3-Dec-15 Quartiles from a Frequency Table Quartiles from a Cumulative Frequency Table Statistics Estimating Quartiles from C.F Graphs

Apr 21, 2023Apr 21, 2023

Quartiles from a Frequency Table

Quartiles from a Cumulative Frequency Table

StatisticsStatisticsw

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Estimating Quartiles from C.F Graphs

Standard Deviation

Scatter Graphs

Standard Deviation from a sample

Probability

Relative Frequency & Probability

S5 Int2

Apr 21, 2023Apr 21, 2023

Starter QuestionsStarter Questions

1. Calculate the mean, median, mode and range

f or the weekly wages £ 200, £ 100, £ 800

£ 160, £ 100, £ 380, £ 120 and £ 180.

2. Make a Cumulative f requency table f or

a batch of eggs graded in sizes 1 - 7.

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S5 Int2

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StatisticsStatistics

Learning IntentionLearning Intention Success CriteriaSuccess Criteria

1.1. Know the term quartiles.Know the term quartiles.1. To explain how to calculate quartiles from frequency tables.

2.2. Calculate quartiles given a Calculate quartiles given a frequency table.frequency table.

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Quartiles from Frequency TablesS5 Int2

Quartiles from Frequency Tables

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Reminder !

S5 Int2

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.


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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 R0

6 number 6 number 6 number 6 numberQ1 Q2 Q3

The quartiles fall in the gaps between Q1 : the 6th and 7th numbersQ2 : the 12th and 13th numbersQ3 : the 18th and 19th numbers.


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6 number 6 number 6 number 6 numberQ1

Q2

Q31 No.

The quartiles fall in the gaps between Q1 : the 6th and 7th

Q2 : the 13th Q3 : the 19th and 20th numbers.


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6 number 6 number 6 number 6 number

Q1

Q2

Q3

1 No.

The quartiles fall in the gaps between Q1 : the 7th numberQ2 : the 13th and 14th numberQ3 : the 20th number.

1 No.


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Q1 Q2 Q3

1 No.

The quartiles fall in the gaps between Q1 : the 7th numberQ2 : the 14th numberQ3 : the 21th number.

1 No.1 No.


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Example 4 :For a ordered list of 34. Describe the quartiles.


Q1

Q2

Q3

1 No.

The quartiles fall in the gaps between Q1 : the 9th numberQ2 : the 17th and 18th numberQ3 : the 26th number.

1 No.

34 ÷ 4 = 8 R2

Apr 21, 2023Apr 21, 2023

Now try Exercise 1Start at 1b

Ch11 (page 162)

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StatisticsStatisticsS5 Int2 Quartiles from Frequency Tables

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Starter QuestionsStarter Questionsw

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1. Multiply out the brackets and simplif y

4(y+ 3) - 3(8- x)

2. Find the gradient and the y - intercept

f or the line with equation 2y = - 4x + 10

3. Find the quartiles f or the ordered 6 numbers

10, 12, 14, 18, 22, 30,32

S5 Int2

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StatisticsStatisticsQuartiles from Cumulative Frequency Table

1. To explain how to calculate quartiles from Cumulative Frequency Table.

1.1. Find the quartile values Find the quartile values from Cumulative Frequency from Cumulative Frequency Table.Table.

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TimeTime Freq.Freq.

(f)(f)

Example 1 :

The frequency table shows the length of phone calls ( in minutes) made from an office in one day.

2

5

4

8

3

1

2

3

4

5

2

10

22

18

5

Cum. Freq.Cum. Freq.

S5 Int2


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StatisticsStatisticsQuartiles from Cumulative Frequency TableS5 Int2

We use a combination of quartiles from a frequency table and the Cumulative Frequency Column.

For a list of 22 numbers, 22 ÷ 4 = 5 R2


Q1

Q2

Q3

1 No.

The quartiles fall in the gaps between Q1 : the 6th number Q1 : 3 minutes

1 No.

Q2 : the 11th and 12th number Q2 : 4 minutesQ3 : the 17th number. Q3 : 4 minutes

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No. Of No. Of SectionsSections

Freq.Freq.

(f)(f)

Example 2 :

A selection of schools were asked how many 5th year sections they have.Opposite is a table of the results.

Calculate the quartiles for the results.

3

8

8

9

5

4

5

6

7

8

3

16

33

25

8

Cum. Freq.Cum. Freq.

S5 Int2


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StatisticsStatisticsQuartiles from Cumulative Frequency TableS5 Int2

We use a combination of quartiles from a frequency table and the Cumulative Frequency Column.

The quartiles fall in the gaps between Q1 : the 8th and 9th numbers Q1 : 5.5Q2 : the 17th number Q2 : 7

Q3 : the 25th ad 26th numbers. Q3 : 7.5


8 number 8 number 8 number 8 numberQ1

Q2

Q31 No.

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Now try Exercise 2Ch11 (page 163)

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S5 Int2


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1. Find the area of the triangle.

2. Write down the two conditions

f or using the cosine rule.

3. Find the length of AC.

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A

B

C

8cm53o

70o

4cm

2cm3cm29o

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

1.1. Know the terms quartiles.Know the terms quartiles.

2.2. Estimate quartiles from Estimate quartiles from cumulative frequency cumulative frequency graphs.graphs.

S5 Int2

Quartiles fromQuartiles fromCumulative FrequencyCumulative Frequency

GraphsGraphs


GraphsGraphs

Number of sockets

Cumulative

Frequency

10 2

20 9

30 24

40 34

50 39

60 40

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Q3

0

5

10

15

20

25

30

35

40

45

0 10 20 30 40 50 60 70

Cum

ulati

ve F

requ

ency

Number of Sockets

Cumulative Cumulative FrequencyFrequency

GraphsGraphs

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Quartiles

40 ÷ 4 =10

Q1

Q2

Q1 =21

Q2 =27

Q3 =36

New Term

Interquartile rangeSemi-interquartile range

(Q3 – Q1 )÷2 = (36 - 21)÷2=7.5


GraphsGraphs

Km travelled on 1 gallon (mpg)

Cumulative

Frequency

20 3

25 11

30 30

35 53

40 69

45 76

50 80

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0

10

20

30

40

50

60

70

80

90

0 10 20 30 40 50 60

Cum

ulat

ive

Fre

quen

cy

Km travelled on 1 gallon (mpg)


GraphsGraphs

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Q3


GraphsGraphs

Quartiles

80 ÷ 4 =20

Q1

Q2

=28

= 32

= 37

New Term

Interquartile rangeSemi-interquartile range

(Q3 – Q1 )÷2 = (37 - 28)÷2=4.5

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GraphsGraphs

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11 28

2. Find the volume of a cone 15cm in height

and 10cm in diameter.

x x 21. Factorise

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1.1. Know the term Standard Know the term Standard Deviation.Deviation.

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

Standard DeviationStandard DeviationS5 Int2

1.1. Calculate the Standard Calculate the Standard Deviation for a collection Deviation for a collection of data.of data.

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Standard DeviationStandard DeviationFor a FULL set of DataFor a FULL set of Data

The range measures spread. Unfortunately any big change in either the largest value or smallest scorewill mean a big change in the range, even though

onlyone 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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S5 Int2


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.

Score Deviation(Deviation)2

70

72

75

78

80

Totals 375

Example 1 :Find the standard deviation of these fivescores 70, 72, 75, 78, 80.

S5 Int2


Step 1 : Find the mean

375 ÷ 5 = 75Step 3 : (Deviation)2

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-5

-3

0

3

5

0

25

9

0

9

25

68

Step 2 : Score - MeanStep 4 : Mean square deviation

68 ÷ 5 = 13.6

Step 5 :

Take the square root of step 4

√13.6 = 3.7

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

Example 2 :Find the standard deviation of these sixamounts of money £12, £18, £27, £36, £37, £50.

S5 Int2


Step 1 : Find the mean

180 ÷ 6 = 30

Apr 21, 2023Apr 21, 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.ww

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Step 2 : Score - Mean

Step 3 : (Deviation)2

Step 4 : Mean square deviation

962 ÷ 6 = 160.33

Score Deviation(Deviation)2

12

18

27

36

37

50

Totals 180

-18

-12

-3

6

7

20

324

144

9

36

49

400

0 962

Step 5 :

Take the square root of step 4

√160.33 = 12.7 (to 1d.p.)

Standard Deviation is £12.70

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When Standard Deviationis LOW it means the data values are close to the

MEAN.

When Standard Deviationis HIGH it means the data values are spread out from

the MEAN.

Mean Mean

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Standard DeviationStandard Deviation

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10 2 - 5

2. Solve the simultaneous equations

x y and x y

1. Construct a cumulative frequency table

For the data below.

S5 Int2

Waist Sizes Frequency

28” 7

30” 12

32” 23

34” 14

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1.1. Construct a table to Construct a table to calculate the Standard calculate the Standard Deviation for a sample of Deviation for a sample of data.data.

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

S5 Int2

Standard DeviationStandard DeviationFor a Sample of DataFor a Sample of Data

2.2. Use the table of values to Use the table of values to calculate Standard calculate Standard Deviation of a sample of Deviation of a sample of data.data.

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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.

2( )

1

x xs

n

s = standard deviationn = number in sample∑ = The sum of

22

1

xx

nsn

x = sample mean

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

practice !

Example 1a : Eight athletes have heart rates 70, 72, 73, 74, 75, 76, 76 and 76.

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S5 Int2


Heart rate (x)

x2

70

72

73

74

75

76

76

76

Totals

4900

5184

5329

5476

5625

5776

5776

5776

∑x2 = 43842∑x = 592

Step 2 :

Square all the values and find the

total

Step 3 :

Use formula to calculate sample deviation

22

1

xx

nsn

259243842

88 1

s

43842 43808

7s

4.875s

2.2 ( 1 . .) s to d p

Step 1 :

Sum all the values

Q1a. Calculate the mean :

592 ÷ 8 = 74

Q1a. Calculate the sample deviation

Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.

Heart rate (x)

x2

80

81

83

90

94

96

96

100

Totals

6400

6561

6889

8100

8836

9216

9216

10000

65218 64800

7s

418s

20.4 1 . .) ( s to d p

Example 1b : Eight office staff train as athletes. Their Pulse rates are 80, 81, 83, 90, 94, 96, 96 and 100 BPM

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∑x = 720

22

1

xx

nsn

272065218

88 1

s

Q1b(ii) Calculate the sample deviation

Q1b(i) Calculate the mean :

720 ÷ 8 = 90

∑x2 = 65218

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

Compare meansAthletes are fitter

StaffAthletes

2.2 1 . .) ( s to d p

74 Mean BPM 90 Mean BPM

20.4 1 . .) ( s to d p

Q1b(iv) What does the deviation tell us.Staff data is more

spread out.

Apr 21, 2023Apr 21, 2023

Now try Ex 5 & 6Ch11 (page 171)

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2

1. I f lines have the same gradient

What is special about them.

2. Factorise x +8x +15

3. Find the missing angles.

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33o

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1. To construct and interpret Scattergraphs.

1. Construct and understand the Key-Points of a scattergraph.

Scatter GraphsScatter GraphsConstruction of Scatter Graphs

2. Know the term positive and negative correlation.

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Scatter GraphsScatter GraphsConstruction of Scatter Graph

Team

100

120

140

160

180

0 20 40 60

Weight (kg)

Hei

ght

(cm

) Sam

Jim

Tim

GaryJoe

Dave

Bob

This scattergraph shows the heights and weights of a

sevens football team

Write down height and weight of each

player.

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xxx x

x x

Strong positive correlation

xx

x xx

x

Strong negative correlation

Best fit line

Best fit line

When two quantities are strongly connected we say there is a strong correlation between them.

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Key steps to:Drawing the best fitting straight line to a scatter

graph1. 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 throughmean values.

Apr 21, 2023Apr 21, 2023

0

2

4

6

8

10

12

0 2 4 6 8 10 12

Ages (Years)

Car

pri

ces

(£1000)

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Is therea

correlation?If yes, what

kind?

AgePrice

(£1000)

3

1

1

2

3

3

4

4

5

9

8

87

6

5

5

4

2

Strong negative correlation

Draw in the best fit line

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Mean Age = 2.9

Mean Price = £6000

Find the mean for

theAge and Prices values.

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Key steps to:Finding the equation of the straight line.

1. Pick any 2 points of graph ( pick easy ones to work with).

2. Calculate the gradient using :

3. Find were the line crosses y–axis this is b.

4. Write down equation in the form : y = ax + b

2 1

2 1

y ya

x x

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0

2

4

6

8

10

12

0 2 4 6 8 10 12

Ages (Years)

Car

pri

ces

(£1000)

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Scatter GraphsScatter Graphs

Pick points (0,10) and

(3,6)

y = 1.38x + 10

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Crosses y-axis at 10

10 61.38

3 0a

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2h - 49

1. Write the five figure summary for the data.

1, 1, 2, 3, 8, 3, 2

2. Factorise

S5 Int2

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ProbabilityProbability


1.1. Understand the probability Understand the probability line. line.

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

2.2. Calculate simply Calculate simply probabilities.probabilities.

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ProbabilityProbabilityLikelihood LineLikelihood Line

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10.50

CertainEvensImpossibleNot very

likelyVerylikely

Winning theLottery

School Holidays

Baby BornA Boy

Seeing a butterfly

In July

Go backin time

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10.50


likelyVerylikely

Everyone getting100 % in test

HomeworkEvery week

Toss a coinThat land

Heads

It willSnow in winter Going without

Food for a year.

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ProbabilityProbability

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To work out a probability

P(A) = number of outcomes

Total number of possible outcomes

Probability is ALWAYS in the range 0 to 1

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We can normally attach a value to the probability of an event happening.

ProbabilityProbabilityNumber Likelihood Number Likelihood

LineLine

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10.50

CertainEvensImpossible

1 2 3 54 76 8

0.1 0.2 0.3 0.4 0.6 0.7 0.8 0.9

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

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

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

88

= 1

48

= 0.5

18 = 0.125

P =

P(E) =

P(1) =

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10.50


likelyVerylikely

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

Q. What is the chance of picking a diamond ?

Q. What is the chance of picking ace ?

52= 0.5

1352

= 0.25452

= 0.08

26

0.1 0.2 0.3 0.4 0.6 0.7 0.8 0.9

P (Red) =

P (D) =

P (Ace) =

52 cards in a pack of cards

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Apr 21, 2023Apr 21, 2023

Now try Ex 8Ch11 (page 177)

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Probability Probability

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Starter QuestionsStarter Questions

21. Factorise 16x - 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

af ter 3 years.

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1.1. Know the term relative Know the term relative frequency.frequency.

1. To understand the term relative frequency.

2.2. Calculate relative Calculate relative frequency from data frequency from data given.given.

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Relative FrequenciesRelative Frequencies

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Relative FrequencyHow often an event happens compared

to the total number of events.

CountryCountry FrequencyFrequency Relative FrequencyRelative Frequency

FranceFrance 180180

ItalyItaly 9090

SpainSpain 9090

TotalTotal

Example : Wine sold in a shop over one week

180 ÷ 360 = 180 ÷ 360 =

90 ÷ 360 = 90 ÷ 360 =

90 ÷ 360 = 90 ÷ 360 =

360 360

0.25 0.25

0.25 0.25

0.5 0.5

1 1

Relative Frequency

always added up to

1

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Relative FrequenciesRelative Frequenciesw

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BoyBoyss

GirlsGirls TotalTotal

FrequencyFrequency 300300 200200

Relative Relative FrequencyFrequency

Example Calculate the relative frequency for boys and girlsborn in the Royal Infirmary hospital in December 2007.

500

0.6 0.4 1

Relative Frequency adds up to 1

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Now try Ex 9Ch11 (page 179)

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13, 19, 25, 25, 28, 32, 34, 36

2. The population of Scotland was 6 Million in 2000.

I t increased by 3% each year for 4 years.

What is the population after the 4 years.

S5 Int2

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Probability from Probability from Relative FrequencyRelative Frequency


1.1. Know the term relative Know the term relative frequency.frequency.

1. To understand the connection of probability and relative frequency.

2.2. Estimate probability from Estimate probability from the relative frequency.the relative frequency.

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S5 Int2


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handedness in a school. Results are given below

Number of Number of Left - Hand Left - Hand StudentsStudents

Total Total AskedAsked

RelativeRelative

FrequencFrequencyy

SeanSean 22 1010

KarenKaren 33 2525

DanielDaniel 2020 200200

Example 1

2 = 0.2

10

S5 Int2

Apr 21, 2023Apr 21, 2023

3 = 0.12

2520

= 0.1200

When the sum of the frequencies is LARGE the

relative frequency is a good estimate of the

probability of an outcome


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Number of Number of Alarmed Alarmed HousesHouses

Total Total AskedAsked

RelativeRelative

FrequencFrequencyy

PaulPaul 77 1010

AmyAmy 1212 2020

MeganMegan 4040 100100

Example 2

7 = 0.7

10

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12 = 0.6

2040

= 0.4100

What is the probability

that a house is alarmed ?

0.4

Who’s results would you use as a estimate of the probability of a

house being alarmed ?

Megan’s

Three students carry out a survey to study how many houses had an alarm system in a particular area. Results are given below

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Now try Ex 10Ch11 Start at Q2

(page 181)

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3-Dec-15 Quartiles from a Frequency Table Quartiles from a Cumulative Frequency Table Statistics Estimating Quartiles from C.F Graphs