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Transcript of 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
Type of Data: Interval
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
Tax Revenue Collection in Rs. Crore
Frequency Polygon
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OGIVE(LESS THAN)
Type of Data: Interval
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
Tax Revenue Collection in Rs. Crore
Ogive(Less Than)
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OGIVE (MORE THAN)
Type of Data: Interval
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
Tax Revenue Collection in Rs. Crore
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.
Type of Data: Numerical
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
Source:http://www.finmin.nic.in/reports/IPFStat200910.pdf retrieved on 23-07-2011.
Type of Data: Numerical
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
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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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
Source:http://www.finmin.nic.in/reports/IPFStat200910.pdf retrieved on 23-07-2011.
Type of Data: Numerical
Concept Name: WeightedArithmetic Mean
Selection of variable: Total Tax Revenue Collected of Centre and the States: 1950-51 to 2009-10
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
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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Median
Source:http://www.finmin.nic.in/reports/IPFStat200910.pdf retrieved on 23-07-2011.
Type of Data: Numerical
Concept Name: Median
Selection of variable: Total Tax Revenue Collected of Centre and the States: 1950-51 to 2009-10
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
Total Revenue Collected
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
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 Revenue Collected
Raw Data
Total Revenue Collected
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
Source:http://www.finmin.nic.in/reports/IPFStat200910.pdf retrieved on 23-07-2011.
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
Source:http://www.finmin.nic.in/reports/IPFStat200910.pdf retrieved on 23-07-2011.
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
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.pdfhttp://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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Mean Absolute Deviation
Source:http://www.finmin.nic.in/reports/IPFStat200910.pdf retrieved on 23-07-2011.
Type of Data:Numerical
Concept Name:Mean Absolute Deviation
Variable Selected:Total Tax revenues of Centre and State
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
Source:http://www.finmin.nic.in/reports/IPFStat200910.pdf retrieved on 23-07-2011.
Type of data:Numerical
Concept Name:Variance
Variable Selected:Total Tax revenues of Centre and State
Formula and Calculation:
For the above data set,Variance is found to be58848680358.4099.
Findings and Interpretation:
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
Source:http://www.finmin.nic.in/reports/IPFStat200910.pdf retrieved on 23-07-2011.
Type of data: Numerical
Concept Name: Standard Deviation
Variable Selected: Total Tax revenues of Centre and State
Formula and Calculation:
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
Source:http://www.finmin.nic.in/reports/IPFStat200910.pdf retrieved on 23-07-2011.
Type of data: Numerical
Concept Name: Coefficient of Variation
Variable Selected: Central Taxes Gross and States own taxes
Formula and Calculation:
Findings and Interpretation:
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
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.pdfhttp://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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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
Source:http://www.finmin.nic.in/reports/IPFStat200910.pdf retrieved on 23-07-2011.
Type of data: Numerical
Concept Name: Skewness
Variable Selected: Total Tax revenues of Centre and State
Formula and Calculation:
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
Source:http://www.finmin.nic.in/reports/IPFStat200910.pdf retrieved on 23-07-2011.
Type of data: Numerical
Concept Name: Kurtosis
Variable Selected: Total Tax revenues of Centre and State
Formula and Calculation:
For the data set, the kurtosis value is 4.79782333743203.
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.pdfhttp://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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Findings and Interpretation:
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
Source:http://www.finmin.nic.in/reports/IPFStat200910.pdf retrieved on 23-07-2011.
Type of data: Numerical
Concept Name: Percentile
Variable Selected: Total Tax revenues of Centre and State
Formula and Calculation:
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
Findings and Interpretation:
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
Source:http://www.finmin.nic.in/reports/IPFStat200910.pdf retrieved on 23-07-2011.
Type of data: Numerical
Concept Name: Decile
Variable Selected: Total Tax revenues of Centre and State
Formula and Calculation:
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
Findings and Interpretation:
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.
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