Statistics Version 2012

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AdditionalMathematicsForm 4

Topic: 7

DECISIVE(Version 2012)

by

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


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edmet-nklpunya.blogspot.com

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, 59, 60, 53, 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 8

3 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 15

6 - 7 8

8 - 9 7

Mode = The value which is repeated

the most number of times in a set of

data.

Example:

Mode of the masses of six pupils in

kilogram:; 50, 52, 59, 60, 53, 59.

Mode = 59

Mode = The value of data which

has the highest frequency.

Modal Class = The class with

highest frequency.

Mode is obtained from the highest

bar of a histogram with the

procedure as shown below.

frequency Modal class

mode

Mean,_

x=

N

x

x = sum of all the values ofdata.

N= number of values of data.

Mean,_

x=

f

fx

x = value of data

f= frequency

Mean,_

x=

f

fx

x = class mid-point

f= frequency

2


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56=+

2

5953


Median, m = the value in the middle

position of a set of data after the data

are arranged in ascending order.

Example:

The median of the masses of six

pupils in kilogram:

50, 52, 53, 59, 59, 60.

Median =

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 the median 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.

1. Find the mode, mean and median of the following sets ofungrouped 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 (d) 3, 2, 1, 4, 5, 9

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

(a)Pocket money (RM),x 25 30 35 40 45 50

Total

Number of Students, f 2 4 4 6 5 5

Cumulative frequency, F

fx

3

Exercise 7.1

(c)No. Goals,

xNo. Players,f fx F

3 12

4 10

5 9

6 7

7 5

mode =

mean, x =

median =


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3. Determine (i) the modal class of each of the following grouped data.

(i) Modal class =

(ii) On a graph paper and by using a scale of

2 cm to 5 unit on x-axis and 2 cm to 2 units

on frequency,faxis, draw a histogram of the

grouped data. Hence, from the graph, estimate

the mode of the data.

(b) Marks, x No. Pupils,f fx F

0 3

1 8

2 6

3 4

4 3

5 1

f fx=

(d)Score,x

No. pupils,

ffx F

8 4

9 8

12 11

15 10

20 5

21 2

(a)Height / cm,

xNo.pupils,

fLB UB

141 145 7

146 150 9

151 155 16

156 160 6

161 165 2

4

(e)Marks,x

No. ofSudents,f

fx F

13 6

14 8

15 1216 10

17 5

18 3

mode= mean, x = median= mode= mean, x = median=

mode= mean, x = median= mode= mean, x = median=


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(i) Modal class =

(ii) On a graph paper and by using a scale of

2 cm to 5 unit on x-axis and 2 cm to 2 units

on frequency,faxis, draw a histogram of the

grouped data. Hence, from the graph, estimatethe mode of the data.

(a) Marks,x No.pupils,f LB UB

20 29 2

30 39 5

40 49 7

50 59 10

60 69 6

5


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

(a)Height / cm,

No. pupils,

fMid-point, x fx

141 145 7

146 150 9

151 155 16

156 160 6

161 165 2

f = fx=

(b)Marks

Number of

pupils, fx fx

20 -29 2

30 39 4

40 49 5

50 59 10

60 69 6

70 79 3

f = fx=

(c)Mass / kg

No. of

pupils, f

30 39 8

40 49 10

50 59 7

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60 69 15

70 79 10

(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 / minutes

No. of

Calls, f

1 3 2

4 6 4

7 10 5

11 13 10

14 17 6

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

(a) Height / cm,Number of

pupils, f

141 145 7

146 150 9

151 155 16

156 160 6

161 165 2

(b) Marks,xNo. of

pupils, f

20 29 2

30 39 4

40 49 5

50 59 10

60 69 670 799 3

(c)Mass /

kg

Number of

pupils

30 39 8

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40 49 10

50 59 8

60 69 14

70 79 10

(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

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 cumulative frequency of a set of grouped data against its

frequency class of upper boundary.

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

Cumulative frequency, F

4

N3 N = Sum of frequency

2

NQ1 = First quartile

m = Median

4

NQ3 = Third quartile

Upper boundary

Q1m Q3

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

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

Example:

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

1. The table below shows marks scored by 30 pupils in a test. Draw an ogive, hence determine the

median, m, first quartile, Q1, andthird quartile, Q3 of the test.

MarksNumber of

pupils, fF UB

20 29 2

30 39 4

40 49 5

50 59 10

60 69 6

70 79 3

Mass / kgFrequency,

f

Cumulative frequency,

FUpper boundary

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

9

Exercise 7.2


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1. 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.

2. 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.

3. 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.

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Exercise 7.3 Effect of Uniform Chances in a Set of Data on the Mode, Mean and Median


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7.3 MEASURE OF DISPERSION

Ungrouped DataUngrouped Data

(in a Frequency Table)Grouped Data

Range = largest value

smallest value of data.

Range = largest value

smallest value of data.

Range = midpoint of the higest

class midpoint of

the lowest class.

Inter quartile range

= Q3 Q1


= Q3 Q1


= Q3 Q1

Variance,

2

= N

x2

_

x

2

where;

2x

= sum of square of thevalues of data.

N = number of value of data

x = mean

Variance,

2 = ffx

2

_

x

2

where;

f = frequency.

x = value of data.

x = mean

Variance,

2

=

f

fx2

_

x

2

where;

f = frequency.

x = class midpoint.

x = mean

Standard deviation,

=_

2

2

xN

x

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 variance.

* the new standard deviation = kx original standard deviation.

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

11

Exercise 7.3 a


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

2. Find the rangeand inter quarter range of each of the following data.

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

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

(a)

ScoreNo. of

Pupils, fF

1 3

2 6

3 12

4 20

5 18

6 11

(b)No. of

book

No. of

pupils

0 10

1 14

2 20

3 26

4 18

5 12

12


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(c) Mass /kg

No. ofpupils

50 2

51 3

52 1053 20

54 8

55 7

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

No. of chickens,

x

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 thefirst quartile, Q1,, the third quartile, Q3 and the inter quartile range.

(c) Draw anogive, hence determine thefirst quartile, Q1,,third quartile,Q3 and the inter quartile rangefrom

the ogive.

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

13

Exercise 7.3 b :


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

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

(a)

Score,

x

No. of

pupils, f

1 3

2 6

3 12

4 20

5 18

6 11

(b)No. of

book

No. of

pupils

0 10

1 14

2 20

3 26

4 18

5 12

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(c) Mass /kg

No. ofpupils

50 2

51 3

52 1053 20

54 8

55 7

(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 ofCalls / min

Numberof Calls

2 3 9

4 5 12

6 7 10

8 9 7

10 11 2

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Exercise 7.3 c :


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

MarksNumber of

pupils

20 29 2

30 39 4

40 49 5

50 59 10

60 - 69 6

70 - 79 3

4. The table below shows the lengths of 60 mature long beans in a field study. Find the mean, variance

andstandard 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

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

16

Exercise 7.3(d) Effect of Uniform Chances in a Set of Data on the Measures of Dispersions


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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.

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 isthe possible value ofp?

4. A set of data x1, x2, x3, x4, x5 has a mean of 10 and a variance of 4. A value of x 6 is added to the set of data, the

mean remains unchanged. Determine

(a) the value of x6,

(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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Exercise 7.4: Problem Solving I


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1. The diagram below is a histogram which represents the distribution of the marks obtained by 40 pupilsin a test.

Number of Pupils

Marks0.5 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/Paper 2)

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

18

Past SPM Papers

14

12

10

8

6

4

2

0


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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/Paper 2)

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,x

2

. [3 marks]

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

(i) the mean,

(ii) the variance. [4 marks]

(SPM 2003/Section A/Paper 2)

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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 of m[4

marks]

(SPM 2006/Paper 1)

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]

(SPM2007/Paper 1)

6. Table 1 shows the cumulative frequency distribution for the scores of 32 students in a competition.

Score < 10 < 20 < 30 < 40 < 50

Number of students 4 10 20 28 32

Table 1

(a) Based on Table 1, copy and complete Table 2.

Score 0 9 10 19 20 29 30 39 40 49

Number of students

Table 2

(b) Without drawing an ogive, find the interquartile range of the distribution.

[5 marks](SPM2007/Section A/Paper 2)

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

Score Number of pupils

10 19 1

20 29 2

30 39 8

40 49 12

50 59 k

60 69 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 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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Statistics Version 2012