Maths File

8/13/2019 Maths File

1/17

1. Presentation of discrete data through frequency table, and different types of charts

39 3 84 94 27 29 49 28 95 68

78 57 52 87 71 68 23 86 68 90

7 20 76 91 11 15 95 33 7 81

82 28 19 90 11 79 28 15 81 5

81 74 86 63 77 67 99 55 55 51

76 16 75 97 85 53 12 7 8 23

32 72 78 12 97 98 79 22 67 36

96 16 66 38 70 35 22 93 17 40

24 58 57 9 84 10 46 56 70 26

43 39 75 22 50 60 35 46 1 39

99 89 87 50 81 50 94 33 48 57

33 46 81 49 78 50 65 30 61 55

Solution :

Command Used =frequency(range ,bin range)

Classes Frequency

0-10 9

10-20 11

20-30 13

30-40 1240-50 11

50-60 12

60-70 11

70-80 13

80-90 16

90-100 12


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BAR CHART

COLUMN CHART

0

2

4

6

8

10

12

14

16

18

0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80 80-90 90-100

0 5 10 15 20

0-10

10-20

20-30

30-40

40-50

50-60

60-70

70-80

80-90

90-100


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PIE CHART

2. Presentation of data through Multiple line chart, Frequency polygon, Ogive

Data For Multiple line chart

Class Freq(1) Freq(2)

0-10 1 10

10-20 2 9

20-30 3 8

30-40 4 7

40-50 5 6

50-60 6 5

60-70 7 4

70-80 8 3

80-90 9 2

90-100 10 1

Solution:

0-10

10-20

20-30

30-40

40-50

50-60

60-70

70-80

80-90

90-100


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Multiple line chart

DATA For Frequency Polygon & Ogive

Class 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80 80-90 90-

Freq 29 79 54 19 35 96 95 65 69 3

Frequency Polygon

Ogive

0

2

4

6

8

10

12

Series1

Series2

0

20

40

60

80

100

120

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105

0

100

200

300

400

500

600

700

0 10 20 30 40 50 60 70 80 90 100


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3. Presentation bivariate of data through Scatter-Plot diagram

X 1 2 3 4 5 6 7 8 9

Y 55 89 15 19 25 97 43 99 64

Solution:

0

20

40

60

80

100

120

0 2 4 6 8 10 12

Series1


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4. Computation of measures of central tendency (Arithmetic mean, median, mode, harmonic

mean and geometric mean).

5 76 27 29 26 60 19 33 768 82 68 71 74 57 91 27 16

40 88 91 57 96 86 97 3 39

29 72 24 30 27 72 62 77 68

37 97 7 52 76 44 33 27 37

30 43 55 62 27 29 17 82 26

41 89 26 61 43 42 30 11 49

97 54 50 66 19 70 67 88 92

90 10 27 38 73 53 89 97 77

32 94 93 60 27 55 74 34 35

46 49 64 6 35 18 27 39 1121 67 43 3 4 91 53 47 50

Solution:

To Calculate Formula Excel Formula Answer

Total Observation Number of Observation (N) =count(range) 108

Arithmetic Mean1 + 2 ++

=average(range) 50.101

Median 54 + 552 =median(range) 48Mode Observation with Highest frequency =mode(range) 27

Geometric Mean 1 .21 =geomean(range) 40.22Harmonic Mean

11 + 12+. . .+ 1 =harmean(range) 26.27


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5. Measures of dispersion (Variance, Standard deviation, mean deviation, range, minimum,

maximum, coefficient of variation).

4 18 80 46 40 52 97 40 37 9259 40 45 5 51 55 53 20 98 75

28 34 21 66 65 41 77 59 98 90

75 99 95 8 97 21 41 68 54 34

42 82 24 42 79 7 75 24 35 19

58 57 15 52 30 84 48 16 45 55

8 18 75 30 16 64 44 82 47 3

95 76 24 14 72 98 90 46 3 63

49 74 13 48 53 61 15 51 77 69

41 57 44 8 23 65 81 40 5 82

76 77 66 71 44 12 53 33 59 4878 77 45 47 48 39 38 39 54 43

Solution:

To Calculate Formula Excel Formula Answer

Total Observation (N) Number of Observation =count(range) 120

Arithmetic Mean () 1 + 2 ++

=average(range) 50.31667

Variance1 2=1 =varp(range) 668.7664

Standard Deviation () 1 2=1 =stdevp(range) 25.86052Mean Deviation

1 | |=1 Create table | |=Sum(range)/Cell(N) 21.24389Coefficient Variance 100 = (Cell()/Cell())*100 51.39553

Maximum Maximum Observation =max(range)99

Minimum Minimum Observation =min(range) 3


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Range(Maximum Observation) - (Minimum

Observation)=max(range) -min(range) 96

Table | |46.31667 32.31667 29.68333 4.316667 10.31667 1.683333 46.68333 10.31667 13.31667 41.683

8.683333 10.31667 5.316667 45.31667 0.683333 4.683333 2.683333 30.31667 47.68333 24.68322.31667 16.31667 29.31667 15.68333 14.68333 9.316667 26.68333 8.683333 47.68333 39.683

24.68333 48.68333 44.68333 42.31667 46.68333 29.31667 9.316667 17.68333 3.683333 16.316

8.316667 31.68333 26.31667 8.316667 28.68333 43.31667 24.68333 26.31667 15.31667 31.316

7.683333 6.683333 35.31667 1.683333 20.31667 33.68333 2.316667 34.31667 5.316667 4.6833

42.31667 32.31667 24.68333 20.31667 34.31667 13.68333 6.316667 31.68333 3.316667 47.316

44.68333 25.68333 26.31667 36.31667 21.68333 47.68333 39.68333 4.316667 47.31667 12.683

1.316667 23.68333 37.31667 2.316667 2.683333 10.68333 35.31667 0.683333 26.68333 18.683

9.316667 6.683333 6.316667 42.31667 27.31667 14.68333 30.68333 10.31667 45.31667 31.683

25.68333 26.68333 15.68333 20.68333 6.316667 38.31667 2.683333 17.31667 8.683333 2.3166

27.68333 26.68333 5.316667 3.316667 2.316667 11.31667 12.31667 11.31667 3.683333 7.31666. Calculation of first four raw and central moments through formula-bar

X 1 2 3 4 5 6 7 8 9

Frequency 21 34 45 54 65 44 37 20 18

Solution

M=

1

=1 =1730

350 = 4.9429

x f fx fx^2 fx^3 fx^4 f(x-M) f(x-M)^2 f(x-M)^3 f(x-M

1 21 21 21 21 21 -82.80000 326.4686 -1287.218939 5075.32

2 34 68 136 272 544 -100.05714 294.4539 -866.5356968 2550.09

3 45 135 405 1215 3645 -87.42857 169.8612 -330.0160933 641.174

4 54 216 864 3456 13824 -50.91429 48.0049 -45.26176093 42.6753

5 65 325 1625 8125 40625 3.71429 0.212245 0.01212828 0.00069

6 44 264 1584 9504 57024 46.51429 49.17224 51.98208746 54.9524

7 37 259 1813 12691 88837 76.11429 156.578 322.1032303 662.612

8 20 160 1280 10240 81920 61.14286 186.9224 571.4486297 1747.009 18 162 1458 13122 118098 73.02857 296.2873 1202.080093 4877.01

10 12 120 1200 12000 120000 60.68571 306.8963 1552.018566 7848.77

SUM 350 1730 10386 70646 524538 0.000 1834.857 1170.612 23499.


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Raw Moments

'1=1

=1

=1730350 = 4.9429

'2=1 2=1 =10386350 = 29.67429

'3=1 3=1 =70646350 = 201.8457

'4=1 4=1 =524538350 = 1498.68

Central Moments

1=1 =1 ( ) = 0350= 0

2= 1 =1 ( )2 =1834.857350 = 5.2424493=

1 =1 ( )3 =1170.612350 = 3.3446064144=

1 =1 ( )4=23499.617350 = 67.14176167. Measure of skewness and kurtosis

X 1 2 3 4 5 6 7 8 9 1F 9 16 42 50 72 46 37 15 10

Solution

X f fixi fi(x-M) fi(x-M)^2 fi(x-M)^3 fi(x-M)^4

1 9 9.000 -36.060 144.480 -578.885 2319.398

2 16 32.000 -48.107 144.641 -434.886 1307.558

3 42 126.000 -84.280 169.122 -339.371 681.005

4 50 200.000 -50.333 50.669 -51.007 51.347

5 72 360.000 -0.480 0.003 0.000 0.0006 46 276.000 45.693 45.389 45.086 44.786

7 37 259.000 73.753 147.015 293.050 584.146

8 15 120.000 44.900 134.401 402.306 1204.236

9 10 90.000 39.933 159.467 636.805 2542.976

10 3 30.000 14.980 74.800 373.502 1865.020

SUM 300 1502 0.000 1069.987 346.600 10600.472


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M=1 =1 =1502300 = 5.007

Central Moments

1=1 =1 ( ) = 0300= 0

2=1 =1 ( )2 =1069.987300 = 3.567

3=1 =1 ( )3 =346.600300 = 1.155

4=1 =1 ( )4=10600.472300 = 35.335

Skewness and Kurtosis

1 = 3223 =1.15523.5673 =0.0292 =

422 =35.3353.5672 =2.778=1=0.029 = 0.172= 2- 3= 2.778 -3 = -0.2228. Fitting of binomial distribution and graphical representation of probabilities

x 0 1 2 3 4 5 6 7 8 9

f 12 19 27 32 45 38 32 24 15

Solution

x f fx

0 12 0

1 19 19

2 27 543 32 96

4 45 180

5 38 190

6 32 192

7 24 168

8 15 120


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Mean =1 =1 =1073250 = 4.292

Since: n.p = mean and where n = 9

=> n.p = 4.292

=> p = 0.476889

Using function =BINOMDIST(number_s ,trials ,probability_s ,cumulative) create column P(X=x) and

multiply N with it to create another column N*P(X=x)

x P(X=x) N*P(X=x)

0 0.002933226 1

1 0.024066408 6

2 0.087759577 22

3 0.186678613 474 0.25527547 64

5 0.232719269 58

6 0.141437426 35

7 0.055259992 14

8 0.012594301 3

9 0.001275718 0

SUM 1 250

Graphical Representation of expected Probability of Binomial distribution

9 6 54

SUM 250 1073


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9. Fitting of Poisson distribution and graphical representation of probabilities

x 0 1 2 3 4 5 6 7 8 9

f 12 27 52 32 25 20 15 9 7

Solution

x f fx0 12 0

1 27 27

2 52 104

3 32 96

4 25 100

5 20 100

6 15 90

7 9 63

8 7 56

9 1 9SUM 200 645

Mean =1 =1 =645200= 3.225

Since: = mean

0.000

0.050

0.100

0.150

0.200

0.250

0.300

0 1 2 3 4 5 6 7 8 9


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=> = 4.292

Using function =POISSON(x, mean, cumulative) create column P(X=x) and multiply N with it to create

another column N*P(X=x)

x P(X=x) N*P(X=x)

0 0.039756 7.951156

1 0.128212 25.64248

2 0.206742 41.3485

3 0.222248 44.44963

4 0.179188 35.83752

5 0.115576 23.1152

6 0.062122 12.42442

7 0.028621 5.724108

8 0.011538 2.307531

9 0.004134 0.826865SUM 1 200

Graphical Representation of expected Probability of Poisson distribution

10. Fitting of Normal distribution and graphical representation of probabilities (expected normal

frequencies)

Classes 30-40 40-50 50-60 60-70 70-80 80-90 90-100

0

0.05

0.1

0.15

0.2

0.25

0 1 2 3 4 5 6 7 8 9


14/17

f 41 68 92 141 136 71 35

Solution

Class x f fx fx^2

30-40 35 41 1435 50225

40-50 45 68 3060 137700

50-60 55 92 5060 278300

60-70 65 141 9165 595725

70-80 75 136 10200 765000

80-90 85 71 6035 512975

90-100 95 35 3325 315875

SUM 584 38280 2655800

Mean = = 1 =1 =38280584 = 65.54795Standard Deviation = 1 2=1 2 = 2655800584 65.547952 = 15.84518=> = 65.54795 ; = 15.84518Using function =NORMDIST(x ,mean ,standard_dev ,cumulative) create column P(X


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11. Calculation of cumulative distribution function for Normal distribution

Classes 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80 80-90 90-

f 31 25 45 78 85 65 51 24 12

Solution

Class x f fx fx^2

0-10 5 31 155 77510-20 15 25 375 5625

20-30 25 45 1125 28125

30-40 35 78 2730 95550

40-50 45 85 3825 172125

50-60 55 65 3575 196625

60-70 65 51 3315 215475

70-80 75 24 1800 135000

80-90 85 12 1020 86700

90-100 95 5 475 45125

SUM 421 18395 981125

Mean = = 1 =1 =18395421 = 43.69359Standard Deviation = 1 2=1 2 = 981125421 43.693592 = 20.52641

0

0.05

0.1

0.15

0.2

0.25

0.3

30-40 40-50 50-60 60-70 70-80 80-90 90-100


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=> = 43.69359 ; = 20.52641Using function =NORMDIST(x ,mean ,standard_dev ,cumulative) create column P(X


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Mean = = 1 =1 =3900100 = 39Since: Mean =

=> =

= 0.025641

Using function =EXPONDIST(x,lambda, cumulative) create column P(X

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