
Find the arithmetic average or mean of the following sets of the numbers:
a)
12, 13, 14, 15, 16, 17, 18
b)
18.2, 20.7, 32.5, 50, 78.6
c)
0.76, 0.09, 0.35, 0.54, 1.36
d)
38.2, 17.6, 63.5, 80.7
e)
34, 14, 39, 20, 16, 45
What is the mode of the following sets of numbers:
a)
58, 56, 59, 62, 56, 63, 54, 53
b)
10, 8, 12, 14, 12, 10, 12, 8, 10, 12, 4
c)
3, 9, 7, 9, 5, 4, 8, 2, 4, 3, 5, 9
d)
1.2, 1.8, 1.9, 1.2, 1.8, 1.7, 1.4, 1.3, 1.8
e)
5.9, 5.6, 5.8, 5.7, 5.9, 5.9, 5.8, 5.7
f)
26.2, 26.8, 26.4, 26.7, 26.5, 26.4, 26.6, 26.5, 26.4
Find the median of each of the following sets of numbers:
a)
13, 24, 19, 13, 6, 36, 17
b)
1.2, 3.4, 3.2, 6.5, 9.8, 0.4, 1.8
c)
1.92, 1.84, 1.89, 1.86, 1.96, 1.98, 1.73, 1.88
d)
4, 8, 32, 16, 9, 7, 29
e)
34, 42, 16, 85, 97, 24, 18, 38
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Page 9
Name of Lecturer: Mr. J.Agius
Course: HVAC1
Example 30B
John’s examination percentages in 8 subjects were 83,47,62,49,55,72,58 and 62. What
was his mean mark?
Answer
Mean mark for 8 subjects
=
83  47  62  49  55  72  58  62
488
=
= 61
8
8
Exercise 30B
1)
In the Christmas term examinations Lisa scored a total of 504 in 8 subjects. Find her
mean mark.
2)
A bowler took 110 wickets for 1815 runs. Calculate his average number of runs per
wicket.
3)
In six consecutive English examinations, Jane’s percentage marks were 83, 76, 85, 73,
64 and 63. Find her mean mark.
4)
A football team scored 54 goals in 40 league games. Find the average number of goals
per game.
5)
In an ice-dancing competition the recorded score for the winners were 5.8, 5.9, 6.0,
5.8, 5.8, 5.8, 5.6 and 5.7. Find their mean score.
Example 30C
On average my car travels 28.5 miles on each gallon of petrol. How far will it travel on 30
gallons?
Answer
If the car travels 28.5 miles on 1 gallon of petrol it will travel 30  28.5 miles, i.e. 855
miles, on 30 gallons.
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Name of Lecturer: Mr. J.Agius
Course: HVAC1
Exercise 30C
1)
My father’s car travels on average 33.4 miles on each gallon of petrol. How far will it
travel on 55 gallons?
2)
The average daily rainfall in Puddletown during April was 2.4mm. How much rain
fell during the month?
Example 30D
Elaine’s average mark after 7 subjects is 56 and after 8 subjects it has risen to 58. How
many does she score in her eight subject?
Answer
Total scored in 7 subjects is 56  7 = 392
Total scored in 8 subjects is 58  8 = 464
Score in her eight subject
= total for 8 subjects  total for 7 subjects
= 464  392
= 72
Therefore Elaine scores 72 in her eight subject.
Exercise 30D
1)
David Gower’s batting average after 11 completed innings was 62. After 12
completed innings it had increased to 68. How many runs did he score in his twelfth
innings?
2)
Anne’s average mark after 8 results was 54. This dropped to 49 when she received her
ninth result which was for French. What was her French mark?
3)
After six examination results Tom’s average mark was 57. His next result increased
his average to 62. What was his seventh mark?
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Name of Lecturer: Mr. J.Agius
Course: HVAC1
Example 30E
In five consecutive frames in the World Championships, a snooker player scored 62, 0, 13,
92 and 53. Find his average score per frame. How many did he score in the next frame if
his average increased to 57?
Answer
Average score for 5 frames =
62  0  13  92  53 220
=
= 44
5
5
If the average score after 6 frames is 57:
Total scored in 6 frames = 57  6 = 342
But the total scored in 5 frames = 220
 score in sixth frame
= total score for 6 frames  total score for 5 frames
= 342  220
= 122
Therefore the sixth frame score was 122.
Exercise 30E
1)
In seven consecutive innings a batsman scored 53, 4, 73, 104, 66, 44 and 83.
i)
What was his average?
ii)
What does he score in his next innings if his average falls to 56?
2)
A paperboy’s sales during a certain week were: Monday 84, Tuesday 112, Wednesday
108, Thursday 95 and Friday 131.
i)
Find his average daily sales.
ii)
When he included his sales on Saturday his daily average increased to
128. How many papers did he sell on Saturday?
3)
The number of hours of sunshine in Rhodes for successive days during a certain week
were 10.9, 11.9, 9.9, 7.7, 11.7, 9.3 and 12.1.
i)
Find the daily average.
ii)
The following week the daily average was 11 hours. How many more
hours of sunshine, were there the second week than the first?
4)
The heights of the 11 girls in a hockey team are 162cm, 152cm, 166cm, 149cm,
153cm, 165cm, 169cm, 145cm, 155cm, 159cm and 163cm.
i)
Find the average height of the team.
ii)
If the girl who was 145cm tall were replaced by a girl 156cm tall, what
difference would this make to the average height of the team?
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Name of Lecturer: Mr. J.Agius
Course: HVAC1
Example 30F
In a rugby XV the average weight of the eight forwards is 85kg and the average weight of
the seven backs is 70kg. Find the average weight of the team.
Answer
Total weight of 8 forwards = 85  8kg = 680kg
Total weight of 7 backs = 70  7kg = 490kg
 total weight of the 15 members of the team
= (680  490)kg
= 1170kg
1170
 average weight of the team =
kg
15
= 78kg
Exercise 30F
1)
The average weight of the 15 girls in a class is 54.4kg while the average weight of the
10 boys is 57.4kg. Find the average weight of the class.
2)
In a school the average size of the 14 lower school forms is 30, the average size of the
16 middle school forms is 25 and the average size of the 20 upper school forms is 24.
Find the average size of form for the whole school.
Mixed Exercises
1)
A darts player scored 2304 in 24 visits to the board. What was his average number of
points per visit?
2)
Olga’s car travels on average 12.6km on each litre of petrol. How far will it travel on
205 litres?
3)
Peter’s examination percentages in 7 subjects were 64, 43, 86, 74, 55, 53 and 66.
What was his mean mark?
4)
During a certain week the number of lunches served in a school canteen were:
Monday 213, Tuesday 243, Wednesday 237 and Thursday 239.
i)
Find the average number of meals served daily over the four days.
ii)
If the daily average for the week (Monday-Friday) was 225, how many
meals were served on Friday?
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Name of Lecturer: Mr. J.Agius
Course: HVAC1
5)
Richard was collecting money for a charity. The average amount collected from the
first 15 houses at which he called was 30p, while the average amount collected after
16 houses was 35p. How much did he collect from the sixteenth house?
6)
The recorded rainfall each day at a holiday resort during the first week of my holiday
was 3mm, 0, 4.5mm, 0, 0, 5mm and 1.5mm. Find the mean daily rainfall for the week.
7)
The daily average number of hours of sunshine during my 14 day holiday in Greece
was 9.4. For how many hours did the sun shine while I was on holiday?
8)
The weights of the members of a rowing eight were 82kg, 85kg, 86kg, 86kg, 84kg,
88kg, 92kg and 85kg. Find the average weight of the “eight”. If the cox weighed
41kg, what was the average weight of the crew?
9)
Jean’s marks in the end of term examinations were 46, 80, 59, 83, 54, 67, 79, 82 and
62.
i)
Find her average mark.
ii)
It was found that there had been an error in her mathematics mark. It
should have been 74, not 83. What difference did this make to her average?
10)
The first Hockey XI scored 14 goals in their first 16 matches. What was the average
number of goals per match?
11)
During the last five years the distances I travelled in my car, in miles, were 10 426, 12
634, 11 926, 14 651 and 13 973.
How many miles did I travel in the whole period?
What was my yearly average?
How many miles should I travel this year to reduce the average annual mileage over
the six years to 11 984?
12)
The average weight of the 15 girls in a class is 54.4kg while the average weight of the
10 boys is 57.4kg. Find the average weight of the class.
13)
The average weight of the 18 boys in a class is 63.2kg. When two new boys join the
class the average weight increases to 63.7kg. What is the combined weight of the two
new boys?
14)
Northshire has an area of 400 000 hectares and last year the annual rainfall was
274cm, while Southshire has an area of 150 000 hectares and last year the annual
rainfall was 314cm. What was the annual rainfall last year for the combined area of
the two counties?
15)
After playing 10 three-day matches and 8 one-day matches, the average daily
attendances for a County Cricket club were 2160 for three-day matches and 4497 for
one-day matches. Calculate the average daily attendance for the 18 matches.
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Name of Lecturer: Mr. J.Agius
16)
Course: HVAC1
Find a) the mean b) the mode c) the median d) the range of each of the following sets
of numbers:
a)
67, 71, 69, 82, 70, 66, 81, 66, 67
b)
84, 93, 13, 16, 28, 13, 32, 63, 45
17)
The marks, out of 100, in a geography test for the members of a class were: 64, 50,
35, 85, 52, 47, 72, 31, 74, 49, 36, 44, 54, 48, 32, 52, 53, 48, 71, 52, 56, 49, 81, 45, 52,
80, 46, 34, 69, 68, 86, 53, 45, 63, 53, 56, 81.
Find a) the mean mark
b) the modal mark
c) the median mark
d) the range of the marks
18)
a)
The mean content of 11 boxes of matches is 46. How many matches are there
altogether?
b)
The next box checked contains 49 matches. What is the mean content when
this box is included?
19)
What mark could Leo get in his next test to get the median stated?
a)
marks so far: 1, 8, 5, 2
: median 5
b)
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marks so far: 8, 2, 4, 5, 3, 8, 4, 2, 9
: median 4
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