Transcript S5_Int2_Statistics.pps

Statistics
S5 Int2
www.mathsrevision.com
Quartiles from a Frequency Table
Quartiles from a Cumulative Frequency Table
Estimating Quartiles from C.F Graphs
Standard Deviation
Standard Deviation from a sample
Scatter Graphs
Probability
Relative Frequency & Probability
6-Apr-17
Created by Mr. Lafferty Maths Dept.
Starter Questions
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S5 Int2
1. Calculate the mean, median, mode and range
for the weekly wages £200, £100, £800
£160, £100, £380, £120 and £180.
2. Make a Cumulative frequency table for
a batch of eggs graded in sizes 1 - 7.
6-Apr-17
Created by Mr Lafferty Maths Dept
Statistics
Quartiles from Frequency Tables
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S5 Int2
Learning Intention
1. To explain how to calculate
quartiles from frequency
tables.
6-Apr-17
Success Criteria
1. Know the term quartiles.
2. Calculate quartiles given a
frequency table.
Created by Mr Lafferty Maths Dept
Statistics
Quartiles from Frequency Tables
S5 Int2
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Reminder !
Range :
The difference between highest and Lowest
values. It is a measure of spread.
Median :
The middle value of a set of data.
When they are two middle values
the median is half way between them.
Mode :
The value that occurs the most in a set
of data. Can be more than one value.
Quartiles : The median splits into lists of equal length.
The medians of these two lists are called quartiles.
6-Apr-17
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Statistics
Quartiles from Frequency Tables
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S5 Int2
To find the quartiles of an ordered list you consider its
length. You need to find three numbers which break the
list into four smaller list of equal length.
Example 1 : For a list of 24 numbers, 24 ÷ 6 = 4
6 number
Q1
6 number
Q2
6 number
Q3
The quartiles fall in the gaps between
Q1 : the 6th and 7th numbers
Q2 : the 12th and 13th numbers
Q3 : the 18th and 19th numbers.
6-Apr-17
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R0
6 number
Statistics
Quartiles from Frequency Tables
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S5 Int2
Example 2 : For a list of 25 numbers, 25 ÷ 4 = 6
6 number Q1
6 number
1 No.
6 number
Q3 6 number
Q2
The quartiles fall in the gaps between
Q1 : the 6th and 7th
Q2 : the 13th
Q3 : the 19th and 20th numbers.
6-Apr-17
Created by Mr Lafferty Maths Dept
R1
Statistics
Quartiles from Frequency Tables
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S5 Int2
Example 3 : For a list of 26 numbers, 26 ÷ 4 = 6
6 number
1 No.
6 number
Q2
6 number
Q1
1 No.
Q3
The quartiles fall in the gaps between
Q1 : the 7th number
Q2 : the 13th and 14th number
Q3 : the 20th number.
6-Apr-17
Created by Mr Lafferty Maths Dept
R2
6 number
Statistics
Quartiles from Frequency Tables
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S5 Int2
Example 4 : For a list of 27 numbers, 27 ÷ 4 = 6
6 number
1 No.
Q1
6 number
1 No.
6 number
Q2
1 No.
Q3
The quartiles fall in the gaps between
Q1 : the 7th number
Q2 : the 14th number
Q3 : the 21th number.
6-Apr-17
Created by Mr Lafferty Maths Dept
R3
6 number
Statistics
Quartiles from Frequency Tables
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S5 Int2
Example 4 : For a ordered list of 34.
Describe the quartiles.
34 ÷ 4 = 8
8 number
1 No.
8 number
R2
Q2
8 number
Q1
6-Apr-17
1 No.
Q3
The quartiles fall in the gaps between
Q1 : the 9th number
Q2 : the 17th and 18th number
Q3 : the 26th number.
Created by Mr Lafferty Maths Dept
8 number
Statistics
Quartiles from Frequency Tables
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S5 Int2
Now try Exercise 1
Start at 1b
Ch11 (page 162)
6-Apr-17
Created by Mr Lafferty Maths Dept
Starter Questions
S5 Int2
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1. Multiply out the brackets and simplify
4(y + 3) - 3(8 - x)
2. Find the gradient and the y - intercept
for the line with equation 2y = - 4x + 10
3. Find the quartiles for the ordered 6 numbers
10, 12, 14, 18, 22, 30,32
6-Apr-17
Created by Mr. Lafferty Maths Dept.
Statistics
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S5 Int2
Quartiles from Cumulative Frequency Table
Learning Intention
1. To explain how to
calculate quartiles from
Cumulative Frequency
Table.
6-Apr-17
Success Criteria
1. Find the quartile values from
Cumulative Frequency Table.
Created by Mr. Lafferty Maths Dept.
Statistics
Quartiles from Cumulative Frequency Table
S5 Int2
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Example 1 :
Time
Freq.
(f)
1
2
2
2
3
5
3
5
10
4
8
18
5
4
22
The frequency table shows the length
of phone calls ( in minutes) made from
an office in one day.
6-Apr-17
Created by Mr. Lafferty Maths Dept.
Cum. Freq.
Statistics
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S5 Int2
Quartiles from Cumulative Frequency Table
We use a combination of quartiles from a frequency table
and the Cumulative Frequency Column.
For a list of 22 numbers, 22 ÷ 4 = 5
5 number
1 No.
5 number
Q2
5 number
Q1
1 No.
R2
5 number
Q3
The quartiles fall in the gaps between
Q1 : the 6th number Q1 : 3 minutes
Q2 : the 11th and 12th number Q2 : 4 minutes
6-Apr-17
Q3 :
the 17th number. Q3 : 4 minutes
Created by Mr. Lafferty Maths Dept.
Statistics
Quartiles from Cumulative Frequency Table
S5 Int2
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Example 2 :
A selection of schools were asked
how many 5th year sections they have.
Opposite is a table of the results.
No. Of
Sections
Calculate the quartiles for the results.
6-Apr-17
Created by Mr. Lafferty Maths Dept.
Freq. Cum. Freq.
(f)
4
3
3
5
5
8
6
8
16
7
9
25
8
8
33
Statistics
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S5 Int2
Quartiles from Cumulative Frequency Table
We use a combination of quartiles from a frequency table
and the Cumulative Frequency Column.
Example 2 : For a list of 33 numbers, 33 ÷ 4 = 8
8 number Q1
8 number
1 No.
8 number
Q3 8 number
Q2
The quartiles fall in the gaps between
Q1 : the 8th and 9th numbers Q1 : 5.5
Q2 : the 17th number Q2 : 7
6-Apr-17
Q3 :
the 25th ad 26th numbers. Q3 : 7.5
Created by Mr. Lafferty Maths Dept.
R1
Statistics
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S5 Int2
Quartiles from Cumulative Frequency Table
Now try Exercise 2
Ch11 (page 163)
6-Apr-17
Created by Mr Lafferty Maths Dept
Starter Questions
S5 Int2
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1. Find the area of the triangle.
3cm
2cm
29o
4cm
2. Write down the two conditions
for using the cosine rule.
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C
70o
3. Find the length of AC.
6-Apr-17
A
53o
B
8cm
Quartiles from
Cumulative Frequency
Graphs
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S5 Int2
Learning Intention
1. To show how to estimate
quartiles from cumulative
frequency graphs.
6-Apr-17
Success Criteria
1. Know the terms quartiles.
2. Estimate quartiles from
cumulative frequency graphs.
Created by Mr. Lafferty Maths Dept.
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S5 Int2
Quartiles from
Cumulative Frequency
Graphs
Number of
sockets
10
20
30
40
50
60
Cumulative
Frequency
2
9
24
34
39
40
New Term
Cumulative Frequency
Interquartile range
Graphs range
Semi-interquartile
S5 Int2
(Q3 – Q1 )÷2 = (36 - 21)÷2
=7.5
Cumulative Frequency
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45
Quartiles
40
35
Q3
30
25
Q2
20
10
Q3 =36
Q2 =27
Q1
15
40 ÷ 4 =10
Q1 =21
5
0
0
10
20
30
40
50
Number of Sockets
60
70
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S5 Int2
Quartiles from
Cumulative Frequency
Graphs
Km travelled on
1 gallon (mpg)
20
25
30
35
40
45
50
Cumulative
Frequency
3
11
30
53
69
76
80
New Term
Cumulative Frequency
Interquartile range
Graphs range
Semi-interquartile
Cumulative Frequency
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S5 Int2
(Q3 – Q1 )÷2 = (37 - 28)÷2
=4.5
90
80
Q3 = 37
70
60
Quartiles
80 ÷ 4 =20
Q2 = 32
50
40
Q1 =28
30
20
10
0
0
10
20
30
40
50
Km travelled on 1 gallon (mpg)
60
Quartiles from
Cumulative Frequency
Graphs
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S5 Int2
Now try Exercise 3
Ch11 (page 166)
6-Apr-17
Created by Mr. Lafferty Maths Dept.
Starter Questions
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S5 Int2
1. Factorise x2  11x  28
2. Find the volume of a cone 15cm in height
and 10cm in diameter.
6-Apr-17
Created by Mr. Lafferty Maths Dept.
Standard Deviation
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S5 Int2
Learning Intention
1. To explain the term and
calculate the Standard
Deviation for a collection
of data.
6-Apr-17
Success Criteria
1. Know the term Standard
Deviation.
1. Calculate the Standard
Deviation for a collection of
data.
Created by Mr. Lafferty Maths Dept.
Standard Deviation
For a FULL set of Data
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S5 Int2
The range measures spread. Unfortunately any big
change in either the largest value or smallest score
will mean a big change in the range, even though only
one number may have changed.
The semi-interquartile range is less sensitive to a single
number changing but again it is only really based on two
of the score.
6-Apr-17
Created by Mr. Lafferty Maths Dept.
Standard Deviation
For a FULL set of Data
S5 Int2
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A measure of spread which uses all the data is the
Standard Deviation
The deviation of a score is how much the score differs
from the mean.
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Created by Mr. Lafferty Maths Dept.
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S5 Int2
Step25::Score - Mean
Deviation
Step
Step 1 : FindStandard
the mean
Step 4 : Mean square deviation
For a Take
FULL
set
of
Data
the square root of step 4
375 ÷ 5 = 75
2
Step 3 : (Deviation)68
÷ 5 = 13.6
√13.6 deviation
= 3.7
Example 1 : Find the standard
of these five
scores 70, 72, 75, 78, 80.
Standard Deviation is 3.7 (to 1d.p.)
Score
Deviation
(Deviation)2
70
-5
25
72
-3
9
75
78
80
Totals
6-Apr-17
375
0
3
5
0
Created by Mr. Lafferty Maths Dept.
0
9
25
68
5Deviation
: square deviation
Step 1 : FindStandard
the
mean
Step
4Step
: Mean
Step
2 : Score - Mean
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S5 Int2
For
a
FULL
set
of
Data
2
Take the square root of step 4
Step180
3 : ÷(Deviation)
6 = 30
962 ÷ 6 = 160.33
= 12.7
(to 1d.p.)
Example 2 √160.33
: Find the
standard
deviation of these six
amounts of money £12, £18, £27, £36, £37, £50.
Standard Deviation is £12.70
Score
Deviation
(Deviation)2
12
-18
324
18
-12
144
27
36
37
Totals
6-Apr-17
-3
6
7
20
50
Created by Mr. Lafferty Maths Dept.
0
180
9
36
49
400
962
Standard Deviation
For a FULL set of Data
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S5 Int2
When Standard Deviation
is LOW it means the data
values are close to the
MEAN.
When Standard Deviation
is HIGH it means the data
values are spread out from
the MEAN.
Mean
6-Apr-17
Mean
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Standard Deviation
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S5 Int2
Now try Exercise 4
Ch11 (page 169)
6-Apr-17
Created by Mr. Lafferty Maths Dept.
Starter Questions
S5 Int2
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1. Construct a cumulative frequency table
For the data below.
Waist Sizes
Frequency
28”
7
30”
12
32”
23
34”
14
2. Solve the simultaneous equations
x  y  10
6-Apr-17
and
2x - y  5
Created by Mr. Lafferty Maths Dept.
Standard Deviation
For a Sample of Data
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S5 Int2
Learning Intention
1. To show how to calculate
the Standard deviation
for a sample of data.
Success Criteria
1. Construct a table to calculate
the Standard Deviation for a
sample of data.
2. Use the table of values to
calculate Standard Deviation
of a sample of data.
6-Apr-17
Created by Mr. Lafferty Maths Dept.
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S5 Int2
Standard Deviation
For a Sample ofWe
Data
will use this
version because it is
easier
to use
in a sample
In real life situations it is normal
to work
with
practice ).
!
of data ( survey / questionnaire
We can use two formulae to calculate the sample deviation.
s
 ( x  x)
2
n 1
s = standard deviation
x = sample mean
6-Apr-17
 x 
2
s
x



n 1
∑ = The sum of
n = number in sample
Created by Mr. Lafferty Maths Dept.
n
2
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2:
Q1a. Calculate the mean : Q1a.Step
Calculate
the
Standard
Deviation
Step592
1 : ÷ 8 = 74
Step 3 :sample deviation
all the values
For a SampleSquare
of Data
S5 Int2
find the total
Sum all the valuesUse formula toand
calculate
sample have
deviation
Example 1a : Eight athletes
heart rates
70, 72, 73, 74, 75, 76, 76 and 76.
s
s
6-Apr-17

2
x


x



n 1
 43842
Heart rate (x)
2
8 1
8
2
4900
72
 43842   43808
5184
73
7
s
n
592 


70
x2
5329
74
5476
75
5625
76
s  4.857
5776
s76 2.2 (to 1 d . p5776
.)
76
5776
Totals ∑x = 592
∑x2 = 43842
Created by Mr. Lafferty Maths Dept.
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S5 Int2
Q1b(i) Calculate the mean :
Standard
Deviation
Q1b(ii) Calculate the
720 ÷ 8 = 90
sample
deviation
For a Sample of
Data
Example 1b : Eight office staff train as athletes.
Their Pulse rates are 80, 81, 83, 90, 94, 96, 96 and 100 BPM
s
s
6-Apr-17

x
2

x



n 1
 65218
2
s 81
90
2
x2
6400
 65218  64800
7
83
720 


8 1
80
n
8
Heart rate (x)
6561
6889
8100
94
s  59.714 8836
96
9216
96
s  7.7
100
Created by Mr. Lafferty Maths Dept.
Totals ∑x = 720
(to 1d. p
.)
9216
10000
∑x2 = 65218
Standard
Q1b(iii) WhoDeviation
are fitter
Q1b(iv) What does the
athletes
or of
staff.
Forthe
adeviation
Sample
Data
tell us.
Compare means
Staff data is more spread
Athletes are fitter
out.
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S5 Int2
Athletes
Staff
Mean  74 BPM
Mean  90 BPM
s  2.2 (to 1d. p.)
s  20.4 (to 1d. p.)
6-Apr-17
Created by Mr. Lafferty Maths Dept.
Standard Deviation
For a Sample of Data
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S5 Int2
Now try Ex 5 & 6
Ch11 (page 171)
6-Apr-17
Created by Mr. Lafferty Maths Dept.
Starter Questions
S5 Int2
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1. If lines have the same gradient
What is special about them.
2. Factorise x2 + 8x + 15
33o
3. Find the missing angles.
6-Apr-17
Created by Mr. Lafferty Maths Dept.
Scatter Graphs
Construction of Scatter Graphs
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S5 Int2
Learning Intention
1. To construct and interpret
Scattergraphs.
Success Criteria
1. Construct and understand
the Key-Points of a
scattergraph.
2. Know the term positive and
negative correlation.
6-Apr-17
Created by Mr Lafferty Maths Dept
S5 Int2
This scattergraph
shows the heights
and weights of a
sevens football team
Write down height and
Scatter Graphs
weight of each player.
Construction of Scatter Graph
180
Bob
Tim
160
Height (cm)
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Team
Sam
140
Joe
Gary
Jim
Dave
120
100
0
6-Apr-17
20
40
Weight (kg)
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60
Scatter Graphs
Construction of Scatter Graph
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S5 Int2
When two quantities are strongly connected we say there is a
strong correlation between them.
Best fit line
x x
x x
x x
Strong positive
correlation
6-Apr-17
x
x x
x
x
x
Best fit line
Strong negative
correlation
Created by Mr Lafferty Maths Dept
Scatter Graphs
Construction of Scatter Graph
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S5 Int2
Key steps to:
Drawing the best fitting straight line to a scatter graph
1. Plot scatter graph.
2. Calculate mean for each variable and plot the
coordinates on the scatter graph.
3. Draw best fitting line, making sure it goes through
mean values.
6-Apr-17
Created by Mr. Lafferty Maths Dept.
Draw in the
best fit line
FindMean
the mean
Age = 2.9
for theAge
and Prices
Mean
Price = £6000
S5 Int2 values.
Scatter Graphs
Price
Age (£1000)
1
1
9
8
2
3
3
3
4
4
5
8
7
6
5
5
4
2
6-Apr-17
12
Is there
a correlation?
If yes, what
kind?
10
Car prices (£1000)
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Construction of Scatter Graph
8
6
4
2
0
0
2
4
6
Ages (Years)
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8
10
12
Scatter Graphs
Construction of Scatter Graph
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S5 Int2
Key steps to:
Finding the equation of the straight line.
1. Pick any 2 points of graph ( pick easy ones to work with).
y2  y1
2. Calculate the gradient using : a 
x2  x1
3. Find were the line crosses y–axis this is b.
4. Write down equation in the form : y = ax + b
6-Apr-17
Created by Mr. Lafferty Maths Dept.
Scatter Graphs
Crosses y-axis at 10
S5 Int2
Pick points
(0,10) and (3,6)
10
a
10  6
 1.33
03
Car prices (£1000)
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12
8
6
4
2
0
0
2
y = -1.33x + 10
6-Apr-17
4
6
Ages (Years)
Created by Mr Lafferty Maths Dept
8
10
12
Scatter Graphs
Construction of Scatter Graph
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S5 Int2
Now try Exercise 7
Ch11 (page 175)
6-Apr-17
Created by Mr. Lafferty Maths Dept.
Starter Questions
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S5 Int2
1. Write the five figure summary for the data.
1, 1, 2, 3, 8, 3, 2
2. Factorise
6-Apr-17
h2 - 49
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Probability
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S5 Int2
Learning Intention
1. To understand probability
in terms of the number
line and calculate simple
probabilities.
Success Criteria
1. Understand the probability
line.
2. Calculate simply probabilities.
6-Apr-17
Created by Mr Lafferty Maths Dept
Probability
Likelihood Line
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S5 Int2
0
Impossible
Seeing
a butterfly
In July
6-Apr-17
0.5
Not very
likely
School
Holidays
Evens
Winning the
Lottery
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1
Very
likely
Baby Born
A Boy
Certain
Go back
in time
Probability
Likelihood Line
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S5 Int2
0
Impossible
It will
Snow in winter
6-Apr-17
0.5
Not very
likely
Evens
Homework Everyone getting
Every week
100 % in test
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1
Very
likely
Certain
Toss a coin Going without
That land
Food
Heads
for a year.
Probability
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S5 Int2
We can normally attach a value
to the probability of an event happening.
To work out a probability
number of outcomes
P(A) =
Total number of possible outcomes
Probability is ALWAYS in the range 0 to 1
6-Apr-17
Created by Mr Lafferty Maths Dept
Probability
Number Likelihood Line
S5 Int2
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1
0
2
0.1
Impossible
3
0.2
4
0.3
5
0.4
6
0.5
0.6
8
7
0.7
0.8
0.9
Evens
Q. What is the chance of picking a number between 1 – 8 ?
1
Certain
P=
8
=1
8
4
P(E) =
= 0.5
8
Q. What is the chance of picking the number 1 ?
1
P(1)
=
= 0.125
6-Apr-17
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8
Q. What is the chance of picking a number that is even ?
Probability
Likelihood Line
S5 Int2
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52 cards in a pack of cards
0
0.1
Impossible
0.2
0.3
Not very
likely
0.4
0.5
0.6
0.7
Evens
0.8
Very
likely
0.9
Q. What is the chance of picking a red card ?
26
P (Red) =
52
Q. What is the chance of picking a diamond ?
P (D) =
Q. What is the chance of picking ace ?
4
52
6-Apr-17
P (Ace) =
Created by Mr Lafferty Maths Dept
1
Certain
= 0.5
13
= 0.25
52
= 0.08
Probability
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S5 Int2
Now try Ex 8
Ch11 (page 177)
6-Apr-17
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Starter Questions
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S5 Int2
1. Factorise 16x2 -36
2. The average price of a two wek holiday is £1000.
The prices depreciates @ 2% each year.
How much is the average price of a holiday
after 3 years.
6-Apr-17
Created by Mr Lafferty Maths Dept
Relative Frequencies
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S5 Int2
Learning Intention
1. To understand the term
relative frequency.
Success Criteria
1. Know the term relative
frequency.
2. Calculate relative frequency
from data given.
6-Apr-17
Created by Mr Lafferty Maths Dept
Relative Frequencies
S5 Int2
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Relative Frequency
Relative
Frequency
always added
up to 1
How often an event happens compared
to the total number of events.
Example : Wine sold in a shop over one week
Country
Frequency
France
180
Italy
90
90 ÷ 360 = 0.25
Spain
90
90 ÷ 360 =
Total
360
1
6-Apr-17
Relative Frequency
180 ÷ 360 = 0.5
Created by Mr Lafferty Maths Dept
0.25
Relative Frequencies
S5 Int2
www.mathsrevision.com
Example
Calculate the relative frequency for boys and girls
born in the Royal Infirmary hospital in December 2007.
Boys Girls
Total
Frequency
300
200
500
Relative Frequency
0.6
0.4
1
6-Apr-17
Created by Mr Lafferty Maths Dept
Relative
Frequency
adds up
to 1
Relative Frequencies
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S5 Int2
Now try Ex 9
Ch11 (page 179)
6-Apr-17
Created by Mr. Lafferty Maths Dept.
Starter Questions
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S5 Int2
1. Write the five figure summary for the data below.
13, 19, 25, 25, 28, 32, 34, 36
2. The population of Scotland was 6 Million in 2000.
It increased by 3% each year for 4 years.
What is the population after the 4 years.
6-Apr-17
Created by Mr. Lafferty Maths Dept.
Probability from
Relative Frequency
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S5 Int2
Learning Intention
1. To understand the
connection of probability
and relative frequency.
6-Apr-17
Success Criteria
1. Know the term relative
frequency.
2. Estimate probability from
the relative frequency.
Created by Mr Lafferty Maths Dept
www.mathsrevision.com
S5 Int2
When the sum of the
Probability
from
frequencies is LARGE the
relative
frequency is a good
Relative
Frequency
estimate of the probability
of an outcome
Example 1
Three students carry out a survey to study left
handedness in a school. Results are given below
Number of
Left - Hand
Students
6-Apr-17
Total
Asked
Sean
2
10
Karen
3
25
Daniel
20
200
Created by Mr Lafferty Maths Dept
Relative
Frequency
2
= 0.2
10
3
= 0.12
25
20
= 0.1
200
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S5 Int2
Who’s results
would you from
Probability
Megan’s
use as a estimate of the
Relative Frequency
probability of a house
being alarmed ?
Example
Three2students carry out a survey to study how many
houses
had an alarm system in a particular area.
What
is the
Results are given below
probability
0.4
that a house is
Number of
Relative
Total
alarmed ?
Alarmed
Asked Frequency
Houses
6-Apr-17
Paul
7
10
Amy
12
20
Megan
40
100
Created by Mr Lafferty Maths Dept
7
= 0.7
10
12
= 0.6
20
40
= 0.4
100
Probability from
Relative Frequency
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S5 Int2
Now try Ex 10
Ch11 Start at Q2
(page 181)
6-Apr-17
Created by Mr. Lafferty Maths Dept.