Module 1 Higher GCSE Mathematics Revision

1
Module 1 Higher GCSE Mathematics Revision
2
Advice

• Do not spend too long on 1 question If you
  cannot see what to do leave it out return to it
  at the end if you have time.
• Show working whenever you can You may get
  method marks even if your final answer is wrong.
• Remember to look at the formula page There may
  be something there that could help you.
• Dont panic! You can do this!!!

3
Cumulative frequency
4
A golf club has 200 members. Their ages are shown
in the frequency table below.
A cumulative frequency (or a less than) table
can be drawn from this data.
8
8
26
34
32
66
45
111
37
148
29
177
16
193
7
200
How old is the youngest/oldest person?
5
We can now use this to draw a cumulative
frequency graph.
8
8
10
34
20
34
66
30
66
111
40
111
50
148
148
60
177
177
70
193
193
80
200
200
6
We can now use this to draw a cumulative
frequency graph.
8
34
66
111
148
177
193
200
7
Cumulative Frequency graph.
We can now use this to find the following
information..
Median
37
Lower quartile
25
Upper quartile
51
Lowest Value
0
Highest Value
80
Interquartile range
51 - 25
26
This information can now be used to draw a box
and whisker diagram..
8
Averages
Connect these together
Mean
Most common
Add them together and divide by how many there are
Median
Mode
Put in order then find the middle
Difference between biggest and smallest
Range
Which is the odd one out?
9
Averages
Find the mean, median, mode and range of these
values.
2
8
7
4
5
6
9
7
Mean (2 8 7 4 5 6 9 7) 8
48 8
6
Median
6.5
Mode
7
Range
7
9 - 2
10
Moving Averages
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Why use them?

• Moving Averages, when graphed, allow us to see
  any trends in data that are cyclical
• By calculating the average of 2 or more items in
  the data, any peaks and troughs are smoothed out.

12
265
269.25
265.25
270.75
4 Point Moving Average
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4 period Moving Average
14
500
x
400
x
x
300
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
200
x
x
x
100
1
2
3
4
1
2
3
4
1
2
3
4
1998
1996
1997
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Mean from frequency table
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ABC Motoring

• Here are the number of tests taken before
  successfully passing a driving test by 40
  students of ABC Motoring

2, 4, 1, 2, 1, 3, 7, 1, 1, 2, 2, 3, 5, 3, 2, 3,
4, 1, 1, 2, 5, 1, 2, 3, 2, 6, 1, 2, 7, 5, 1, 2,
3, 6, 4, 4, 4, 3, 2, 4
It is difficult to analyse the data in this form.
We can group the results into a frequency table.
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ABC Motoring

• Thats better!

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Finding the Mean
Remember, when finding the mean of a set of data,
we add together all the pieces of data.
This tells us that there were nine 1s in our
list. So we would do 111111111 9
It is simpler to use 1x9!!
We can do this for every row.
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Finding the Mean
107
40
So we have now added all the values up. What do
we do now?
We divide by how many values there were.
So we divide by the total number of people.
We now need to add these together
20
107
40
The Mean is
107 40 2.7 (1dp)
21
ABC Motoring

• Students who learn to drive with ABC motoring,
  pass their driving test after a mean number of
  2.7 tests.

22
Mean from Grouped frequency table
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Grouped data
Here are the Year Ten boys javelin scores.
How could you calculate the mean from this data?
How is the data different from the previous
examples you have calculated with?
Because the data is grouped, we do not know
individual scores. It is not possible to add up
the scores.
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Midpoints
It is possible to find an estimate for the
mean. This is done by finding the midpoint of
each group. To find the midpoint of the group
10 d lt 15 10 15 25 25 2
12.5 m
Find the midpoints of the other groups.
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Estimating the mean from grouped data
1 7.5
7.5
7.5
8 12.5
100
12.5
12 17.5
210
17.5
10 22.5
225
22.5
3 27.5
82.5
27.5
1 32.5
32.5
32.5
1 37.5
37.5
37.5
36
695
TOTAL
Estimated mean 695 36
19.3 m (to 1 d.p.)
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1 Find the modal group, median and estimate the
mean from the table below.

• Michelle keeps a record of the number of minutes
  her train is late each day. The table shows her
  results for a period of 50 days.

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Probability
28
Three boys ring Jane and ask her for a date
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Probability tree diagrams

• The probability that it will rain on Monday is
  0.2. The Probability it will rain on Tuesday is
  0.3.
• What is the probability that it will rain on
  Monday and Tuesday?

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Tuesday
It rains
0.3
Monday

• We can solve this problem by drawing a tree
  diagram.

We know that the probability is 0.2
It rains
0.2
There are two possible events here It rains or
It does not rain
0.7
It does not rain
Now lets look at Tuesday
The probability that it rains on Tuesday was
given to us as 0.3. We can work out that the
probability of it not raining has to be 0.7,
because they have to add up to 1.
It rains
0.3
0.8
It does not rain
What is the probability?
Well, raining and not raining are Mutually
Exclusive Events. So their probabilities have to
add up to 1.
1 0.2 0.8
We now have the required Tree Diagram.
0.7
It does not rain
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We wanted to know the probability that it rained
on Monday and Tuesday.
0.2 x 0.3 0.06
We can work out the probability of both events
happening by multiplying the individual
probabilities together
This is the only path through the tree which
gives us rain on both days
So the probability that it rains on Monday and
Tuesday is 0.06
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Actually, we can work out the probabilities of
all the possible events
0.2 x 0.7 0.14
The probability of rain on Monday, but no rain on
Tuesday is 0.14
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0.8 x 0.3 0.24
The probability that it will not rain on Monday,
but will rain on Tuesday is 0.24
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0.8 x 0.7 0.56
The probability that it does not rain on both
days is 0.56
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Lets look at the completed tree diagram
The end probabilities add up to 1. Remember this!
It can help you check your answer!
What do you notice?
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Stem and Leaf Diagrams

• Numerical bar charts

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Stem and Leaf Diagrams
Here is a set of data 14, 28, 17, 21, 23, 23,
19, 16, 26, 24, 24, 13, 20, 16, 18, 15, 22, 29,
21, 25 Decide on suitable groups tens are too
big, fives would give four groups - ok
4
2
5
1
9
5
8
6
0
3
4
4
6
6
9
3
3
1
7
8
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
10 to 14
1 1 2 2
15 to 19
20 to 24
25 to 29
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Stem and Leaf Diagrams
Now we need to put the leaves in numerical order
1 3 4 1 5 6 6 7 8 9 2 0 1 1 2
3 3 4 4 2 5 6 8 9
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Interpreting Stem and Leaf Diagrams
1 3 4 1 5 6 6 7 8 9 2 0 1 1 2
3 3 4 4 2 5 6 8 9
Mode most common 16, 21, 23 and 24 (all
happen twice) Median middle one (they are
already in order) 20 pieces of data looking
for 10th and 11th both are 21 Range biggest
smallest 29 13 16 Dont usually get asked
for the mean.
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Your Turn

• Draw a stem and leaf diagram of this data
• 2.3, 3.6, 4.5, 3.1, 5.8, 6.2, 4.2, 5.0, 3.1, 3.7,
  2.2, 2.8, 5.1, 6.0, 3.0, 4.9, 4.4, 5.3
• From the diagram, find the mode, median and range

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Good Luck!!!