Topic 7 Statistics

7/28/2019 Topic 7 Statistics

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TOPIC 7: STATISTICS *bring graph paper & french curve* *paper 2

Mean, Mode & Median of Ungrouped Data

mean,

f

xx mode = most frequent

median = middle

EXAMPLE 1 :Find the mode, median and mean for

3, 4, 0, 2, 2, 1, 5, 6, 7

Answer : Mode = 2 ascending orderMedian = 3 0, 1, 2, 2, 3, 4, 5, 6, 7

Mean ,

f

xx =

9

765432210

= 3.333

EXAMPLE 2 :

Find the mode, median and mean for2, 3, 2, 2, 6, 8, 9, 3, 2, 3

Answer : Mode = 2 ascending order

Median =2

33 2, 2, 2, 2, 3, 3, 3, 6, 8, 9

= 3

Mean ,

f

xx =

10

9863332222

= 4

1. Find the mode, median and mean for4, 5, 0, 2, 2, 1, 5, 5, 3

[Ans : Mode = 5, Median = 3, Mean = 3]2. Find the mode, median and mean for

52, 60, 70, 55, 56, 56

[Ans : Mode = 56, Median = 56, Mean = 58.167]3. If the mean of the set of data 4, k, 5, 6, 5, 6 and 4is 5, where k is a constant, find the mode.

[Ans : Mode = 5]Mean, Mode & Median of Grouped Data

mean,

f

xfx

mode = using histogram

median =c

f

Ff

Lm

2

1

L = lower boundary of the median class

f= total frequency (jumlah kekerapan)F= cumulative frequency before the median class

c = size of class interval(selang kelas)fm = frequency of the median classRange(julat) = highest value - lowest value

First quartile, Q1 = thf4

1 observation

=

1

1

1

41

Q

mQ

Q cf

Ff

L

Third quartile, Q3 = thf4

3 observation

=

3

3

3

4

3

Q

mQ

Q cf

Ff

L

Interquartile range = Q3Q1(julat antara kuartil)

EXAMPLE 3 :

Find the mode, median and mean for the data inthe table below.

Answer : Mode = 3 Median = thf2

1 observation

= 20th observation

= 3

Mean ,

f

xfx =

40

107

= 2.675

Score 0 1 2 3 4 5

Frequency 2 5 8 17 5 3

Score,x 0 1 2 3 4 5

Frequency,f 2 5 8 17 5 3 f 40fx 0 5 16 51 20 15 fx 107

(min) (mod)

(sempadan

bawah)

highest frequency

for class interval

Modal class =

(kuartil pertama)

(kuartil ketiga)


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4. Find the mode, median and mean for the data in

the table below.

[Ans : Mode = 4, Median = 6, Mean = 7.2]

Score 2 4 6 9 12 13

Frequency 1 3 2 1 2 1

5. The table below shows the scores obtained

by number of children in a game. If the meanscore is 27, calculate the value of p.

[Ans : p = 3]

Score Frequency

20 5

30 p

40 2

Mode from Histogram

EXAMPLE 4 :Table 1 shows the frequency distribution of thescores of a group of pupils in a game.

Score Number of pupils

1 - 10 5

11 - 20 8

21 - 30 12

31 - 40 441 - 50 6

51 - 60 5

Table 1(a) Use the graph paper to answer this question.Using a scale of 2 cm to 10 scores on thehorizontal axis and 2 cm to 2 pupils on the vertical

axis, draw a histogram to represent the frequencydistribution of the scores.(b) Find the mode score.

Answer : (a)

Score Number of pupils Midpoint1 - 10 5 5.5

11 - 20 8 15.5

21 - 30 12 25.5

31 - 40 4 35.5

41 - 50 6 45.5

51 - 60 5 55.5

(b) Mode score = 24


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6. Table 2 shows the frequency distribution of the

scores of a group of pupils in a test.


50 - 54 2

55 - 59 3

60 - 64 7

65 - 69 9

70 - 74 4

75 - 79 1Table 2

(a) Use the graph paper to answer this question.Using a scale of 2 cm to 5 scores on the horizontalaxis and 2 cm to 1 pupils on the vertical axis, draw

a histogram to represent the frequency distributionof the scores.

(b) Find the mode score.

[Ans : Mode = 66]

7. Table 3 shows the frequency distribution of the

scores of a group of pupils in a game.


65 - 69 2

70 - 74 8

75 - 79 16

80 - 84 12

85 - 89 6

90 - 94 495 - 99 2

Table 3(a) Use the graph paper to answer this question.

Using a scale of 2 cm to 5 scores on the horizontalaxis and 2 cm to 2 pupils on the vertical axis, draw ahistogram to represent the frequency distribution ofthe scores.(b) Find the mode score.

[Ans : Mode = 78]EXAMPLE 5 :Find the modal class and calculate the median for the data in the table below.

Marks 40 - 45 46 - 51 52 - 57 58 - 63 64 - 69 70 - 75 7681

Frequency 3 6 8 14 10 5 4

Answer :

Marks Frequency Upper Boundary Cumulative Frequency

40 - 45 3 45.5 3

46 - 51 6 51.5 9

52 - 57 8 57.5 17

58 - 63 14 63.5 31

64 - 69 10 69.5 41

70 - 75 5 75.5 46

76 - 81 4 81.5 50

Modal class = 5863 Median =c

f

Ff

Lm

2

1

= thf2

1 observation

= 25th observation

=6

14

17)50(2

1

5.57

= 57.5 + 3.429= 60.929

L = 57.5

f= 50F= 17

c = 63.5 - 57.5 = 6

m = 14


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8. Find the modal class and calculate the median for the data in the table below.

Marks 1 - 15 16 - 30 31 - 45 46 - 60 61 - 75 76 - 90

f 8 11 25 34 16 6

[Ans : Modal class = 46-60, Median = 48.147]9. Find the modal class and calculate the medianfor the data in the table below.

[Ans : Modal class = 10-14, Median = 10.75]

Points 0 - 4 5 - 9 10 - 14 15 - 19

frequency 2 5 8 3

10. The table below shows the marks obtained by34 students in a SPM trial examination forMathematics. Find the modal class and calculate the

median mark.

[Ans : Modal class = 41-60, Median mark = 55.167]

Marks Number of students

01 - 20 2

21 - 40 4

41 - 60 1561 - 80 10

81 - 100 3


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Ogive

EXAMPLE 6 :Table below shows the frequency distribution of the ages of a group of workers in an office.

Age(years) 20 - 24 25 - 29 30 - 34 35 - 39 40 - 44 45 - 49 5054

Number of workers 2 3 7 14 9 4 1

(a) Use the graph paper to answer this question.Using a scale of 2 cm to 5 years on the horizontal axis and 2 cm to 5 workers on the vertical axis, draw an

ogive to represent the frequency distribution of the ages.

(b) From the ogive, find the(i) median ages of the workers.

(ii) interquartile range.

Answer :

Age(years) Number of workers Upper Boundary Cumulative Frequency

< 20 0 19.5 0

20 - 24 2 24.5 2

25 - 29 3 29.5 5

30 - 34 7 34.5 12

35 - 39 14 39.5 26

40 - 44 9 44.5 35

45 - 49 4 49.5 39

5054 1 54.5 40

(a)

(b) (i) Median = thf2

1 observation (ii) Q3 = thf4

3 observation Q1 = thf4

1 observation

= 20th observation = 30th observation = 10th observation= 37.75 = 41.25 = 33.75

Interquartile range = Q3Q1= 41.25 - 33.75= 7.5


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11. Table below shows the frequency distribution of the score of a group of students in a game.

Score 5 - 9 10 - 14 15 - 19 20 - 24 25 - 29 30 - 34 3539

Number of students 2 3 7 12 14 9 3

(a) Use the graph paper to answer this question.Using a scale of 2 cm to 5 score on the horizontal axis and 2 cm to 5 students on the vertical axis, draw an

ogive to represent the frequency distribution of the scores.(b) From the ogive, find the

(i) median scores of the students.(ii) interquartile range.

[Ans : (i) Median scores = 25 (ii) Interquartile range = 9.5]

12. Table below shows the frequency distribution of the score of a group of students in a game.

Score 1 - 10 11 - 20 2130 31 - 40 41 - 50 51 - 60 6170 7180 8190 91100

Number ofstudents

3 5 7 12 15 19 16 10 8 5

(a) Use the graph paper to answer this question.Using a scale of 2 cm to 20 scores on the horizontal axis and 2 cm to 10 students on the vertical axis,draw an ogive to represent the frequency distribution of the scores.(b) From the ogive, find the

(i) median scores of the students.

(ii) interquartile range.

[Ans : (i) Median scores = 54.5 (ii) Interquartile range =2 9]


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Measures of Dispersion for Ungrouped Data

Variance,

22

2

f

x

f

x

Standard deviation, variance

EXAMPLE 7 :22, 20, 26, 25, 28, 30, 38, 35, 45, 40, 32

From the data given above, find the(a) range (d) mean

(b) median (e) variance(c) interquartile range (f) standard deviation

Answer : ascending order20, 22, 25, 26, 28, 30, 32, 35, 38, 40, 45

Q1 Q2 Q3(a) range = 45 20 = 25(b) median = 30(c) interquartile range = Q3Q1 = 3825 = 13

(d) mean,

f

xfx

=11

4540383532302826252220

= 31

(e) variance,

22

2

f

x

f

x

=11

454038353230282625222022222222222

-

2

114540383532302826252220

= 1017 - 312= 56

(f) standard deviation, variance

= 56

= 7.483

13. 5, 1, 2, 3, 4, 6, 3, 8, 2, 5, 9

From the data given above, find the(a) range (d) mean(b) median (e) variance

(c) interquartile range (f) standard deviation

[Ans : (a)Range = 8 (b)Median = 4 (c)Interquartile range =4

(d)Mean = 4.364 (e)Variance = 5.868 (f)Standard deviation = 2.422]


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EXAMPLE 8 :

12, 8, 10, 6, 6, 7, 13, 5, 8, 10, 13, 15From the data given above, find the(a) range (d) mean

(b) median (e) variance(c) interquartile range (f) standard deviation

Answer : ascending order

5, 6, 6, 7, 8, 8, 10, 10, 12, 13, 13, 15

Q1 Q2 Q3(a) range = 15 5 = 10

(b) median =2

108 = 9

(c) interquartile range = Q3Q1

=2

1312 2

76

= 12.56.5

= 6

(d) mean,

f

xf

x

=12

151313121010887665

= 9.417

(e) variance,

22

2

f

x

f

x

=12

151313121010887665222222222222

-

2

12

151313121010887665

= 117.1679.4172

= 28.487

(f) standard deviation, variance

= 487.28 = 5.337

14. 12, 17, 13, 19, 15, 8, 12, 11

From the data given above, find the(a) range (d) mean(b) median (e) variance

(c) interquartile range (f) standard deviation

[Ans : (a)Range = 11 (b)Median = 12.5 (c)Interquartile range =4.5

(d)Mean = 13.375 (e)Variance = 10.734 (f)Standard deviation = 3.276]


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Measures of Dispersion for Grouped Data

Variance,

22

2

f

fx

f

fx

Standard deviation, variance

EXAMPLE 9 :Calculate the interquartile range, mean, variance and the standard deviation for the data.

Class interval 6 - 8 9 - 11 12 - 14 15 - 17 18 - 20

Frequency 3 8 14 11 4

Answer :

Classinterval

Frequency UpperBoundary

CumulativeFrequency

< 6 0 5.5 06 - 8 3 8.5 39 - 11 8 11.5 11

12 - 14 14 14.5 25

15 - 17 11 17.5 3618 - 20 4 20.5 40

Q3 = thf4

3 observation Q1 = thf4

1 observation

= 30th observation = 10th observation

=3

3

3

4

3

Q

mQ

Q cf

Ff

L

=

1

1

1

4

1

Q

mQ

Q cf

Ff

L

=3

11

25)40(4

3

5.14

=

38

3)40(4

1

5.8

= 14.5 + 1.364 = 8.5 + 2.625

= 15.864 = 11.125

Interquartile range = Q3Q1= 15.86411.125= 4.739

Classinterval

Frequency,f

Midpoint,x

fx fx2

6 - 8 3 7 21 1479 - 11 8 10 80 800

12 - 14 14 13 182 236615 - 17 11 16 176 281618 - 20 4 19 76 1444

f= 40 xf = 2xf =535 7573

L = 14.5

f= 40F= 25c = 17.5 - 14.5 = 3

m = 11

L = 8.5

f= 40F= 3c = 3

m = 8

Mean,

f

xfx =

40

535= 13.375

Variance,22

2

f

fx

f

fx

=40

7573 -

2

40

535

= 189.325178.891

= 10.434

Standard deviation, variance = 434.10 = 3.23


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15. Calculate the interquartile range, mean, variance and the standard deviation for the data.

Class interval 1 - 5 6 - 10 11 - 15 16 - 20 21 - 25

Frequency 3 13 23 9 2

[Ans : Mean = 12.4, 2 = 20.64, = 4.543]


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Class interval 1 - 3 4 - 6 7 - 9 10 - 12 13 - 15

Frequency 1 3 8 6 2

[Ans : Mean = 8.75, 2 = 8.888, =2.981]


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Class interval 10 - 19 20 - 29 30 - 39 40 - 49

Frequency 5 7 5 3

[Ans : Mean = 27.5, 2 = 101, = 10.05]

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Topic 7 Statistics