Statistics Module Form 4

7/29/2019 Statistics Module Form 4

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My

AdditionalMathematics

ModulesForm 4Topic: 7

DECISIVE(Version 2011)

by

NgKL(M.Ed.,B.Sc.Hons.,Dip.Ed.,Dip.Edu.Mgt.&Lship,Cert.NPQH.)


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THIS MODUL BELONGS TO

NAME: .

CLASS:

MY PROMISEMY RESOLUTION

MY AIMIS TO SCORE

INMY SPM ADDITIONAL MATHEMATICS

ANDWILL WORK HARD, SMART AND EFFICIENTLY FOR IT

THIS IS MY SIGNATURETO SIGNIFY MY DETERMINATION

...

( )

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7.1 - MEASURES OF CENTRAL TENDENCY

IMPORTANT POINTS:

Ungrouped DataUngrouped Data

(in a Frequency Table)Grouped Data

Data sets which are not

grouped into classes.

Example:

The masses of six pupils

in kilogram:

50, 52, 55, 60, 55, 59.

Data sets which are not

grouped into classes but

are presented in

Frequency Table.

Example:

Number

of Books

Read

Number

of

Students

0 5

1 6

2 83 4

4 2

Data sets which are

grouped into classes and

presented in Frequency

Table.

Example:

Number

of Books

Read

Number

of

Students

0 1 11

2 - 3 12

4 - 5 156 - 7 8

8 - 9 7

3


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Median, m = the value

in the middle position of

a set of data after the

data are arranged in

ascending order.

Median, m = the value

in the middle position of

a set of data after the

data are arranged in

ascending order.

Median, m

=Lm +

mf

F2

N

c

Lm = lower boundary of

the median class.

N = sum of frequency.

F = cumulative

frequency of the

class before the

median class.

fm = frequency of themedian class.

c = size of the median

class.

Effects of uniform changes in a set of data on the mode, mean and median:

1. When a constant numberkis added or subtracted to each data in a set, then

* the new mode = original mode k* the new mean = original mean k* the new median = original median k

2. When a constant numberkis multiplied to each data in a set, then

* the new mode = kx original mode.

* the new mean = kx original mean.

* the new median = kx original median.

When two sets of data, i.e. set X and set Y are combined, then

Combined mean =yx NN

yx

++

1. Find the mode, mean and median of the following sets of data.

(a) 9, 5, 3, 3, 7, 13, 9

5

Exercise 7.1.1


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(b) 2, 8, 11, 9, 6, 5, 12, 11

(c) 3, 4, 11, 3, 10, 11, 2, 3, 7

2. Find the mode, mean and median of the following sets of data.

(a

)

Pocket money

(RM),x25 30 35 40 45 50

Number of

students, f2 4 4 6 5 5

fx

(b)

No. of

absentees

No. of

classes, f

0 3

1 8

2 6

3 4

4 3

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5 1

(c) No. ofgoals,x No. ofPlayers, f

3 12

4 10

5 9

6 7

7 5

(d)Score,x

No. of

pupils,f

8 4

9 8

12 11

15 10

20 5

21 2

3. Determine the modal classby drawing a histogram, hence estimate the

mode for each set of group data below.

(a)Height /

cm

No. of

pupils /fLB UB

Modal class =

Mode =

141 145 7

146 150 9

151 155 16

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156 160 6

161 165 2

(b)Marks

No.. of

pupils /f

Modal class =

Mode =20 29 2

30 39 4

40 49 5

50 59 10

60 69 6

70 79 3

(c)

Mass /

kg

No. of

pupils /f

Modal class =

Mode =.30 39 3

40 49 8

50 59 12

60 69 15

70 79 10

4. Find the mean of each grouped data of the following.

(a)

Height /

cm

No. of

pupils , f

Mid Point ,

x / cmfx

141 145 7

146 150 9

151 155 16

156 160 6

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161 165 2

(b)Marks

Number of

pupils

20 -29 2

30 39 4

40 49 5

50 59 10

60 - 69 6

70 - 79 3

(c)

Mass /

kg

No. of

pupils, f

30 39 8

40 49 10

50 59 7

60 69 15

70 - 79 10

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(d) The table below shows the duration of telephone calls received in an office

on a certain day for 40 calls. Determine the mean of the duration of calls.

Duration

of Calls /

min

No. ofCalls, f

1 2 2

3 4 4

5 6 5

7 8 10

9 10 6

5. For each of the following sets of data, without drawing an ogive, calculate

the median of the set of data.

(a)

Height /

cm

Number of

pupils, fF LB

141 145 7

146 150 9

151 155 16

156 160 6

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161 165 2

(b)Marks

No. of

Pupils, fF LB

20 29 2

30 39 4

40 49 5

50 59 10

60 69 6

70 - 799 3

(c)

Mass /

kg

Number of

pupils, f

30 39 8

40 49 10

50 59 8

60 69 14

70 - 79 10

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(d) The table below shows the duration of telephone calls received in an office

on a certain day for 40 calls. Without drawing an ogive, determine the

median of the duration of calls.

Duration of

Calls / min

Number

of Calls, f

2 3 9

4 5 12

6 7 10

8 9 7

10 11 2

7.2 OGIVE

An ogive is a statistical graph which is drawn of cumulativefrequency of a set of grouped data against its frequency class of upper

boundary.

An ogive can be used to estimate the median, m, first quartile, Q1and third quartile, Q3 of the grouped data.

Cumulative frequency, F

4

N3N = Sum of frequency

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2

NQ1 = First quartile

m = Median

4

NQ3 = Third quartile

Upper boundary, UB

Q1 m Q3

To draw an ogive, a Cumulative Frequency & Upper Boundarytable has to be built.

A class with zero frequency and its upper boundary also need to becreated.

Example:

A graph is then plotted with its cumulative frequency against upperboundary to give an ogive.

1. The table below shows the duration of telephone calls received in an office

on a certain day for 40 calls. Draw an ogive, hence determine the median,

m, first quartile, Q1, and third quartile, Q3of the duration of calls.

Duration of

Calls / min

Number

of Calls, fF LB

2 3 9

4 5 12

6 7 10

8 9 7

Mass / kg Frequency, fCumulative

frequency, F

Upper

boundary, UB

20 29 0 0 29.5

30 39 8 8 39.5

40 49 10 18 49.550 59 8 26 59.5

60 69 14 40 69.5

70 79 10 50 79.5

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Exercise 7.2.1


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10 11 2

2. The table below shows marks scored by 30 pupils in a test. Draw an ogive,

hence determine themedian, m, first quartile, Q1, andthird quartile, Q3of

the test.

MarksNumber of

pupils

20 29 2

30 39 4

40 49 5

50 59 10

60 69 6

70 79 3

1. (a) The mode, mean and median of a set of numbers are 6, 8.5 and 7.8

respectively. Determine the new mode, mean and median if each of the

numbers in the set is;

(i) added by 3 and then divided by 2.

(ii) subtracted by 5 and then multiplied by 4.

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Exercise 7.2.2 - Effects of uniform chan es in a set of data:


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(b) The mode, mean and median of a set of data are 32.5, 30 and 31.5

respectively. Find the new mode, mean and median if each value in the

data is;

(i) added by 3 and then multiplied by .,(ii) subtracted by 1.2.

(c) A set of data with 6 numbers has a mean of 21. When a new number is

added to the set, the mean becomes 20. Find the value of the number

added.

7.3 MEASURE OF DISPERSION

Ungroup DataUngroup Data

(in a Frequency Table)Group Data

Range = largest value of

data smallest

value of data.

Range = largest value of

data smallest

value of data.

Range = midpoint of the

highest class

midpoint of the

lowest class.

Inter quartile range

= Q3 - Q1


= Q3 - Q1


= Q3 - Q1

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Variance,

2 =N

x2

-

_

x2

where;

x2

= sum of square of

the values of

data.

N = number of

values of data.

x = mean

Variance,

2 =

f

fx2

_

x2

where;f = frequency.

x = value of data.

x = mean

Variance,

2 =

f

fx2

_

x2

where;f = frequency.

x = class midpoint.

x = mean

Standard deviation,

=_

2

2

xNx

Standard deviation,

=_

2

2

xf

fx

Standard deviation,

=_

2

2

xf

fx

Effects of uniform changes in a set of data on the range, inter quartile

range, variance and standard deviation.

1. When a constant numberkis added or subtracted to each data in a set, then

* the new range, interquartile range, variance and standard deviation =

original range range, interquartile range, variance and standard deviation

respectively.

2. When a constant numberkis multiplied to each data in a set, then

* the new range = kx original range.

* the new interquartile range = kx original interquartile range..

* the new variance = k2 x original varaince.

* the new standard deviation = kx original standard deviation.

When two sets of data, i.e. set X and set Y are combined, then;

Combined standard deviation =

2

yxyx

22

NNyx

N Nyx

+++ +

1. Find therange and inter quartile range of each set of the following data.

(a) 46, 35, 41, 40, 32, 38, 44, 40 (b) 17, 4, 6, 10, 12, 12

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Exercise 7.3.1


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(c ) 22, 20, 25, 19, 24 (b) 3, 12, 8, 4, 10, 6, 7

2. Find the range and inter quartile range of each set of the following data.

(a

)

Score

x

No. of

Pupils, f

1 32 6

3 12

4 20

5 18

6 11

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(b)No. of

Book,x

No. of

Pupils, f

0 10

1 14

2 20

3 26

4 18

5 12

(c) Mass,x /

kg

No. of

pupils, f

50 2

51 3

52 10

53 20

54 8

55 7

(d)No. of

Children

No. ofFamily,

f

0 1

1 2

2 8

3 2

4 1

5 1

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3. The table below shows the incomes of 40 workers in a factory.

Income /

RM

No. of

workers, f

301 400 5

401 500 9

501 600 12

601 700 8

701 800 6

(a) Find the range of incomes of the workers.

(b) Calculate the first quartile, Q1,, the third quartile, Q3 and the inter

quartile range.

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(c) Draw anogive, hence determine thefirst quartile, Q1,,third quartile,Q3and the inter quartilerangefrom the ogive.

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4. The table below shows the number of chicken sold over a period of 60 days.

No. of

chickens

No.

of days, f

11 15 11

16 20 16

21 25 19

26 30 8

31 - 35 6

(a) Find the range of incomes of the workers.

(b) Calculate the first quartile, Q1,, the third quartile, Q3 and the interquartile range.

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(c) Draw anogive, hence determine thefirst quartile, Q1,,third quartile,Q3and the inter quartile rangefrom the ogive.

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1. Find the mean,variance and standard deviation of each set of the following

data.

(a) 9, 5, 3, 3, 7, 13, 9

(b) 2, 8, 11, 9, 6, 5, 12, 11

(c) 3, 4, 11, 3, 10, 11, 2, 3, 7

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Exercise 7.3.2


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2. Find the mean, variance and standard deviation of each of the following

data.

(a

)

Score,

x

No. of

pupils, ffx fx2

1 3

2 6

3 12

4 20

5 18

6 11

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(b)No. of

Book,x

No. of

Pupils,f

0 10

1 14

2 20

3 26

4 18

5 12

(c) Mass /

kg

No. of

pupils

50 2

51 3

52 10

53 20

54 8

55 7

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(d)

No.

of

children

No. of

family

0 1

1 2

2 8

3 2

4 1

5 1

1. The table below shows the duration of telephone calls received in an office

on a certain day for 40 calls. Find the mean, variance and standard

deviationof the duration of calls.

Duration of

Calls / min

Number

of Calls, f

Midpoint

,

x

fx fx2

2 3 9

4 5 12

6 7 10

8 9 7

10 11 2

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Exercise 7.3.3


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2. The table below shows marks scored by 30 pupils in a test. Find the mean,varianceandstandard deviation of the test.

MarksNumber of

pupils, f

20 29 2

30 39 4

40 49 550 59 10

60 - 69 6

70 - 79 3

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3. The table below shows the lengths of 60 mature long beans in a field study.Find the mean, varianceandstandard deviationof the lengths of the beans.

Length / cmNumber

of Beans

10 14 8

15 19 15

20 24 19

25 29 13

30 34 5

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1. The range and the variance of a set of data are 12 and 13 respectively. Each

value in the set of data is multiplied by 3 and then subtracted by 5. Find

(a) the new range,

(b) the new variance

2. A set of data has a range of 30, an inter quartile range of 5 and a standard

deviation of 8. Each value in the set of the data is divided by 4 and then

added by 3. Find

(a) the new range,

(b) the new inter quartile range,(c) the new standard deviation.

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Exercise 7.3.4


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3. Determine the range, the inter quartile range and the variance of the set ofdata, 3, 5, 6, 8 11, 13. What will be the range, the inter quartile range and the

variance when values of the data is changed to the following

(a) 4, 6, 7, 9, 12, 14.

(b) 9, 15, 18, 24, 33, 39.

(c) 1.5, 2.5, 3.0, 4.0, 5.5, 6.5.

(d) 2.5, 4.5, 5.5, 7.5, 11.5, 13.5

1. Given the mode and the mean of the following set of data, 9,p, 14, q, 33, q

are 33 and 20 respectively. Determine the values ofp and q.

2. The median of the set data 4, 5, 6, 8, k, 9, is 7. Determine the value ofk.

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Exercise 7.4: Problem Solvin I


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3. A set of data has seven numbers. Its mean is 9. If a numberp is added to the

set, the new mean is 12. What is the possible value ofp?

4. A set of datax1, x2, x3, x4,x5 has a mean of 10 and a variance of 4. A value ofx6 is added to the set of data, the mean remains unchanged. Determine

(a) the value ofx6,

(b) the variance of the new set of data.

5. A set of data consists of 6 numbers. The sum of the numbers is 39 and the

sum of the squares is 271.

(a) Find the mean and variance of the set of data.(b) If a number 5 is taken out from the set of data, find the new mean and

standard deviation of the new data.

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6. The mean and variance of two sets of data are as following;

Set X:x1,x2,x3,x4,x5,x6,x7; mean = 11, variance =16.

Set Y: y1,y2,y3,y4,y5; mean =12, variance = 9.

Find the mean and the variance when the two set of data are combined into

one.

1. The diagram below is a histogram which represents the distribution of the

marks obtained by 40 pupils in a test.

Number of Pupils

32

SPM Papers (2003 2010)

14

12

10

8

6

4

2

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Marks

0.5 10.5 20.5 30.5 40.5 50.5

(a) Without using an ogive, calculate the median mark. [3 marks]

(b) Calculate the standard deviation of the distribution. [4 marks]

(SPM 2005/SectionA/Paper2)

2. A set of data consists of 10 numbers. The sum of the numbers is 150 and the

sum of the squares of the numbers is 2472.

(a) Find the mean and variance of the 10 numbers. [3 marks]

(b) Another number is added to the set of data and the mean is increased by

1. Find

(i) the value of this number,

(ii) the standard deviation of the set of 11 numbers. [4 marks]

(SPM 2004/SectionA/Paper2)

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3. A set of examination marksx1,x2,x3,x4,x5,x6 has a mean of 5 and a standard

deviation of 1.5.

(a) Find

(i) the sum of the marks, x,(ii) the sum of the squares of the marks, x2. [3 marks]

(b) Each mark is multiplied by 2 then 3 is added to it. Find, for the new set

of marks,

the mean,

the variance. [4 marks]

(SPM 2003/Section A/Paper2)

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4. The positive integers consists of 2, 5 and m. The variance for this set of

integers is 14. Find the value ofm. [4 marks]

(SPM 2006/Paper1)

Answer:

5. A set of data consists of five numbers. The sum of the numbers is 60 and the

sum of the squares of the numbers is 800.

Find, for the five numbers

(a) the mean,

(b) the standard deviation. [3 marks]

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8. A set of seven numbers has a mean of 9.

(a) Find x.

(b) When a numberkis added to this set, the new mean is 8.5.

Find the value ofk. [3 marks] (SPM 2008/Paper1)

Answer:

9. A set of 12 numbers,x1,x2,.... x12, has a variance of40 and it is given that

x2 = 1080. Find

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(a) the mean,_

x. [3 marks]

(b) the value of x. SPM2009/Paper1

Answer:

10. Table 1 shows the frequency distribution of the scores of a group of pupils in

a game.

Score Number of pupils10 19 1

20 29 2

30 39 8

40 49 12

50 59 k

60 69 1

Table 1

(a) It is given that the median score of the distribution is 42.

Calculate the value ofk. [3 marks]

(b) Use the graph paper provided by the invigilator to answer this question.

Using a scale of 2 cm to 10 cm scores on the horizontal axis and 2 cm to

2 pupils on the vertical axis, draw a histogram to represent the frequency

distribution of the scores.

Find the mode score. [4 marks]

(c) What is the mode score if the score of each pupil is increased by 5?

[1 mark]

(SPM2006/Section A/Paper 2)

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11. A set of data consists of 2, 3, 3, 4, 5, 7 and 9. Determine the interquartile

range of the data [3 marks]

SPM2010/Paper1

Answer:

12. Table 6 shows the frequency distribution of the marks of a group of students

Score Number of pupils

1 10 5

11 20 821 30 20

31 40 10

41 50 7

Table 6

(a) Use the graph paper to answer this question.

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Using a scale of 2 cm to 10 cm scores on the horizontal axis and 2 cm to

2 pupils on the vertical axis, draw a histogram to represent the

frequency distribution of the scores.

Find the mode score. [4 marks]

(b) Calculate the standard deviation of the marks. [4 marks]

(SPM2010/Section A/Paper2)

Documents

Statistics Module Form 4