Transcript S3_3_Credit_Ch12_Sta..

Statistics
S3
Credit
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Mean
Mean from a Frequency Table
Median and Mode
Range of a Set of Data
Semi-Interquartile Range ( SIQR )
Quartile Graphs ( S – Curves )
Standard Deviation / Sample Standard Deviation
Probability
Estimating Probability from Relative Frequency
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1
Starter Questions
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S3
Credit
Q1.
Round to 2 significant figures
(a)
52.567
(b)
626
Q2.
Why is 2 + 4 x 2 = 10 and not 12
Q3.
Solve for x
2x  20  8  x
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2
Frequency Tables
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S3
Credit
Working Out the Mean
Learning Intention
1. Explain the meaning of the
term Mean
Success Criteria
1. Know the term mean.
2. Calculate the mean for a
given set of data.
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The mean
The mean is the most commonly used average.
To calculate the mean of a set of values we add together
the values and divide by the total number of values.
Mean =
Sum of values
Number of values
For example, the mean of 3, 6, 7, 9 and 9 is
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3+6+7+9+9
34

5
5
 6.8
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4
Two dice were thrown 10 times and their scores were
added together and recorded. Find the mean for this
data.
7, 5, 2, 7, 6, 12, 10, 4, 8, 9
Mean  7 + 5 + 2 + 7 + 6 + 12 + 10 + 4 + 8 + 9
10
70

10
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7
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5
Average / Mean
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S3
Credit
Now try Exercise 2.1
Ch12 (page 228)
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6
Starter Questions
S3
Credit
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1. Mutliply out 2y( y - 4)
2. Explain why 9 x 2  36 factorises to 9( x - 2)( x  2)
3.
12% of £22
4.
Tidy up the expression
7 - (-10)  3
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Frequency Tables
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S3
Credit
Working Out the Mean
Learning Intention
1. To explain how to work
out the Mean by adding in
a third column to a
Frequency Table.
Success Criteria
1. Add a third column to a
frequency table.
2. Calculate the Mean from a
frequency Table.
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Frequency Tables
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Credit
Working Out the Mean
Example : This table shows the number
of light bulbs used in people’s living rooms
Adding a third column to this table
will help us find the total number of
bulbs and the ‘Mean’.
No of
Bulbs
(c)
Freq.
(f)
1
7
7x1=7
2
5
5 x 2 = 10
3
5
5 x 3 = 15
4
2
2x4=8
5
1
1x5=5
Totals
20
Mean Number of bulbs
45
= 2.25 bulbs per room
20
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(f) x (B)
45
Frequency Tables
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Credit
Working Out the Mean
Example : This table shows the number
of brothers and sisters of pupils in an
S3 class.
No of
Sibling
s (S)
Adding a third column to this table
will help us find the total number of
siblings and the ‘Mean’.
Mean Number of siblings
=
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33
= 1.1 siblings
30
Sxf
0
9
1
13
1 x 13 = 13
2
6
2 x 6 = 12
3
1
3x1=3
5
1
5x1=5
Totals
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Freq.
(f)
30
0 x 9 =0
33
Frequency Tables
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S3
Credit
Working Out the Mean
Now try Ex 2.2
Ch12 (page 229)
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Starter Questions
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S3
Credit
Q1.
1.5% of £500
Q2.
Find the ratio of cos 60o
Q3.
75.9 x 70
Q4.
Explain why
the length a = 36m
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30m
24m
a
12
Different Averages
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Learning Intention
1. To define the terms
Median and Mode for a set
of data.
6-Apr-17
Success Criteria
1. Know the terms Median and
Mode.
2. Work out values for the
Median and Mode for given
set of data
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Statistics
S3
Credit
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Reminder !
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.
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Different Averages
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Example :
Find the median and mode for the set of data.
10, 2, 14, 1, 14, 7
Median = 1,2, 7,10,14,14
7 + 10 17
Median =
=
= 8.5
2
2
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Mode = 14
Different Averages
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Credit
Now try Exercise 3.1
Ch12 (page 231)
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Lesson Starter
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S3
Credit
Q1.
Explain why 2.5% of £800 = £20
Q2.
Calculate sin 90o
Q3.
Factorise 5y2 – 10y
Q4.
A circle is divided into 10 equal pieces.
Find the arc length of one piece of the circle
if the radius is 5cm.
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17
Different Averages
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Learning Intention
1. To define the term Range
for a set of data.
Success Criteria
1. Know the term Range.
2. Calculate the value for the
Range for given set of data
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Finding the range
The range of a set of data is a measure of how the data
is spread across the distribution.
To find the range we subtract the lowest value in the set
from the highest value.
Range = highest value – lowest value
When the range is small; the values are similar in size.
When the range is large; the values vary widely in size.
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The Range
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Example : find the range for the following
(a)
3, 1, 4, 10
10 – 1 = 9
(b)
-3, 8, -6, 1, 7, 5
7 – (-6) = 13
– (-15.5)
(c) The highest and lowest every recorded35.3
temperature
= 50.8oC
for Glasgow are 35.3oC and -15.5oC respectively.
Find the value of the range.
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20
Statistics
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Working Out Statistics
Now try Ex 4.1
Ch12 (page 232)
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Starter Questions
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S3
Credit
1. Calculate the mean, median, mode and range
for the weekly wages £200, £100, £800
£160, £100, £380, £120 and £180.
2. Find the angle cos-1 (0).
3. Show that y2  5 y  6 factorises to ( y - 6)( y  1)
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Statistics
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Credit
Semi- Inter Quartile Range
Learning Intention
1. To explain the term
semi-interquartile range.
Success Criteria
1. Know the term semiinterquartile range.
2. Calculate
semi-interquartile range.
( Q3 – Q1 ) ÷ 2
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Statistics
S3
Credit
Semi- Inter Quartile Range
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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 : Splits a dataset into 4 equal lengths.
Q1 = 25%
Q2 = 50%
Q3 = 75%
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Statistics
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Semi-interquartile Range
(SIQR) = ( Q3 – Q1 ) ÷ 2
= ( 9 – 3 ) ÷ 2Semi- Inter Quartile Range
=3
Example 2 : For a list of 9 numbers find the SIQR
R1
3, 3, 7, 8, 10, 9, 1, 5, 9 9 ÷ 4 = 2
1
3
2 number
Q1
3 5
7
2 number
1 No.
Q2
8
9
9
2 number
2 number
Q3
The quartiles fall in the gaps between
Q1 : the 3rd and 4th numbers 3
Q2 : the 5th number
7
Q3 : the 7th and 8th number. 9
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Statistics
Semi-interquartile
Range
SemiInter Quartile Range
(SIQR) = ( Q3 – Q1 ) ÷ 2
= ( 10 – 3 ) ÷ 2
= 3.5
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Example 3 : For the ordered list find the SIQR.
R3
3, 6, 2, 10, 12, 3, 4 7 ÷ 4 = 1
2
3
1 number
4
1 number
Q1
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3
6
10
1 number
Q2
1 number
Q3
The quartiles fall in the gaps between
Q1 : the 2th number 3
Q2 : the 4th number 4
Q3 : the 6th number. 10
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12
Statistics
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Semi- Inter Quartile Range
Now try Ex 5.1
Ch12 (page 235)
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Starter Questions
S3
Credit
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1. Is the following statment true?
4(y + 3) - 3(8 - x) = 4y -12 + 3x
2. Find the angle for sin -1 (0.7)
3. Explain why I can simply pick out
the quartiles from this dataset.
10, 12, 14, 18, 22, 30,32
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Quartiles from
Cumulative Frequency
Graphs
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Credit
Learning Intention
1. To show how to estimate
quartiles from cumulative
frequency graphs.
Success Criteria
1. Know the terms quartiles.
2. Estimate quartiles from
cumulative frequency graphs.
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Quartiles from
Cumulative Frequency
Graphs
Number of
sockets
Frequency
10
20
30
40
50
60
2
7
15
10
5
1
Cumulative
Frequency
2
9
24
34
39
40
Cumulative Frequency
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Interquartile Range = (36 - 21) = 15
Cumulative
Frequency
Semi-interquartile
range
SIQR = (QGraphs
3 – Q1 )÷2
= (36 - 21)÷2
=7.5
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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Quartiles from
Cumulative Frequency
Graphs
Km travelled on
1 gallon (kmpg)
20
25
30
35
40
45
50
Cumulative
Frequency
3
11
30
53
69
76
80
Interquartile range = (37 - 28) = 9
Cumulative
Frequency
Semi-interquartile
range
= (Q
3 – Q1 ) ÷2
Graphs
= (37 - 28) ÷2
= 4.5
Cumulative Frequency
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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
Statistics
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S3
Credit
Working Out Statistics
Now try Ex 5.2
Ch12 (page 238)
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Starter Questions
S3
Credit
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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. A circle is divided into 6 equal pieces.
Calculate the area of one of the pieces
when the diameter of the circle is 20cm.
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Standard Deviation
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S3
Credit
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.
2. 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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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.
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Created by Mr. Lafferty Maths Dept.
Standard Deviation
For a FULL set of Data
S3
Credit
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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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Credit
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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Credit
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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Credit
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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Relative Frequencies
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Credit
Now try Ex 6.1
Ch12 (page 240)
6-Apr-17
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Standard Deviation
For a Sample of Data
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Credit
Standard deviation
Learning Intention
1. To show how to calculate
the Sample Standard
deviation for a sample of
data.
6-Apr-17
Success Criteria
1. Know the term Sample
Standard Deviation.
2. Calculate the Sample
Standard Deviation for a
collection of data.
Created by Mr. Lafferty Maths Dept.
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Credit
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
Square
all the values
For
a
Sample
of
Data
S3
find the total
Credit Sum all the values
Use 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  4.875
5776
s76 2.2 (to 1 d . p5776
.)
76
Created by Mr. Lafferty Maths Dept.
Totals ∑x = 592
5776
∑x2 = 43842
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Credit
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
94
96
96
7
418
s
7
s  7.7
100
Created by Mr. Lafferty Maths Dept.
Totals ∑x = 720
x2
6400
 65218  64800
83
720 


8 1
80
n
8
Heart rate (x)
6561
6889
8100
8836
9216
9216
(to 1d10000
. p.)
∑x2 = 65218
Standard
Q1b(iii) WhoDeviation
are fitter
Q1b(iv) What does the
athletes
or of
staff.
Forthe
adeviation
Sample
Data
tell us.
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Credit
Compare means
Staff data is more spread
Athletes are fitter
out.
Athletes
Staff
Mean  74 BPM
Mean  90 BPM
s  2.2 (to 1d. p.)
s  7.7 (to 1d. p.)
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Standard Deviation
For a Sample of Data
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Standard deviation
Now try Ex 7.1 & 7.2
Ch12 (page 243)
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Starter Questions
S3
Credit
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1. If a triangle is right angled and two sides
have length 10 and 9.
What are the possible sizes of the third side.
2. Factorise x2 + 8x + 15
3. The missing angles are 90 and 57. Explain why?
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33o
Probability
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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.
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Probability
Likelihood Line
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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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Credit
0
Impossible
It will
Snow in winter
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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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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
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Probability
Number Likelihood Line
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Credit
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1
0
2
0.1
Impossible
3
0.2
4
0.3
5
0.4
6
0.5
0.6
7
0.7
8
0.8
Evens
P=
Q. What is the chance of picking a number that is even ?P(E) =
Q. What is the chance of picking the number 1 ?
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1
Certain
Q. What is the chance of picking a number between 1 – 8 ?
6-Apr-17
0.9
P(1) =
8
=1
8
4
= 0.5
8
1
= 0.125
8
Probability
Likelihood Line
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Credit
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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
Q. What is the chance of picking a red card ?
P (Red) =
Q. What is the chance of picking a diamond ?
P (D) =
Q. What is the chance of picking ace ? P (Ace) =
4
52
6-Apr-17
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0.9
1
Certain
26
52
= 0.5
13
= 0.25
52
= 0.08
Probability
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Credit
Now try Ex 8.1
Ch12 (page 246)
6-Apr-17
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Starter Questions
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S3
Credit
1. Is it true that 16x2 -36 factorises to 4(2x -3)(2x + 3)
2. Write down what you understand by the term
(SOH)(C AH)(T OA)
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Probability from
Relative Frequency
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S3
Credit
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
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Credit
When the sum of the
frequencies
is LARGE Frequencies
the
Relative
Relative
relative frequency is a good
Frequency
estimate of the probability
always added
Relative Frequency
of an outcome
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
S3
Credit
www.mathsrevision.com
Example
When the sum of the
frequencies
is LARGE the
Relative
Frequencies
relative frequency is a good
estimate of the probability
of an outcome
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
www.mathsrevision.com
S3
Credit
Now try Ex 8.2
Ch12 (page 248)
6-Apr-17
Created by Mr. Lafferty Maths Dept.