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### Statistics

### Statistics

### Statistics

### Statistics

### Statistics

### Statistics

### Probability

### ProbabilityLikelihood Line

### ProbabilityLikelihood Line

### Probability

### ProbabilityNumber Likelihood Line

### ProbabilityLikelihood Line

### Probability from Relative Frequency

### Relative Frequencies

Credit

Mean

Mean from a Frequency Table

Median and Mode

Range of a Set of Data

Semi-Interquartile Range ( SIQR )

Quartile Graphs ( S – Curves )

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Standard Deviation / Sample Standard Deviation

Probability

Estimating Probability from Relative Frequency

Created by Mr. Lafferty

S3

Credit

Q1. Round to 2 significant figures

(a) 52.567 (b) 626

Q2. Why is 2 + 4 x 2 = 10 and not 12

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Q3. Solve for x

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S3

Credit

Working Out the Mean

Learning Intention

Success Criteria

- Know the term mean.

1. Explain the meaning of the term Mean

- 2. Calculate the mean for a given set of data.

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Created by Mr. Lafferty Maths Dept.

Mean=

Number of values

The mean

Themeanis 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.

For example, the mean of 3, 6, 7, 9 and 9 is

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Two dice were thrown 10 times and their scores were added together and recorded. Find themeanfor this data.

7, 5, 2, 7, 6, 12, 10, 4, 8, 9

Mean

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S3

Credit

Now try Exercise 2.1 Ch12 (page 228)

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S3

Credit

Working Out the Mean

Learning Intention

Success Criteria

- Add a third column to a frequency table.

- 1. To explain how to work out the Mean by adding in a third column to a Frequency Table.

- 2. Calculate the Mean from a frequency Table.

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Created by Mr. Lafferty Maths Dept.

Working Out the Mean

Example : This table shows the number

of light bulbs used in people’s living rooms

(f) x (B)

Adding a third column to this table

will help us find the total number of

bulbs and the ‘Mean’.

1

7

7 x 1 = 7

2

5

5 x 2 = 10

3

5

5 x 3 = 15

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4

2

2 x 4 = 8

5

1

1 x 5 = 5

Totals

20

45

Created by Mr. Lafferty Maths Dept.

Working Out the Mean

Example : This table shows the number

of brothers and sisters of pupils in an

S3 class.

S x f

Adding a third column to this table

will help us find the total number of

siblings and the ‘Mean’.

0

9

0 x 9 =0

1

13

1 x 13 = 13

2

6

2 x 6 = 12

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3

1

3 x 1 = 3

5

1

5 x 1 = 5

33

Totals

30

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Working Out the Mean

Now try Ex 2.2

Ch12 (page 229)

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S3

Credit

Q1.

Q2. Find the ratio of cos 60o

Q3. 75.9 x 70

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30m

Explain why

the length a = 36m

Q4.

24m

a

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S3

Credit

Learning Intention

Success Criteria

- To define the terms Median and Mode for a set of data.

- 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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Created by Mr. Lafferty Maths Dept.

Credit

Reminder !

Median : The middle value of a set of data.

When they are two middle values

the median is half way between them.

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Mode : The value that occurs the most in a set

of data. Can be more than one value.

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Example :

Find the median and mode for the set of data.

10, 2, 14, 1, 14, 7

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Now try Exercise 3.1

Ch12 (page 231)

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S3

Credit

Q1.

Q2. Calculate sin 90o

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Q3. Factorise 5y2 – 10y

A circle is divided into 10 equal pieces.

Find the arc length of one piece of the circle

if the radius is 5cm.

Q4.

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S3

Credit

Learning Intention

Success Criteria

- To define the term Range for a set of data.

- 1. Know the term Range.

- 2. Calculate the value for the Range for given set of data

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Range = highest value – lowest value

Finding therange

Therangeof 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.

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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S3

Credit

Example : find the range for the following

10 – 1 = 9

(a) 3, 1, 4, 10

7 – (-6) = 13

(b) -3, 8, -6, 1, 7, 5

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35.3 – (-15.5)

= 50.8oC

(c) The highest and lowest every recorded temperature

for Glasgow are 35.3oC and -15.5oC respectively.

Find the value of the range.

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Working Out Statistics

Now try Ex 4.1

Ch12 (page 232)

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Credit

Semi- Inter Quartile Range

Learning Intention

Success Criteria

- Know the term semi-interquartile range.

- To explain the term
- semi-interquartile range.

- Calculate
- semi-interquartile range.
- ( Q3 – Q1 ) ÷ 2

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Credit

Semi- Inter Quartile Range

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.

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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%

Created by Mr Lafferty Maths Dept

Semi-interquartile Range (SIQR) = ( Q3 – Q1 ) ÷ 2

= ( 9– 3) ÷ 2

= 3

S3

Credit

Semi- Inter Quartile Range

Example 2 : For a list of 9 numbers find the SIQR

3, 3, 7, 8, 10, 9, 1, 5, 9

9 ÷ 4 = 2 R1

1 3 3 5 7 8 9 9 10

2 number

2 number

2 number

1 No.

2 number

Q1

Q2

Q3

The quartiles fall in the gaps between

Q1 : the 3rd and 4th numbers

Q2 : the 5th number

Q3 : the 7th and 8th number.

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3

7

9

Created by Mr Lafferty Maths Dept

Semi-interquartile Range (SIQR) = ( Q3 – Q1 ) ÷ 2

= ( 10 – 3 ) ÷ 2

= 3.5

S3

Credit

Semi- Inter Quartile Range

Example 3 : For the ordered list find the SIQR.

3, 6, 2, 10, 12, 3, 4

7 ÷ 4 = 1 R3

2 3 3 4 6 10 12

1 number

1 number

1 number

1 number

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Q1

Q2

Q3

The quartiles fall in the gaps between

Q1 : the 2th number

Q2 : the 4th number

Q3 : the 6th number.

3

4

10

Created by Mr Lafferty Maths Dept

Semi- Inter Quartile Range

Now try Ex 5.1

Ch12 (page 235)

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Quartiles fromCumulative FrequencyGraphs

S3

Credit

Learning Intention

Success Criteria

- Know the terms quartiles.

- 1. To show how to estimate quartiles from cumulative frequency graphs.

- 2. Estimate quartiles from cumulative frequency graphs.

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Interquartile Range = (36 - 21) = 15

Semi-interquartile range

SIQR = (Q3 – Q1 )÷2

= (36 - 21)÷2

=7.5

Cumulative FrequencyGraphs

S3

Credit

Quartiles

40 ÷ 4 =10

Q3

Q3 =36

Q2

Q2 =27

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Q1

Q1 =21

Interquartile range = (37 - 28) = 9

Semi-interquartile range

= (Q3 – Q1 ) ÷2

= (37 - 28) ÷2

= 4.5

Cumulative FrequencyGraphs

Cumulative FrequencyGraphs

S3

Credit

Q3

= 37

Quartiles

80 ÷ 4 =20

Q2

= 32

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Q1

=28

Working Out Statistics

Now try Ex 5.2

Ch12 (page 238)

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S3

Credit

Learning Intention

Success Criteria

- Know the term Standard Deviation.

- 1. To explain the term and calculate the Standard Deviation for a collection of data.

- 2. Calculate the Standard Deviation for a collection of data.

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Created by Mr. Lafferty Maths Dept.

For a FULL set of Data

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.

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The semi-interquartile range is less sensitive to a single number changing but again it is only really based on two of the score.

Created by Mr. Lafferty Maths Dept.

For a FULL set of Data

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.

375 ÷ 5 = 75

Step 5 :

Take the square root of step 4

√13.6 = 3.7

Standard Deviation is 3.7 (to 1d.p.)

Step 2 : Score - Mean

Step 4 : Mean square deviation

68 ÷ 5 = 13.6

Standard Deviation

For a FULL set of Data

Step 3 : (Deviation)2

Example 1 : Find the standard deviation of these five

scores 70, 72, 75, 78, 80.

-5

25

-3

9

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0

0

3

9

5

25

0

68

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180 ÷ 6 = 30

Step 5 :

Take the square root of step 4

√160.33 = 12.7 (to 1d.p.)

Standard Deviation is £12.70

Step 2 : Score - Mean

Step 4 : Mean square deviation

962 ÷ 6 = 160.33

Step 3 : (Deviation)2

Standard Deviation

For a FULL set of Data

Example 2 : Find the standard deviation of these six

amounts of money £12, £18, £27, £36, £37, £50.

-18

324

-12

144

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-3

9

6

36

7

49

20

400

962

0

Created by Mr. Lafferty Maths Dept.

For a FULL set of Data

When Standard Deviation

is HIGH it means the data

values are spread out from

the MEAN.

When Standard Deviation

is LOW it means the data

values are close to the

MEAN.

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Mean

Mean

Created by Mr. Lafferty Maths Dept.

Now try Ex 6.1

Ch12 (page 240)

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Created by Mr. Lafferty Maths Dept.

For a Sample of Data

S3

Credit

Standard deviation

Learning Intention

Success Criteria

- 1. To show how to calculate the Sample Standard deviation for a sample of data.

- Know the term Sample Standard Deviation.

- 2. Calculate the Sample Standard Deviation for a collection of data.

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Created by Mr. Lafferty Maths Dept.

For a Sample of Data

We will use this version because it is easier to use in practice !

In real life situations it is normal to work with a sample

of data ( survey / questionnaire ).

We can use two formulae to calculate the sample deviation.

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s = standard deviation

∑ = The sum of

x = sample mean

n = number in sample

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592 ÷ 8 = 74

Step 2 :

Square all the values and find the total

Step 3 :

Use formula to calculate sample deviation

Step 1 :

Sum all the values

Q1a. Calculate the sample deviation

Standard Deviation

For a Sample of Data

Example 1a : Eight athletes have heart rates

70, 72, 73, 74, 75, 76, 76 and 76.

4900

5184

5329

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5476

5625

5776

5776

5776

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∑x = 592

∑x2 = 43842

720 ÷ 8 = 90

Q1b(ii) Calculate the sample deviation

Standard 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

6400

6561

6889

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8100

8836

9216

9216

10000

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∑x = 720

∑x2 = 65218

Q1b(iii) Who are fitter the athletes or staff.

Compare means

Athletes are fitter

Q1b(iv) What does the deviation tell us.

Staff data is more spread out.

Standard Deviation

For a Sample of Data

Athletes

Staff

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For a Sample of Data

Standard deviation

Now try Ex 7.1 & 7.2

Ch12 (page 243)

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Credit

Learning Intention

Success Criteria

- Understand the probability line.

- To understand probability in terms of the number line and calculate simple probabilities.

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- Calculate simply probabilities.

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0.5

1

S3

Credit

Impossible

Evens

Certain

Not very

likely

Very

likely

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Seeing

a butterfly

In July

School

Holidays

Winning the

Lottery

Baby Born

A Boy

Go back

in time

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0.5

1

S3

Credit

Impossible

Evens

Certain

Not very

likely

Very

likely

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It will

Snow in winter

Homework

Every week

Everyone getting

100 % in test

Toss a coin

That land

Heads

Going without

Food

for a year.

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Credit

We can normally attach a value

to the probability of an event happening.

To work out a probability

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P(A) =

Probability is ALWAYS in the range 0 to 1

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2

3

4

5

6

8

7

0

0.5

1

S3

Credit

0.1

0.2

0.3

0.4

0.6

0.7

0.8

0.9

Impossible

Evens

Certain

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8

P =

= 1

Q. What is the chance of picking a number between 1 – 8 ?

8

4

Q. What is the chance of picking a number that is even ?

= 0.5

P(E) =

8

Q. What is the chance of picking the number 1 ?

1

= 0.125

P(1) =

8

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0.5

1

S3

Credit

52 cards in a pack of cards

0.1

0.2

0.3

0.4

0.6

0.7

0.8

0.9

Impossible

Evens

Certain

Not very

likely

Very

likely

26

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= 0.5

P (Red) =

Q. What is the chance of picking a red card ?

52

13

Q. What is the chance of picking a diamond ?

= 0.25

P (D) =

52

4

Q. What is the chance of picking ace ?

P (Ace) =

= 0.08

52

Created by Mr Lafferty Maths Dept

Credit

Learning Intention

Success Criteria

- Know the term relative frequency.

- To understand the connection of probability and relative frequency.

- Estimate probability from the relative frequency.

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Created by Mr Lafferty Maths Dept

When the sum of the frequencies is LARGE the relative frequency is a good estimate of the probability of an outcome

Relative Frequencies

Relative Frequency always added up to 1

S3

Credit

Relative Frequency

How often an event happens compared

to the total number of events.

Example : Wine sold in a shop over one week

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0.5

180 ÷ 360 =

0.25

90 ÷ 360 =

0.25

90 ÷ 360 =

1

360

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When the sum of the frequencies is LARGE the relative frequency is a good estimate of the probability of an outcome

S3

Credit

Example

Calculate the relative frequency for boys and girls

born in the Royal Infirmary hospital in December 2007.

Relative Frequency adds up to 1

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500

1

0.4

0.6

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Now try Ex 8.2

Ch12 (page 248)

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