statistic concepts

7/29/2019 Statistic Concepts

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Table of ContentsAbout the Data ............................................................................................................................................................. 2

Frequency Distribution ................................................................................................................................................. 4

Histogram ..................................................................................................................................................................... 6

Frequency Polygon ....................................................................................................................................................... 7

OGIVE(LESS THAN) ........................................................................................................................................................ 8

OGIVE (MORE THAN) .................................................................................................................................................... 9

Pareto Charts .............................................................................................................................................................. 10

Pie Charts .................................................................................................................................................................... 11

Stem and Leaf Charts and Mode ................................................................................................................................ 12

Scatter Plots ............................................................................................................................................................... 13

Geometric Mean ........................................................................................................................................................ 15

Arithmetic Mean ........................................................................................................................................................ 17

Weighted Arithmetic Mean ........................................................................................................................................ 19

Median ....................................................................................................................................................................... 20

Quartiles ..................................................................................................................................................................... 22

Range .......................................................................................................................................................................... 24

Mean Absolute Deviation ........................................................................................................................................... 25

Variance ...................................................................................................................................................................... 25

Standard Deviation ..................................................................................................................................................... 26

Coefficient of Variation .............................................................................................................................................. 26

Skewness .................................................................................................................................................................... 28

Kurtosis ....................................................................................................................................................................... 28

Percentile ................................................................................................................................................................... 29

Decile .......................................................................................................................................................................... 30


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About the Data

TAX REVENUE OF CENTRE AND THE STATES: 1950-51 to 2009-10 (Rs. Crore)

Total Tax Revenue(A+C) Central Taxes Gross(A) States' own Taxes( C)

Year Direct Indirect Total Direct Indirect Total Direct Indirect Total

1950-51 231 396 627 176 229 405 55 167 222

1951-52 244 495 739 190 322 512 54 173 227

1952-53 252 426 678 186 259 445 66 167 233

1953-54 242 430 672 166 254 420 76 176 252

1954-55 240 480 720 161 294 455 79 186 265

1955-56 259 509 768 171 314 485 88 195 283

1956-57 288 602 890 194 376 570 94 226 320

1957-58 327 718 1045 230 462 692 97 256 353

1958-59 344 745 1089 238 463 701 106 282 388

1959-60 378 838 1216 269 525 794 109 313 422

1960-61 402 948 1350 292 603 895 110 345 455

1961-62 449 1094 1543 337 717 1054 112 377 489

1962-63 560 1305 1865 423 862 1285 137 443 580

1963-64 693 1632 2325 550 1084 1634 143 548 6911964-65 743 1856 2599 600 1221 1821 143 635 778

1965-66 734 2188 2922 598 1463 2061 136 725 861

1966-67 767 2494 3261 657 1650 2307 110 844 954

1967-68 780 2676 3456 655 1698 2353 125 978 1103

1968-69 840 2919 3759 698 1812 2510 142 1107 1249

1969-70 963 3237 4200 826 1996 2822 137 1241 1378

1970-71 1009 3743 4752 869 2337 3206 140 1406 1546

1971-72 1171 4404 5575 1047 2826 3873 124 1578 1702

1972-73 1346 5090 6436 1233 3272 4505 113 1818 1931

1973-74 1552 5837 7389 1375 3695 5070 177 2142 2319

1974-75 1834 7389 9223 1650 4672 6322 184 2717 2901

1975-76 2493 8689 11182 2205 5404 7609 288 3285 3573

1976-77 2585 9747 12332 2328 5943 8271 257 3804 4061

1977-78 2680 10557 13237 2405 6453 8858 275 4104 4379

1978-79 2851 12677 15528 2528 7997 10525 323 4680 5003

1979-80 3096 14587 17683 2818 9156 11974 278 5431 5709

1980-81 3268 16576 19844 2997 10182 13179 271 6394 6665

1981-82 4133 20009 24142 3786 12061 15847 347 7948 8295

1982-83 4492 22750 27242 4139 13557 17696 353 9193 95461983-84 4907 26618 31525 4498 16223 20721 409 10395 10804

1984-85 5330 30484 35814 4798 18673 23471 532 11811 12343

1985-86 6252 37015 43267 5620 23050 28670 632 13965 14597

1986-87 6889 42650 49539 6236 26602 32838 653 16048 16701

1987-88 7483 49493 56976 6752 30913 37665 731 18580 19311

1988-89 9758 57168 66926 8830 35644 44474 928 21524 22452

1989-90 11165 66528 77693 10003 41633 51636 1162 24895 26057

1990-91 12260 75462 87722 11030 46547 57577 1230 28915 30145

1991-92 16657 86541 103198 15353 52008 67361 1304 34533 35837

1992-93 19387 94779 114166 18140 56496 74636 1247 38283 39530

1993-94 21713 100248 121961 20299 55443 75742 1414 44805 46219

1994-95 28878 118971 147849 26973 65324 92297 1905 53647 55552


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1995-96 35777 139482 175259 33564 77660 111224 2213 61822 64035

1996-97 41061 159995 201056 38898 90864 129762 2163 69131 71294

1997-98 50538 170121 220659 48282 90938 139220 2256 79183 81439

1998-99 49119 183898 233017 46601 97196 143797 2518 86702 89220

1999-2000 60864 213719 274583 57960 113792 171752 2904 99927 102831

2000-01 71762 233558 305322 68305 120298 188605 3457 113260 116717

2001-02 73109 241426 314535 69198 117862 187060 3911 123564 127475

2002-03 87365 268912 356277 83363 132542 215905 4002 136370 140372

2003-04 109546 304538 414084 105091 149257 254348 4455 155281 159736

2004-05 137093 357277 494370 132183 172774 304957 4910 184503 189413

2005-06 167635 420053 587688 162337 203814 366151 5298 216239 221537

2006-07 231376 505331 736708 225045 248467 473513 6331 256864 263195

2007-08 318839 551490 870329 312220 280927 593147 6619 270563 277182

2008-09(R.E.) 346390 601270 947660 338906 289043 627949 7484 312227 319711

2009-10(B.E.) 372061 624823 996885 363956 277123 641080 8105 347700 355805

Data collected from:http://www.finmin.nic.in/reports/IPFStat200910.pdf

Importance of data:The table shows the income of India of last sixty years and its pattern of

growth.

Type of data: Continuous numerical type data

Rs. Crore

Raw data in arranged array

627

672

678

720

739

768890

1045

1089

1216

1350

1543

1865

2325

2599

2922

3261

3456

3759

4200

4752

5575

6436

7389

9223

Contd..
http://www.finmin.nic.in/reports/IPFStat200910.pdfhttp://www.finmin.nic.in/reports/IPFStat200910.pdfhttp://www.finmin.nic.in/reports/IPFStat200910.pdf


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Frequency Distribution

Max No 996885

Min No 627

Range 996258

No of Classes taken 10Class interval 100000(approx99625)

Rs. Crore

Raw data in arranged array

11182

12332

13237

15528

17683

19844

24142

27242

31525

35814

43267

49539

56976

66926

77693

87722103198

114166

121961

147849

175259

201056

220659

233017

274583

305322

314535

356277

414084

494370

587688

736708

870329

947660

996885


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Class boundary Class Mid-point FrequencyRelative

Freq(%)

Cumulative

Frequency()

100000 0-100000 50000 41 68.333333 41 60

200000100001-

200000150000 5 8.3333333 46 19

300000200001-

300000250000 4 6.6666667 50 14

400000300001-

400000350000 3 5 53 10

500000400001-

500000450000 2 3.3333333 55 7

600000500001-

600000550000 1 1.6666667 56 5

700000600001-

700000650000 0 0 56 4

800000700001-

800000750000 1 1.6666667 57 4

900000800001-

900000 850000 1 1.6666667 58 3

1000000900001-

1000000950000 2 3.3333333 60 2

Here we can see that in class interval (0 to 100000) we maximum no of frequency, almost 68.33 % data

fall in this class interval.


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Histogram

Type of Data: Interval

Concept Name:Histogram

Selection of variable:Tax Revenue collection in Rs. Crore from 1950-51 to 2009-10

Formula and calculation steps:

A graph of the data in a frequency distribution is called a histogram. The class boundaries (or class

midpoints) are shown on the horizontal axis. The vertical axis is either frequency, relative frequency, or

percentage. Bars of the appropriate heights are used to represent the number of observations within

each class.

Findings and Interpretation of results:In class interval (0 to 100000) has maximum no of frequency.

0

10

20

30

40

50

FrequencyDistriution

Tax Revenue Collection in Rs. Crore

Histogram

0-100000

100001-200000

200001-300000

300001-400000

400001-500000

500001-600000

600001-700000

700001-800000


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Frequency Polygon


Concept Name: Frequency Polygon

Selection of variable:Data taken fromTAX REVENUE OF CENTRE AND THE STATES: 1950-51 to 2009-10

(Rs. Crore)

Formula and calculation steps: Midpoints of the interval of corresponding rectangle in a histogram arejoined together by straight lines. It gives a polygon i.e. a figure with many angles

Findings and Interpretation of results:It shows the class interval where maximum no of data fall

Mid point Frequency

0 0

50000 41

150000 5

250000 4

350000 3450000 2

550000 1

650000 0

750000 1

850000 1

950000 2

1050000 0

0

5

10

15

20

25

30

35

40

45

Frequency


Frequency Polygon


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OGIVE(LESS THAN)


Concept Name:Ogive (LessThan)

Selection of variable:Cumulative Frequency Less than Vs Tax Revenue collection

Formula and calculation steps:taking class boundary at X axis and cumulative frequencies in Y axis

Findings and Interpretation of results:It shows no of data fall in bellow that class boundary.

0

20

40

60

80

0 200000 400000 600000 800000 1000000 1200000

CumulativeFrequencylessthan


Ogive(Less Than)


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OGIVE (MORE THAN)


Concept Name:Ogive (More Than)

Selection of variable:Cumulative Frequency Less than Vs Tax Revenue collection

Formula and calculation steps: taking class boundary at X axis and cumulative frequencies in Y axis

Findings and Interpretation of results: It shows no of data fall in beyond that class boundary.

0

20

40

60

80

0 200000 400000 600000 800000 1000000 1200000CumulativeFrequencymore

than


Ogive(More Than)


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Pareto Charts

Source:http://www.who.int/retrieved on 24-07-2011.

Type of Data:Numerical

Concept Name:Pareto Charts

Selection of variable: Top ten causes of death in India according to WHO

Formula and calculation steps:Calculated the relative frequency % of each cause and the cumulative relativefrequency and plotted it.

Findings and Interpretation of results:The chart shows that 80% deaths in India are caused by followingdiseases Coronary Heart Disease, Diarrhoeal diseases, Lung Disease and Stroke. So the Initial focus of Government

should be taking requisite steps to reduce the deaths due to these diseases

Cause of Death No. of Deaths in '000Relative Frequency (%)

((2) 6767)*100

Cumulative

Relative

Frequency

Coronary Heart Disease 1416 20.93 20.93

Diarrhoeal diseases 1231 18.19 39.12Lung Disease 1122 16.58 55.70

Stroke 940 13.89 69.59

Influenza & Pneumonia 760 11.23 80.82

Tuberculosis 317 4.68 85.50

Low Birth Weight 279 4.12 89.63

Suicide 243 3.59 93.22

Liver diseases 236 3.49 96.70

Road Traffic Accidents 223 3.30 100.00

Total 6767 100.00

0.0010.0020.0030.00

40.0050.0060.0070.0080.0090.00100.00

0.00

5.00

10.00

15.00

20.00

25.00

Pareto Chart

Relative Frequency %

Cumulative Relative Frequency
http://www.who.int/http://www.who.int/http://www.who.int/http://www.who.int/


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Stem and Leaf Charts and Mode

Source:http://loksabha.nic.in/ retrieved on 23-07-2011.

Type of Data: Numerical

Concept Name:Stem and Leaf Charts and Mode

Selection of variable: No. of Members of Parliament of LokSabha from each State and Union Territory

Formula and calculation steps: Stems are taken as the digits in Tens place and leaf is unit place

Findings and Interpretation of results: We can see from the Stem and leaf plot that maximum number

of State's and Union territories have less than 10 members of Parliament also the Mode for the given

data set is '1' i.e. maximum no. States and Union Territories have only single representation

Sl.No.Name of State/Union

Territory

No. of Member of

Parliament for Lok Sabha

1 Andhra Pradesh 42

2 Andaman and Nicobar Islands 1 Stem Leaf

3 Arunachal Pradesh 2 0 1 1 1 1 1 1 1 1 1 2 2 2 2 2 4 5 6 7

4 Assam 14 1 0 1 3 4 45 Bihar 40 2 0 1 5 6 7 7 7 7 7 7 7 7 9

6 Chandigarh 1 3 9

7 Chhattisgarh 11 4 0 2 2 7 7 7 7 7 7 7 7

8 Dadra and Nagar Haveli 1 5

9 Daman and Diu 1 6

10 Delhi 7 7

11 Goa 2 8 012 Gujarat 26

13 Haryana 10

14 Himachal Pradesh 4 Mode 115 Jammu and Kashmir 6

16 Jharkhand 14

17 Karnataka 28

18 Kerala 20

19 Lakshadweep 1

20 Madhya Pradesh 29

21 Maharashtra 48

22 Manipur 2

23 Meghalaya 2

24 Mizoram 1

25 Nagaland 1

26 Orissa 21

27 Puducherry 1

28 Punjab 13

29 Rajasthan 25

30 Sikkim 1

31 Tamil Nadu 39

32 Tripura 2

33 Uttar Pradesh 80

34 Uttarakhand 5

35 West Bengal 42
http://loksabha.nic.in/http://loksabha.nic.in/http://loksabha.nic.in/http://loksabha.nic.in/


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Scatter Plots

Source:http://www.finmin.nic.in/reports/IPFStat200910.pdf retrieved on 23-07-2011.


Concept Name: Scatter Plot

Selection of variable: Tax Revenue Collected of Centre and the States: 1950-51 to 2009-10 for both

Direct and indirect Tax

Formula and calculation steps: For Scatter Plot we have taken direct and indirect tax collection over the

period for seeing the correlation between them

Findings and Interpretation of results: From the plot we can see that there is a strong correlation

between the two variables

Total Tax Revenue(All India) in Rs. Crore

Year Direct Indirect Total

1950-51 231 396 627

1951-52 244 495 739

1952-53 252 426 678

1953-54 242 430 672

1954-55 240 480 720

1955-56 259 509 768

1956-57 288 602 890

1957-58 327 718 1045

1958-59 344 745 1089

1959-60 378 838 1216

1960-61 402 948 1350

1961-62 449 1094 1543

1962-63 560 1305 1865

1963-64 693 1632 2325

1964-65 743 1856 2599

1965-66 734 2188 2922

1966-67 767 2494 3261

1967-68 780 2676 3456

1968-69 840 2919 3759

1969-70 963 3237 4200

1970-71 1009 3743 4752

1971-72 1171 4404 5575

1972-73 1346 5090 6436

1973-74 1552 5837 7389

1974-75 1834 7389 9223

1975-76 2493 8689 11182

1976-77 2585 9747 12332

1977-78 2680 10557 13237

1978-79 2851 12677 155281979-80 3096 14587 17683

1980-81 3268 16576 19844

1981-82 4133 20009 24142

1982-83 4492 22750 27242

1983-84 4907 26618 31525

1984-85 5330 30484 35814

1985-86 6252 37015 43267

1986-87 6889 42650 49539

1987-88 7483 49493 56976

1988-89 9758 57168 66926

1989-90 11165 66528 77693

1990-91 12260 75462 87722

1991-92 16657 86541 103198

1992-93 19387 94779 114166
http://www.finmin.nic.in/reports/IPFStat200910.pdfhttp://www.finmin.nic.in/reports/IPFStat200910.pdfhttp://www.finmin.nic.in/reports/IPFStat200910.pdfhttp://www.finmin.nic.in/reports/IPFStat200910.pdf


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Total Tax Revenue(All India) in Rs. Crore

Year Direct Indirect Total

1993-94 21713 100248 121961

1994-95 28878 118971 147849

1995-96 35777 139482 175259

1996-97 41061 159995 201056

1997-98 50538 170121 220659

1998-99 49119 183898 233017

1999-2000 60864 213719 274583

2000-01 71762 233558 3053222001-02 73109 241426 314535

2002-03 87365 268912 356277

2003-04 109546 304538 414084

2004-05 137093 357277 494370

2005-06 167635 420053 587688

2006-07 231376 505331 736708

2007-08 318839 551490 870329

2008-09(R.E.) 346390 601270 947660

2009-10(B.E.) 372061 624823 996885

0

100000

200000

300000

400000

500000

600000

700000

0 100000 200000 300000 400000

In

directTaxRevenue

Direct Tax Revenue

Scatter Plot

Direct Tax Vs Indirect Tax

Revenue


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Geometric Mean



Concept Name: Geometric Mean

Selection of variable: Total Tax Revenue Collected of Centre and the States: 1950-51 to 2009-10

Formula and calculation steps: Calculated the Growth of revenue over the previous year and the

corresponding growth factor and then multiplying and taking nth root to find geometric mean

Findings and Interpretation of results: The Average growth factor comes out to 1.13308 so our annual

rate of increase in tax revenue collection is 13.30%

Total Tax Revenue(All India) Growth over the previous year Growth Factor (x)

Year Rs. Crore % (g) (g100)+1

1950-51 627

1951-52 739 17.86 1.179

1952-53 678 -8.25 0.917

1953-54 672 -0.88 0.991

1954-55 720 7.14 1.0711955-56 768 6.67 1.067

1956-57 890 15.89 1.159

1957-58 1045 17.42 1.174

1958-59 1089 4.21 1.042

1959-60 1216 11.66 1.117

1960-61 1350 11.02 1.110

1961-62 1543 14.30 1.143

1962-63 1865 20.87 1.209

1963-64 2325 24.66 1.247

1964-65 2599 11.78 1.118

1965-66 2922 12.43 1.124

1966-67 3261 11.60 1.1161967-68 3456 5.98 1.060

1968-69 3759 8.77 1.088

1969-70 4200 11.73 1.117

1970-71 4752 13.14 1.131

1971-72 5575 17.32 1.173

1972-73 6436 15.44 1.154

1973-74 7389 14.81 1.148

1974-75 9223 24.82 1.248

1975-76 11182 21.24 1.212

1976-77 12332 10.28 1.103

1977-78 13237 7.34 1.073

1978-79 15528 17.31 1.173

1979-80 17683 13.88 1.139

1980-81 19844 12.22 1.122

1981-82 24142 21.66 1.217

1982-83 27242 12.84 1.128

1983-84 31525 15.72 1.157

1984-85 35814 13.61 1.136

1985-86 43267 20.81 1.208

1986-87 49539 14.50 1.145

1987-88 56976 15.01 1.150

1988-89 66926 17.46 1.175

1989-90 77693 16.09 1.1611990-91 87722 12.91 1.129

1991-92 103198 17.64 1.176

1992-93 114166 10.63 1.106


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Year Rs. Crore Growth over the previous year % Growth Factor (x)

1993-94 121961 6.83 1.068

1994-95 147849 21.23 1.212

1995-96 175259 18.54 1.185

1996-97 201056 14.72 1.147

1997-98 220659 9.75 1.098

1998-99 233017 5.60 1.056

1999-2000 274583 17.84 1.178

2000-01 305322 11.19 1.112

2001-02 314535 3.02 1.030

2002-03 356277 13.27 1.133

2003-04 414084 16.23 1.162

2004-05 494370 19.39 1.194

2005-06 587688 18.88 1.189

2006-07 736708 25.36 1.254

2007-08 870329 18.14 1.181

2008-09(R.E.) 947660 8.89 1.089

2009-10(B.E.) 996885 5.19 1.052

The Geometric Mean is given by

Where xiare the variables for which mean is required and n is the number of the variables.

In above case n= 59 and xiare the growth factor values for different years

Therefore

Geometric Mean (G) 1.13308

The geometric mean is more appropriate than the arithmetic mean for describing proportional growth,

both exponential growth (constant proportional growth) and varying growth; in business the geometric

mean of growth rates is known as the compound annual growth rate (CAGR). The geometric mean of

growth over periods yields the equivalent constant growth rate that would yield the same final amount.


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Year Rs. Crore

1993-94 121961

1994-95 147849

1995-96 175259

1996-97 201056

1997-98 220659

1998-99 233017

1999-2000 2745832000-01 305322

2001-02 314535

2002-03 356277

2003-04 414084

2004-05 494370

2005-06 587688

2006-07 736708

2007-08 870329

2008-09(R.E.) 947660

2009-10(B.E.) 996885

Total 8275357

The Arithmetic Mean is given by

Where xiare the variables for which mean is required and n is the number of the variables.

Here

n = 60

x = 8275357

Therefore

Arithmetic Mean= = = Rs. 140260.288


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Weighted Arithmetic Mean



Concept Name: WeightedArithmetic Mean


Formula and calculation steps: Calculated the sum of product of frequency and midpoint of each classand then divide it by the total frequency. Here frequency are taken as weights

Findings and Interpretation of results: Theweighted average Revenue collected from 1950 to 2009 is Rs.

163333.33

Class (Rs. Crore) Mid-point (x) Frequency (f) f * x

0-100000 50000 41 2050000

100001-200000 150000 5 750000

200001-300000 250000 4 1000000

300001-400000 350000 3 1050000

400001-500000 450000 2 900000

500001-600000 550000 1 550000

600001-700000 650000 0 0

700001-800000 750000 1 750000

800001-900000 850000 1 850000

900001-1000000 950000 2 1900000

Total 60 9800000

Here

f = n = 60

(f*x) = 9800000

Therefore

Weighted Arithmetic Mean of Grouped data = = = Rs. 163333.33


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Median



Concept Name: Median


Formula and calculation steps: Arranged the data in ascending order and found the mean of 30th

and31

stitem to find the median

Findings and Interpretation of results: TheMedian of the collected data is Rs. 18763.50 which means

half of the items lie above this point, and the other half lie below it.

Total Revenue Collected

Raw Data


Raw data in Arranged

Array

627 627

739 672

678 678

672 720

720 739

768 768

890 890

1045 1045

1089 1089

1216 1216

1350 1350

1543 1543

1865 1865

2325 2325

2599 25992922 2922

3261 3261

3456 3456

3759 3759

4200 4200

4752 4752

5575 5575

6436 6436

7389 7389

9223 9223

11182 11182

12332 1233213237 13237

15528 15528

17683 17683 30th item

19844 19844 31st item

24142 24142

27242 27242

31525 31525

35814 35814

43267 43267

49539 49539

56976 56976

66926 66926

77693 77693

87722 87722


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Raw Data


Raw data in Arranged

Array

103198 103198

114166 114166

121961 121961

147849 147849

175259 175259201056 201056

220659 220659

233017 233017

274583 274583

305322 305322

314535 314535

356277 356277

414084 414084

494370 494370

587688 587688

736708 736708

870329 870329947660 947660

996885 996885

The Median is given by

Median = ( th item in the arranged data array

In our case n= 60 therefore the Median is the (60+1)/2th item i.e. 30.5 item or we can take the mean of

the 30th

item and the 31st

item when the data is arranged in ascending or descending order.

In our case 30th

item is 17683 and 31st

item is 19844

Therefore

Median =

= 18763.50


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Total Tax Revenue in Rs. Crore

43267

49539

56976

66926

77693

87722

103198

114166

121961

147849

175259

201056

220659

233017

274583

305322

314535

356277

414084

494370

587688

736708

870329

947660

996885

The first quartile is calculated as ( ) item. Here it is the 15.25th

item. To find that, we calculate it

as,

First QuartileQ1 = 15.25th

item = 15th

item + (1/4) (16th

item 15th

item) = 2679.75

Similarly values for other quartiles can be found out as,

Quartile Value

Q1 2679.75

Q2 18763.5Q3 168406.5


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Range


Type of data: Numerical

Concept Name: Range

Variable selected: Total Tax revenues of Centre and State

Formula and Calculation: Range of a data set is calculated as the difference between the maximum

value and the minimum value.

Findings and Interpretation of results: For the above data set, the range is 996258

Interquartile range


Type of data:Numerical

Concept Name:Interquartile range

Variable Selected:Total Tax revenues of Centre and State

Formula and Calculation:Interquartile range is the difference in value of the third quartile and the firstquartile.

Interquartile range = Q3 Q1

= 168406.5 - 2679.75

= 165726.75

Findings and Interpretation:From the range value, we get a very high dispersion. But on closer look at

the data, the increase in tax collected in one year over the previous year has increased every year and in

the last twenty years, it has sometimes doubled itself. So the range value does not perfectly represent

the spread of the data set, whereas the quartiles and interquartile range are more fair representations

of the dispersion. We infer that 50% of the values lie between 2679.75(Q1) and 168406.5(Q3) and also

that 75% lie below 168406.5(Q3). The very fact that from 168406.5(Q3) to the maximum value just 25%

of the data are present shows the inaccuracy of the range value in determining the spread.

Minimum 627

Maximum 996885

Range 996258
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Mean Absolute Deviation


Type of Data:Numerical

Concept Name:Mean Absolute Deviation


Formula and Calculation:

For the above data set, the Mean Absolute Deviation is found as Rs 168983.9711

Findings and Interpretation:

In the above data set, the income tax collected over all the years has a dispersion of Rs 168983.9711

from the average tax collected of Rs 137922.6167.

Variance


Type of data:Numerical

Concept Name:Variance



For the above data set,Variance is found to be58848680358.4099.


Since variance is ameasure of by how much the values in the data set are likely to differ from the meanof the values, we can see that in the above data set, the data are very widely dispersed from the mean.


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Standard Deviation



Concept Name: Standard Deviation

Variable Selected: Total Tax revenues of Centre and State


For the given data set, Standard Deviation is242587.4695

Findings and Interpretations:

The average tax collected is Rs 137922.6167 and the tax collected for all the years are at a standarddeviation of Rs 242587.4695 away from the mean.

Coefficient of Variation



Concept Name: Coefficient of Variation

Variable Selected: Central Taxes Gross and States own taxes



From the coefficient of variation of the two data sets, we can infer that the dispersion is more or less the

same.

Rs. Crore

States Taxes Centres Taxes

222 405

227 512

233 445

252 420

265 455

283 485

320 570

353 692

388 701

422 794


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Rs. Crore

States Taxes Centres Taxes

455 895

489 1054

580 1285

691 1634

778 1821

861 2061

954 2307

1103 23531249 2510

1378 2822

1546 3206

1702 3873

1931 4505

2319 5070

2901 6322

3573 7609

4061 8271

4379 8858

5003 10525

5709 11974

6665 13179

8295 15847

9546 17696

10804 20721

12343 23471

14597 28670

16701 32838

19311 37665

22452 44474

26057 51636

30145 57577

35837 67361

39530 74636

46219 75742

55552 92297

64035 111224

71294 129762

81439 139220

89220 143797

102831 171752

116717 188605

127475 187060

140372 215905159736 254348

189413 304957

221537 366151

263195 473513

277182 593147

319711 627949

355805 641080

For the data set on the left, (i.e. States taxes)

Mean 49644.05

S.D 85675.55CV % 172.5797


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CV = 85675.55 / 49644.05 = 172.5797

For the data set on the right, (i.e. Centres taxes)

Mean 88278.57

S.D 157277.2

CV % 178.1602

CV = 157277.2/ 88278.57 = 178.1602

Skewness



Concept Name: Skewness



For the above data set, the skewness is 2.30112980468625.

Findings and Interpretations:

It is positive which indicates that more values are present to the left side( or below) the mean value and

the distribution will be skewed to the left. Thus this also supports what we have inferred from quartiles

and interquartile range that 75% of values lay below Q3. Therefore, tax collected by the government is

mostly below the mean value.

Kurtosis



Concept Name: Kurtosis



For the data set, the kurtosis value is 4.79782333743203.


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Kurtosis is a measure of the peakedness of the distribution curve. Taking into account that the kurtosis

for a standard normal distribution is 3, our data set has a higher value.

From this we can conclude that our data if organized in a distribution will be more peaked than a

standard normal distribution as the kurtosis value is higher. Kurtosis values indicate a sharp peak nearthe mean. This can be seen from the histogram that there is a sharp peak near the mean value of

137922.616666667 and then a long tail.

Percentile



Concept Name: Percentile



L/N(100) = P

where L is the number of items less than a value, N is the total number of items (here 60) and P is the

percentile.

So for a tax revenue of Rs 43267 in the year 1985-86 the percentile value is found out as:

L = number of items less than 19844 = 35

N = 60

P = (35/60) * 100 = 58.33


Therefore a value of 43267 is at a percentile of 58.33 which means it is greater than approximately 58%

of the items in the data set.


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Decile



Concept Name: Decile



Deciles are the points in a data set which divide the set into 10 equal parts.

For the above data set, the deciles are:

Decile Value

1 768

2 1543

3 3456

4 7389

5 17683

6 43267

7 103198

8 220659

9 414084


The decile values divide the data into 10 equal parts and a value of Rs 672 is in the first one-tenth of thedata set and less than the first decile.

statistic concepts

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