# Strata-based Quantification of Distributional Uncertainty in Socio-Economic
Indicators: A Comparative Study of Indian States

Abhik Ghosh  Interdisciplinary Statistical Research Unit, Indian Statistical
Institute, Kolkata 700108<EMAIL_ADDRESS>Olivia Mallick  Souvik
Chattopadhay  Banasri Basu  Physics and Applied Mathematics Unit, Indian
Statistical Institute, Kolkata 700108, India<EMAIL_ADDRESS>
###### Abstract

This paper reports a comprehensive study of distributional uncertainty in a
few socio-economic indicators across the various states of India over the
years 2001-2011. We show that the DGB distribution, a typical rank order
distribution, provide excellent fits to the district-wise empirical data for
the population size, literacy rate (LR) and work participation rate (WPR)
within every states in India, through its two distributional parameters.
Moreover, taking resort to the entropy formulation of the DGB distribution, a
proposed uncertainty percentage (UP) unveils the dynamics of the uncertainty
of LR and WPR in all states of India. We have also commented on the changes in
the estimated parameters and the UP values from the years 2001 to 2011.
Additionally, a gender based analysis of the distribution of these important
socio-economic variables within different states of India has also been
discussed. Interestingly, it has been observed that, although the
distributions of the numbers of literate and working people has a direct
(linear) correspondence with that of the population size, the literacy and
work-participation rates are distributed independently of the population
distributions.

Keywords: Entropy; Rank Order Distribution; Population Distribution; Literacy
Rate; Work Participation Rate.

##  I Introduction

Socio-economic development of our world is majorly driven by people and their
governance. The habitable area of the world is divided into distinct units,
namely countries or dependent territories, governed by independent
administrative bodies of different nature. These countries are further
subdivided (first tier division) into smaller units, sometimes called states
or provinces, for the purpose of internal governance, management and better
policy making. For the administrative purpose and the ease of working in a
local manner, these states or provinces are further subdivided (second tier
division) into more smaller strata called counties or districts. Although
further subdivisions are seen in some countries, most large countries have
major administrative divisions up to the second tier. For example, USA is
divided into states (first-tier division) which, in turn, are divided into
counties (second-tier division). China is divided into provinces and direct-
controlled municipalities (first level), which are split into prefectures
(second level), and prefectures are divided into counties (third level). India
is divided into states (first tier) and subsequently in districts (second-
tier). Although there is wide heterogeneity among the administrative units
(states or districts) in a country, these local administrative units can be
seen as socially constructed strata, which serve as spatial scenarios for
social and economic processes lopez2008 . Such administrative divisions are
often regions of a country that are granted a certain degree of autonomy and
manage themselves through local governments. In a growing number of developing
countries, there is an attempt to proliferate sub-national administrative
divisions for decentralization of local governments grossman2014  such that
the reorganization may yield economic benefit for the cities ma2005 . Even in
the absence of large administrative and political reforms, administrative
units are constantly being created, destroyed, merged or split, demonstrating
the fact that the sub-divisional strata within the countries undergo
evolution.

The population distribution that does not distribute randomly eeckhout2004 ;
kgbb ; ag2019  over different regions of a country is one of the key factors
of the high degree of diversity and complexity christenson1980  of the
internal regional administration of various countries and territories. In this
context, the study and understanding of the geographical distribution of the
population within a given country or region becomes relevant, as it is a
necessary step for the development of theories that could accurately describe
the evolution of human agglomerations batty2008 ; battym2013 . However, almost
all studies regarding population distribution focus on city populations (see,
for instance, kgbb ; ag2019 ; soo2005 ; makse2008 ; holmes2010 ; jia2010 ;
rybski2011 ; douady2011 ; masucci2013 ), whereas the literature regarding the
population distribution in administrative divisions is scarce mansilla2009 ;
betten2013 .

Moreover, there are studies predicting strong correlation of the population of
any city with many of the characterizing features of the inhabitants of the
corresponding city: mean income, number of registered patents per capita,
criminality rates, land value and rent prices christenson1980 ; batty2008 ;
ag2019 . In this respect, it is important to study the correlations of various
socio-economic variables with the population distribution of the smaller
strata of a country. In particular, to frame better policies for the whole
country as well as for these subdivisions, it is necessary to understand, in
detail, the distribution and associated uncertainty or inequalities of
different socio-economic indicators within and between these strata along with
their relations with the populations. These would then help us to classify
different strata (subdivisions) within a country into similar groups in terms
of the policy requirements for their developments where the groups are formed
based on their underlying difference in terms of the socio-economic
indicators. Is there any particular stratum within the country that requires
special attention than the rest or all the strata are equal in terms of their
socio-economic characteristics? In an attempt to answer such questions, in
this paper, we will propose and discuss an innovative way of quantifying the
distributional uncertainty of the socio-economic factors within different
strata (first tier subdivisions) of a country based on their rank-size pattern
across the second-tier subdivisions.

To study the uncertainty characteristics of a given socio-economic factor
within a strata (first-tier subdivision) of a country, we first fit an
appropriate distribution to the rank-size values of this factor across all
second-tier subdivisions within the strata. The well known Zipf’s law,
alternatively known as power law or Pareto distribution zipf has been
successfully used to model such rank-size data on socio-economic factors in
its upper range of values, but observed to fail in the lower end of the rank-
size distribution lan2000 ; newman2005 . Very recently, a universal framework
through rank-order distribution has been developed to successfully model a
wide-range of rank-size data from various areas in arts and science Martinez-
Mekler09 ; ag2019 ; Ausloos/Cerqueti:2016 ; Alvarez-Martinez/etc:2011 ;
Alvarez-Martinez/etc:2014 ; Oscar2017 ; Alvarez-Martinez/etc:2018  where a
two-parameter discrete generalized beta (DGB) distribution is used instead of
the one-parameter Pareto distribution. The DGB distribution is also shown to
provide excellent fit to the rank-size data on various socio-economic
parameters within a country and between different countries across the world
ag2019 . Remarkably, this DGB distribution can indeed be obtained as an
appropriate maximum-entropy distribution agmaxent  and hence its entropy can
be used to quantify the maximum uncertainty present in the empirical data
distribution. We will follow this idea to develop an uncertainty measure based
on the entropy value of the fitted DGB distribution to the empirical data on
the given socio-economic factor across all second-tier subdivisions within the
strata.

Our proposed uncertainty measure, which we refer to as the Uncertainty
Percentage (UP), will indicate the distribution of the factor under study
across different parts of the given strata; it will hence take the maximum
value of 100% if the factor value (socio-economic condition) is the same
across all parts of the strata, i.e., in all second-tier subdivision within
it, indicating the equally distributed scenario (least inequality). On the
other hand, our measure UP will take the minimum value of 0% if the inequality
within the strata is extreme in that only one second-tier subdivision has a
non-zero value of the factor under study (socio-economic condition is as
desired in only one subdivision)! Any value of UP in between these two extrema
will indicate how the targeted factor is distributed within different parts
(subdivisions) of the strata with higher values indicating more uniformity
(less inequality). Once this uncertainty measure is computed for all the
strata within a country, they can be classified accordingly and can be used
for their further in-depth comparisons. For example, the fitted DGB
distribution of different socio-economic factor can be linked by established
appropriate relationship between the corresponding parameter estimates which,
along with our UP measure, provide further insights to characterize and
classify different strata within a country for better policy making.

Further, by using our proposed idea, we will analyze uncertainty among the
Indian states (first-tier strata) based on the census data for the years 2011
and 2001. In this paper, we focus on the basic three socio-economic entities,
namely, population, literacy and working population. Like Population,
education is a key element in any society that removes inequality from
society, impacts the growth of employment and improves a country’s gross
national product. On the other hand, the distributional inequality of
unemployment rates (that is given by the size of the non-working population)
plays an important role for structuring the economic development. These three
factors, together controls, to a large extent, the development of the human
resources in any society. For a vast and diverse country like India, which has
so many states with very different cultures, it is very important to analyses
the levels of literacy and employment across various states of India and union
territories (UTs) and investigate their underlying distributional uncertainty
or inequality. We would like to apply our proposed idea ag2019 ; agmaxent  of
fitting the DGB distribution to the empirical data on the said three factors
for all the districts of each Indian states. In a subsequent analysis we
measure the inequality across the second-tier subdivisions (districts) within
a state via our proposed UP measure. The correlation of the distribution of
literacy and employment rate within each state with its population
distribution will be investigated by linking the parameters of the
corresponding fitted DGB distributions. Additionally, to strengthen our
proposition we have also performed a gender based analysis for all these
socio-economic indicators. Finally, our temporal analysis provide an overall
picture about the uncertainty or uniformity of the Indian states in terms of
population, literacy and employment rates, their interrelations and changes
from the year 2001 to 2011.

##  II Methodology

###  II.1 Modelling Empirical Data by DGB distribution

Let us consider a strata (e.g., state of a country) having $N$ second-tier
subdivisions (e.g., districts). Suppose that a socio-economic factor under
study (e.g., population, literacy, work participation rate, etc.) takes the
value $x_{i}$ at the $i$-th subdivision for $i=1,\ldots,N$. We arrange these
data in decreasing order of “importance” (size) and denote the rank of the
$i$-th subdivision as $r_{i}$ for $i=1,\ldots,N$. We model these rank-size
data $\\{(r_{i},x_{i}):i=1,\ldots,N\\}$ by the discrete generalized beta (DGB)
distribution having probability mass function

$f_{(a,b)}(r)=A\frac{(N+1-r)^{b}}{r^{a}},~{}~{}~{}~{}r=1,\ldots,N.$ (1)

Here, in (1), $a,b$ are two real valued model parameters characterizing the
underlying distributional structure and $A$ is the normalizing constant
depending on $(a,b)$ ensuring that $\sum_{r=1}^{N}f_{(a,b)}(r)=1$. Note that,
for the choice $b=0$, the probability mass function in (1) simplifies to that
of the Pareto distribution. With different values of the two parameters
$(a,b)$, the class of DGB distribution in (1) allows us to model a wide range
of rank-size distributions having an inflection point ag2019  and can
successfully model different types of socio-economic factors Martinez-Mekler09
; ag2019 ; Ausloos/Cerqueti:2016 ; Alvarez-Martinez/etc:2011 ; Alvarez-
Martinez/etc:2014 ; Oscar2017 ; Alvarez-Martinez/etc:2018

Given empirical rank-size data $\\{(r_{i},x_{i}):i=1,\ldots,N\\}$ of any
socio-economic factor, we can fit the DGB distribution by estimating the model
parameters $(a,b)$ by maximizing the likelihood of the observed data, given by

$L(a,b)=\prod_{i=1}^{N}f_{(a,b)}(r_{i})^{{x}_{i}}=\prod_{i=1}^{N}\frac{(N+1-r_{i})^{b{x}_{i}}}{r_{i}^{a{x}_{i}}}A^{{x}_{i}}.$
(2)

This maximum likelihood estimator is known to be asymptotically most efficient
and also enjoys several other optimality properties. However, we need to use
numerical optimization method to compute these estimates since the likelihood
function in (2) does not posses an explicit form for its maximizer. Once we
get the maximum likelihood estimator (MLE), $(\widehat{a},\widehat{b})$, of
the model parameters $(a,b)$ we can then verify if the fitted DGB model is a
good fit to the empirical data or not. For this purpose, we use the overall
error in model fit, computed in terms of the Kolmogorov-Smirnov (KS) measure
between the observed and the predicted cumulative rank-sizes. Denoting the
predicted size of rank $r_{i}$ as
$p_{i}=\left(\sum_{i=1}^{N}x_{i}\right)f_{(\widehat{a},\widehat{b})}(r_{i})$
for all $i=1,\ldots,N$, we define the KS goodness-of-fit measure as

$\displaystyle KS=\max_{1\leq i\leq N}\left|\left(\sum_{j:r_{j}\leq
r_{i}}p_{j}\right)-\left(\sum_{j:r_{j}\leq r_{i}}x_{j}\right)\right|.$ (3)

The lower KS values indicate a better fitment of the model, i.e. the
theoretically estimated fitted DGB distribution is very close to the
empirically observed rank-sizes when the KS values are nearly vanishing. Zero
KS value indicates perfect fit with no error for the observed data.

###  II.2  Quantification of Distributional Uncertainty

Once a DGB distribution is found to be well fitted for a socio-economic
indicator within a strata, we propose to quantify the underlying
distributional uncertainty by its entropy. Entropy is a widely used measure of
disorder within any physical system. Although the idea of entropy was
originally used in Thermodynamics long back, its major use in information
science and allied disciplines were started after Shannon’s groundbreaking
work shanon1 ; shanon2  on mathematical formulation of entropy in an
information channel and Jayne’s Maximum entropy principle jaynes .
Subsequently, the idea of entropy and maximum entropy distribution has been
widely used to analyze and assess the uncertainty in several geographical or
socioeconomic variables kapurbook

Here we also consider the DGB distribution, which is shown to be a maximum
entropy rank-order distribution under appropriate utility constraints agmaxent
For this DGB distribution having probability as given in (1), its Shannon
entropy is given by

$\displaystyle S_{N}(a,b)$ $\displaystyle=$
$\displaystyle-\sum_{r=1}^{N}f_{(a,b)}(r)\log f_{(a,b)}(r)=-\log
A-A\sum_{r=1}^{N}\frac{(N+1-R)^{b}}{r^{a}}[b\log(N+1-r)-a\log r].$ (4)

This entropy value $S(a,b)$ then gives the maximum amount of uncertainty
(entropy) lying within the underlying rank-order distribution for the given
utility constraints which are characterized by the model parameters $(a,b)$.
Therefore, given the empirical rank-size data, once we obtain the estimated
parameter value $(\widehat{a},\widehat{b})$, we can estimate the (maximum)
amount of uncertainty in these data by the entropy of the fitted DGB
distribution, i.e., by $\widehat{S}_{N}=S_{N}(\widehat{a},\widehat{b})$.

However, the estimated entropy value $\widehat{S}_{N}$ can vary from 0 to
$\log(N)$ so it cannot be used to compare the uncertainty present in two
distributions having different values of $N$. Since the strata may often have
different number ($N$) of subdivisions, we cannot compare them by just using
$\widehat{S}_{N}$ as a measure of uncertainty. Standardizing by the range, we
then define our proposed uncertainty percentage (UP) measure as the proportion
of entropy estimated from the fitted DGB distribution with respect to its
maximum possible value, i.e.,

$UP=\frac{S_{N}(\widehat{a},\widehat{b})}{\log N}\times 100\%.$ (5)

Note that, clearly the UP measure takes the value 100% if
$S_{N}(\widehat{a},\widehat{b})=\log(N)$ which holds if and only if
$\widehat{a}=0=\widehat{b}$ indicating that the best-fitted rank-sizes are
uniformly distributed over the subdivisions within the given strata; in other
words, there is no inequality of the distribution of the targeted socio-
economic indicator within different parts of the strata. On the other hand,
the UP measure will be zero if $S_{N}(\widehat{a},\widehat{b})=0$ indicating
the extremely dispersed distribution of the rank-sizes within the strata with
only one subdivision having non-zero value of the targeted indicator.
Therefore, the UP can be used to quantify the distributional uncertainty of
the spread of the distribution of any socio-economic indicator within a strata
measuring its $closeness$ to the optimum case of uniformity. Since the UP
measure takes the value from 0 to 100%, irrespective of the value of $N$, it
can then also be used to compare the uncertainty present within different
strata as well as for different indicators.

###  II.3 Inter-relations between the distributions of two socio-economic
variables

In any human agglomeration unit, most of the socio-economic indicators are
highly correlated with each other. In particular, we have noted that the
population distribution has a huge impact on the distribution of various
socio-economic factors that had been explored in the literature via regression
within the power law structure. However, in this paper, we have used the
alternative DGB rank-order distribution that is noted to yield better fitment
of the model to the empirical data. Being consistent with the fitted DGB
distributions, we here propose to investigate the relationship between the
distributions of any socio-economic indicator with the population distribution
within different strata. To be more specific, our approach focus more on the
inter-relation between the distributional structure and uncertainty of the
socio-economic variables rather than the corresponding values of the
indicators.

To investigate the relationship between the DGB distributions fitted to two
variables within a strata, let us recall that the DGB distributions are
characterized by its two model parameters $(a,b)$. Suppose that there are $T$
strata and, for the $j$-th stratum, the MLE of the parameters $(a,b)$
corresponding to the fitted population distribution is
$(\widehat{a}_{Pj},\widehat{b}_{Pj})$ and the same for any targeted socio-
economic indicator $Y$ (e.g., literacy or unemployment rate) is
$(\widehat{a}_{Yj},\widehat{b}_{Yj})$, $j=1,\ldots,T$. Then to study the
relationship between the best-fitted DGB distributions to the targeted
indicator $Y$ and the population across the $T$ strata, it is enough to
investigate the relationship between the corresponding parameters
$(\widehat{a}_{Pj},\widehat{b}_{Pj})$ and
$(\widehat{a}_{Yj},\widehat{b}_{Yj})$ for $j=1,\ldots,T$. The first attempt in
this regard should be the examination of the (Pearson) bivariate correlations
among these estimated parameters across strata. If this correlation is
significantly higher (in absolute value), it indicates the linearity of their
relationship which we may then further explore via appropriate multivariate
regression models. The same can be studied using the UP values and also for
any two socio-economic indicators (other than the population).

However, if there is no linear relationship between the fitted parameters or
the UP values corresponding to two indicators, one should investigate their
relation via appropriate non-parametric correlation coefficients (e.g., rank
correlation). If this non-parametric correlation is found to be significant,
then one may proceed for further analyses of their non-linear relationship
using advanced statistical tools, which will not be considered in the present
paper.

##  III Results: Understanding Uncertainty among Indian States

###  III.1 Data Description and Overall Statistics

We consider the data from Indian census which is conducted in every 10 years
to capture a detailed picture of demographic, economic and social conditions
of all persons in the country pertaining to that specific time. The raw census
data for the years 2011 and 2001 are obtained from the Primary Census data and
Digital library (www.censusindia.gov.in). Following the stratified
administrative structure of India, we use the districts as smaller second-tier
units within each state (first-tier strata) of India. According to the final
census in 2011, there were 29 states and 7 UTs in India, which consist of the
pool of strata in our analysis; for simplicity we will also refer the UTs as
the State. It is important to note that some states, created at a later time,
were not present in the earlier round of census (at the year 2001). The total
number of districts within all the states of India was 640 and 593 in the
years 2011 and 2001, respectively. Additionally, we will also use these census
data separately for male and female populations for further gender-based
uncertainty analysis in Section III.3.

Figure 1: Plot of LR and WPR for different states in the years 2001 and 2011,
along with the corresponding linear fits. The black solid line represent the
$y=x$ line (linear growth)

As mentioned earlier, in this paper, we focus on two important socio-economic
factors, namely literacy and work participation rates, along with the
population of any unit under consideration. For literacy computation, a person
is defined to be literate if (s)he is aged more than 7 years and can both read
and write with understanding in any language. On the other hand, a person who
has worked in any economically productive activity for more than 6 months
during the last one year preceding the data collection date is considered to
be a worker (main or marginal). We analyze the data on the corresponding
variables (in rate) defined as follows:

Literacy rate (LR) $\displaystyle=$ $\displaystyle\frac{\mbox{number of
literate people in a district}}{\mbox{total population in that
district}}\times 100,$ Work Participation Rate (WPR) $\displaystyle=$
$\displaystyle\frac{\mbox{number of Workers in a district}}{\mbox{total
Population in that district}}\times 100.$

A comprehensive summary of state-wise population (P), LR and WPR statistics
(in %) are reported in Supplementary Table S1 (Appendix A) for the years 2011
and 2001. It can easily be seen that the population of each state has
increased over the years (linearly). To understand the changes in LR and WPR
more clearly, we have plotted their values in the years 2001 versus in the
year 2011 in Figure 1, along with the corresponding linear fit. According to
2011 census India has literacy rate (LR) around 84% to 50%, among which Kerala
is highest and Bihar is lowest; but, the WPR for different states of India
varies in between 35% to 50% only. Further, the LR has significantly increased
in all the states whereas the changes in WPR is rather variables. Except for
Mizoram and Kerala, all states has an increase of more than 5% in LR, with as
many as 9 states having an increase of more than 10%. On the other hand, 10
states have no significant changes (only less than 1%) in the WPR. The UT of
Dadra and Nagar Haveli has the highest increase in LR but maximum decrease in
WPR. Nagaland is seen to have the highest increase of 6.64% in the WPR. Among
larger states, Kerala, Himachal Pradesh, Assam, Jharkhand and Odisha has shown
some amount of increase ($>2\%$) in WPR. Overall Tripura seems to have
performed best considering both LR and WPR. On an average, there have been
sublinear growths in both LR and WPR among the Indian states and UTs.

###  III.2 Modelling Population, Literacy and Work participation rates within
each States

Fitting the DGB rank-order Distributions within each States:

We now discuss the results obtained by the application of the DGB distribution
for different socioeconomic variables (Population, LR and WPR) for various
states of India using the district-wise data. The states comprising of
districts less than 5 are excluded from the analysis; this excludes 6 UTs (all
except Delhi) and three states (Goa, Sikkim and Tripura). The values of the
estimated parameters ($a,b$) and the goodness-of-fit measures (KS) for all
three socioeconomic variables for different states, are reported in Table 1
for both the years 2011 and 2001. As expected (see ag2019 ; agmaxent ), the
DGB distribution, with the two distributional parameters $\widehat{a}$ and
$\widehat{b}$, provide excellent fits to the empirical data not only for
Population size but also for LR and WPR with extremely small KS error.

Considering the distributional structures, for most states, both the
parameters $\widehat{a}$ and $\widehat{b}$ tend to have positive values and
are also nature-wise similar. However, there has been significant variations
among the estimated parameter values across different states for all the
variables, indicating differences in their distributions with the states
(across districts). Particularly the UT of Delhi, the capital of India and one
of the crowded states, has a negative $\widehat{a}$ value $-0.157$ and a
positive $\widehat{b}$ value 1.387 for the distribution of populations. On the
other hand, for modelling population, the not so much populated state Mizoram
has a high positive value of $\widehat{a}$ (0.999) than any other states but a
negative $\widehat{b}$ value. While considering the distribution of LR, most
of the states fits to the positive values for both $\widehat{a}$ and
$\widehat{b}$, except for Chandigarh, Delhi, Haryana, Mizoram which yield
negative $\widehat{a}$ value and positive $\widehat{b}$ value. In 2011,
Kerala, the state with 84 percent literate people, corresponds to
$\widehat{a}$ and $\widehat{b}$ values of 0.018 and 0.035, respectively. West
Bengal having 67.4 percent literate people gives $\widehat{a}$ and
$\widehat{b}$ values 0.043 and 0.127. For the distribution of WPR as well,
some states have negative $\widehat{a}$ value indicating their difference from
the majority of the states. In general, disparity of LR among the states are
more than that of WPR.

Table 1: The parameter estimates and the KS measures for for different states along with total numbers (N) of districts  |  | Population | LR | WPR
---|---|---|---|---
States | $N$ | $\widehat{a}$ | $\widehat{b}$ | KS | $\widehat{a}$ | $\widehat{b}$ | KS | $\widehat{a}$ | $\widehat{b}$ | KS
Year 2011
India | 640 | 0.252 | 0.872 | 0.005 | 0.063 | 0.133 | 0.004 | 0.082 | 0.109 | 0.006
Andhra Pradesh | 23 | 0.124 | 0.171 | 0.007 | 0.074 | 0.050 | 0.001 | -0.009 | 0.109 | 0.001
Arunachal Pradesh | 16 | 0.074 | 0.825 | 0.020 | 0.069 | 0.125 | 0.002 | 0.122 | 0.033 | 0.003
Assam | 27 | 0.265 | 0.307 | 0.011 | 0.076 | 0.070 | 0.002 | 0.060 | 0.071 | 0.006
Bihar | 38 | 0.143 | 0.534 | 0.003 | 0.058 | 0.086 | 0.006 | 0.064 | 0.056 | 0.003
Chattishgarh | 18 | 0.390 | 0.664 | 0.015 | -0.022 | 0.285 | 0.005 | 0.044 | 0.059 | 0.002
Delhi | 9 | -0.157 | 1.387 | 0.026 | 0.015 | 0.033 | 0.001 | 0.101 | 0.022 | 0.006
Gujarat | 26 | 0.383 | 0.454 | 0.020 | 0.021 | 0.112 | 0.003 | 0.112 | 0.011 | 0.003
Haryana | 21 | 0.140 | 0.199 | 0.006 | -0.014 | 0.150 | 0.004 | 0.020 | 0.105 | 0.002
Himachal Pradesh | 12 | 0.306 | 0.748 | 0.035 | 0.009 | 0.076 | 0.001 | 0.059 | 0.119 | 0.005
Jammu Kashmir | 22 | 0.368 | 0.487 | 0.011 | 0.141 | 0.049 | 0.003 | 0.175 | 0.046 | 0.004
Jharkhand | 24 | 0.271 | 0.394 | 0.013 | 0.060 | 0.107 | 0.003 | 0.052 | 0.111 | 0.004
Karnataka | 30 | 0.689 | 0.033 | 0.031 | 0.047 | 0.122 | 0.002 | 0.044 | 0.032 | 0.002
Kerala | 14 | 0.056 | 0.592 | 0.011 | 0.018 | 0.035 | 0.002 | 0.074 | 0.107 | 0.006
Madyapradesh | 50 | 0.197 | 0.293 | 0.005 | 0.016 | 0.159 | 0.005 | 0.048 | 0.098 | 0.001
Maharashtra | 35 | 0.491 | 0.288 | 0.014 | 0.027 | 0.077 | 0.001 | 0.052 | 0.038 | 0.002
Manipur | 9 | 0.126 | 0.623 | 0.030 | 0.038 | 0.103 | 0.003 | 0.054 | 0.059 | 0.001
Meghalayas | 7 | 0.376 | 0.477 | 0.014 | 0.055 | 0.110 | 0.007 | 0.036 | 0.025 | 0.002
Mizoram | 8 | 0.990 | -0.119 | 0.020 | -0.046 | 0.239 | 0.006 | -0.008 | 0.184 | 0.003
Nagaland | 11 | 0.241 | 0.563 | 0.012 | 0.050 | 0.171 | 0.004 | 0.107 | 0.109 | 0.005
Oddisha | 30 | 0.239 | 0.488 | 0.007 | 0.027 | 0.230 | 0.005 | 0.035 | 0.167 | 0.008
Punjab | 20 | 0.381 | 0.332 | 0.024 | 0.042 | 0.082 | 0.007 | 0.062 | 0.021 | 0.003
Rajasthan | 33 | 0.367 | 0.211 | 0.012 | 0.062 | 0.066 | 0.003 | 0.063 | 0.049 | 0.001
TamilNadu | 32 | 0.175 | 0.477 | 0.009 | 0.042 | 0.050 | 0.001 | 0.035 | 0.079 | 0.001
Uttarakhand | 13 | 0.478 | 0.542 | 0.037 | 0.014 | 0.061 | 0.002 | -0.001 | 0.191 | 0.006
Uttar Pradesh | 71 | 0.147 | 0.394 | 0.018 | 0.030 | 0.116 | 0.002 | 0.079 | 0.026 | 0.003
West Bengal | 19 | 0.255 | 0.364 | 0.006 | 0.043 | 0.127 | 0.002 | 0.051 | 0.024 | 0.001
Year 2001
India | 593 | 0.236 | 0.808 | 0.007 | 0.077 | 0.192 | 0.004 | 0.074 | 0.123 | 0.005
Andhra Pradesh | 23 | 0.125 | 0.148 | 0.009 | 0.103 | 0.077 | 0.003 | -0.016 | 0.150 | 0.004
Arunachal Pradesh | 13 | 0.130 | 0.490 | 0.011 | 0.103 | 0.131 | 0.005 | 0.082 | 0.080 | 0.002
Assam | 23 | 0.252 | 0.357 | 0.012 | 0.077 | 0.099 | 0.004 | 0.146 | 0.041 | 0.005
Bihar | 37 | 0.152 | 0.461 | 0.004 | 0.102 | 0.140 | 0.006 | 0.081 | 0.081 | 0.006
Chattishgarh | 16 | 0.456 | 0.248 | 0.017 | -0.038 | 0.322 | 0.009 | 0.037 | 0.081 | 0.004
Delhi | 9 | -0.094 | 1.163 | 0.021 | 0.003 | 0.051 | 0.001 | 0.027 | 0.077 | 0.003
Gujarat | 25 | 0.360 | 0.419 | 0.018 | 0.019 | 0.168 | 0.003 | 0.051 | 0.065 | 0.002
Haryana | 19 | 0.263 | 0.168 | 0.006 | 0.031 | 0.086 | 0.003 | 0.018 | 0.112 | 0.003
Himachal Pradesh | 12 | 0.326 | 0.714 | 0.035 | -0.002 | 0.109 | 0.002 | 0.114 | 0.033 | 0.003
Jammu Kashmir | 14 | 0.166 | 0.766 | 0.019 | 0.182 | 0.077 | 0.005 | 0.166 | 0.083 | 0.017
Jharkhand | 18 | 0.083 | 0.661 | 0.014 | 0.134 | 0.173 | 0.006 | 0.060 | 0.130 | 0.003
Karnataka | 27 | 0.464 | 0.120 | 0.020 | 0.083 | 0.125 | 0.002 | 0.030 | 0.065 | 0.001
Kerala | 14 | 0.021 | 0.596 | 0.012 | 0.021 | 0.048 | 0.002 | 0.085 | 0.104 | 0.005
Madyapradesh | 45 | 0.104 | 0.357 | 0.004 | 0.030 | 0.163 | 0.004 | 0.061 | 0.090 | 0.002
Maharashtra | 35 | 0.449 | 0.298 | 0.013 | 0.040 | 0.098 | 0.001 | 0.030 | 0.061 | 0.001
Manipur | 9 | 0.350 | 0.409 | 0.029 | 0.098 | 0.077 | 0.003 | 0.024 | 0.075 | 0.002
Meghalaya | 7 | 0.380 | 0.527 | 0.018 | 0.202 | 0.025 | 0.005 | 0.034 | 0.075 | 0.001
Mizoram | 8 | 1.116 | -0.305 | 0.009 | -0.001 | 0.203 | 0.003 | 0.081 | 0.098 | 0.002
Nagaland | 8 | 0.284 | 0.261 | 0.015 | -0.005 | 0.353 | 0.006 | 0.044 | 0.178 | 0.008
Oddisha | 30 | 0.246 | 0.481 | 0.008 | 0.024 | 0.334 | 0.005 | 0.057 | 0.179 | 0.010
Punjab | 17 | 0.310 | 0.470 | 0.022 | 0.046 | 0.127 | 0.007 | 0.060 | 0.046 | 0.001
Rajasthan | 32 | 0.331 | 0.225 | 0.012 | 0.080 | 0.086 | 0.003 | 0.066 | 0.051 | 0.003
TamilNadu | 30 | 0.176 | 0.495 | 0.008 | 0.059 | 0.056 | 0.001 | 0.055 | 0.108 | 0.001
Uttarakhand | 13 | 0.364 | 0.552 | 0.023 | 0.040 | 0.081 | 0.003 | 0.007 | 0.208 | 0.004
Uttar Pradesh | 70 | 0.142 | 0.383 | 0.016 | 0.054 | 0.161 | 0.002 | 0.095 | 0.051 | 0.004
West Bengal | 18 | 0.312 | 0.395 | 0.010 | 0.063 | 0.177 | 0.002 | 0.077 | 0.042 | 0.002

Next, if we compare the fitted distribution (their estimated parameters) for
the two years 2011 and 2001, they changes more for the smaller states and the
states having significantly change in district numbers. Among the states
having mostly the same numbers of district in both 2001 and 2011, significant
change in the values of the estimated parameters are observed for Bihar and
meghalay while considering the LR distribution but the others remains mostly
the same. However, the distribution of WPR has significant changes in more
states including Assam, Delhi, Gujarat, Mizoram and Tamil Nadu.

Distributional Uncertainty Analysis:

We have revealed that the DGB distribution can completely characterize the
population size, LR and WPR distributions through two $sensors$ or $drivers$
modelled by two parameters ($\widehat{a}$ , $\widehat{b}$). The predicted
distribution fits very well with the observed rank-size data of all the
districts of India as well as the data of the districts within the states. We
now find the $entropy$ (S) values of all the considered distributions and take
resort to the proposed entity $Uncertainty$ $Percentage$ (UP) by standardizing
the $entropy$ $S$ values of the three variables of different Indian province.
The resulting values of UP are plotted in Figure 2 for both the years 2011 and
2001.

(a) Year 2011

(b) Year 2001

Figure 2: Plots of UP for Populations, LR and WPR for different states of
India in 2011 and 2001

One can see from Figure 2 that the UP values changes mostly for the population
and least for the WPR. So, there has been no significant changes in the
distributional uncertainty of the work participation rates in all states on
India. A little significant increase of 0.14% only is observed for the UP of
WPR in Assam whereas it changes less that 0.1% in all other states. The
changes in the UP values for LR is also less than 0.5% indicating no
significant changes in the distributional uniformity of literacy rates in all
the states in India. For the population, however, maximum increase (of 0.72%)
in the UP values is observed in West Bengal indicating greater uniformity of
its distribution within the states. On the other hand, in Karnataka and
Chattisgarh, the UP values of population distribution decreases more than 2%
indicating significant relocations of people within these states (mostly from
rural areas to major cities).

In absolute term, Andhra Pradesh, Madhya Pradesh and Haryana has maximum UP
values and hence more uniform population distribution, whereas those in
Mizoram and Uttarakhand are more sparse. In terms of literacy rate, however,
Delhi, Kerala and Uttarakhand are most evenly distributed states having
highest UP values and Chhattisgarh and Odisha have most uneven distribution of
LR (lowest UP values). Finally, Karnataka, West Bengal and Meghalaya have
highest UP values for WPR indicating most equally distributed work-
participation rates among their districts, whereas Jammu and Kashmir, Nagaland
and Odisha have most uneven workforce within the states.

Relationship among the variables:

We have verified both the Pearson (linear) and the rank (non-parametric)
correlation values between the variables population, LR and WPR in terms of
the estimated parameters and the UP values for the years 2001 and 2011
separately; the rank correlations are illustrated in the supplementary Figure
S1 (Appendix A). Interestingly, none of these correlations are significant at
95% level; the same is also observed for the Pearson correlations. This leads
to the inference that the distributional uncertainty among these three
indicators are mostly uncorrelated with each other across all the states of
India. It is an important observation in contrast to the fact that the
distribution of the numbers of literate and working persons in a population
depends directly (linearly) on the corresponding population distribution.

###  III.3 Gender based analyses

The study on gender composition largely reflect the underlying social,
economic and cultural patterns of the society in different ways. According to
United Nation estimates, the world had 986 females against 1000 males in 2000.
Interestingly the sheer weight of the population of the Asian countries,
particularly China (944) and India (933) with low sex ratio contributes to the
preponderance of males over females in world.

As per the Indian Census 2011, total population of India comprises of
62,37,24,248 males and 58,64,69,174 females with the sex ratio of 940 females
per 1000 males. As per Census 2011, top five states/Union territories which
have the highest sex ratio are Kerela (1,084) followed by Puducherry (1,038),
Tamil Nadu (995), Andhra Pradesh (992) and Chhattisgarh (991). Five states
which have the lowest sex ratio are Daman Diu (618), Dadra Nagar Haveli (775),
Chandigarh (818), NCT of Delhi (866) and Andaman Nicobar Islands (878). The
gender statistics revealed by Census India 2001, 532 million constituting 52
percent are males and 497 million constituting remaining 48 percent are
females in the population. Eighteen states/Uts have recorded sex ratio above
the national average of 933, while remaining seventeen falls below this.
Chandigarh and Daman Diu occupy the bottom positions with less than 800
females per 1000 males. Though the Census has shown an increase in the sex
ratio of total population from 933 in 2001 to 940 in 2011, it still requires
further improvement for the balance of male -female ratio.

In any country, higher literacy rates improve development indicators
consistently. In this respect, it is important to know the female percentage
of the literate persons, as that may lead to better attainment of health and
nutritional status, economic growth of community as a whole. The literacy rate
of India in 2011 is 74.0 per cent. Literacy rate among females is 65.5 per
cent whereas the literacy rate among males is 82.1 per cent showing a gap of
16.6 percent. If we see the data of 2001 census, the gap was 21.6 revealing
development in this index .

Moreover, the male and female working population and work participation rate
highlights the gender biased occupational distribution of a region. The Work
Participation Rate (WPR), which is defined as the percentage of total workers
to the total population, is 39.8 and 39.3 per cent as per the 2011 and 2001
census. While the WPR for males marginally increased from 51.6 percent to 51.9
percent during 1991-2001, for females it improved significantly from 22.7 to
25.7 during the corresponding period. Interestingly, this increase is mainly
due to increase in proportion of marginal workers which registered significant
increase from 3.4 percent to 8.7 percent. According to 2001 census, in the
total population percentage of male non-workers and female non-workers are
given by 48.1 percent and 74.3 percent respectively.

This gender based information of the census data inspired us to perform a
gender-based analysis of the distribution of important socio-economic
variables, such as population, literacy and work-participation rates of
different states across India. According to our expectation, the analysis also
revealed that both the male and female distributions fit very well with the
DGBD. Besides, the DGB distribution fits very well with the male-female (sex)
ratio of LR and WPR data for all the districts within the states of India with
a very low value of KS (RO). For this purpose, we have computed the Sex-Ratio
for the LR and WPR as follow:

Sex-Ratio of LR (SR-LR) $\displaystyle=$ $\displaystyle\frac{\mbox{number of
female literate in a district}}{\mbox{number of male literate in that
district}}\times 100,$ Sex-Ratio of WPR (SR-WPR) $\displaystyle=$
$\displaystyle\frac{\mbox{number of female workers in a
district}}{\mbox{number of male workers in that district}}\times 100.$

(a) SR-LR

(b) SR-WPR

Figure 3: Plot of the UP values for the gender based SR-LR and SR-WPR for different states in the years 2011 vs. 2001, along with the corresponding linear fits. The black solid line represent the $y=x$ line (linear growth) Table 2: The parameter estimates and the KS measures for different states along with total numbers (N) of districts and the UP values for Gender-based Analysis |  | SR-LR | SR-WPR
---|---|---|---
States | $N$ | $\widehat{a}$ | $\widehat{b}$ | KS | UP | $\widehat{a}$ | $\widehat{b}$ | KS | UP
Year 2011
Andhra Pradesh | 23 | 0.079 | 0.027 | 0.003 | 69.24 | 0.010 | 0.292 | 0.004 | 68.77
Arunachal Pradesh | 16 | -0.003 | 0.211 | 0.002 | 69.03 | 0.146 | 0.200 | 0.008 | 68.56
Assam | 27 | 0.027 | 0.018 | 0.001 | 69.30 | 0.125 | 0.255 | 0.022 | 68.48
Bihar | 38 | 0.031 | 0.036 | 0.001 | 69.29 | 0.178 | 0.162 | 0.010 | 68.60
Chhattisgarh | 18 | 0.017 | 0.079 | 0.001 | 69.26 | 0.028 | 0.157 | 0.006 | 69.10
Delhi | 9 | 0.052 | 0.007 | 0.001 | 69.29 | 0.145 | 0.174 | 0.012 | 68.65
Gujarat | 26 | 0.026 | 0.064 | 0.002 | 69.26 | 0.294 | 0.208 | 0.012 | 67.64
Haryana | 21 | -0.007 | 0.126 | 0.005 | 69.22 | 0.106 | 0.292 | 0.012 | 68.39
Himachal Pradesh | 12 | 0.090 | 0.068 | 0.002 | 69.15 | 0.133 | 0.144 | 0.006 | 68.81
Jammu Kashmir | 22 | 0.075 | 0.084 | 0.002 | 69.15 | 0.130 | 0.407 | 0.009 | 67.74
Jharkhand | 24 | 0.051 | 0.044 | 0.001 | 69.26 | 0.062 | 0.332 | 0.007 | 68.44
Karnataka | 30 | 0.064 | 0.046 | 0.001 | 69.24 | 0.065 | 0.162 | 0.003 | 69.01
Kerala | 14 | 0.023 | 0.046 | 0.002 | 69.28 | 0.039 | 0.292 | 0.014 | 68.65
Madhya Pradesh | 50 | 0.046 | 0.059 | 0.001 | 69.25 | 0.073 | 0.329 | 0.006 | 68.50
Maharashtra | 35 | 0.049 | 0.026 | 0.000 | 69.28 | 0.027 | 0.294 | 0.003 | 68.74
Manipur | 9 | 0.065 | 0.025 | 0.005 | 69.26 | 0.056 | 0.111 | 0.005 | 69.13
Meghalaya | 7 | 0.153 | 0.015 | 0.001 | 69.11 | 0.257 | -0.005 | 0.002 | 68.82
Mizoram | 8 | -0.014 | 0.163 | 0.003 | 69.16 | 0.114 | 0.035 | 0.004 | 69.16
Nagaland | 11 | 0.038 | 0.037 | 0.002 | 69.28 | -0.086 | 0.297 | 0.006 | 68.98
Odisha | 30 | 0.046 | 0.062 | 0.002 | 69.24 | 0.023 | 0.702 | 0.014 | 67.03
Punjab | 20 | 0.038 | 0.024 | 0.001 | 69.26 | 0.245 | 0.006 | 0.011 | 68.80
Rajasthan | 33 | 0.034 | 0.081 | 0.001 | 69.24 | 0.086 | 0.141 | 0.003 | 69.01
Tamilnadu | 32 | 0.047 | 0.030 | 0.001 | 69.28 | 0.117 | 0.192 | 0.007 | 68.75
Uttarakhand | 13 | 0.067 | 0.068 | 0.002 | 69.19 | -0.134 | 0.924 | 0.012 | 66.39
Uttar Pradesh | 71 | 0.047 | 0.047 | 0.001 | 69.26 | 0.142 | 0.314 | 0.019 | 68.28
West Bengal | 19 | -0.009 | 0.100 | 0.004 | 69.25 | 0.180 | 0.197 | 0.008 | 68.41
Year 2001
Andhra Pradesh | 23 | 0.117 | 0.045 | 0.003 | 69.13 | -0.027 | 0.387 | 0.005 | 68.57
Arunachal Pradesh | 13 | 0.038 | 0.203 | 0.006 | 68.95 | 0.101 | 0.231 | 0.003 | 68.63
Assam | 23 | 0.026 | 0.037 | 0.003 | 69.29 | 0.193 | 0.314 | 0.014 | 67.81
Bihar | 37 | 0.045 | 0.091 | 0.002 | 69.20 | 0.153 | 0.292 | 0.014 | 68.22
Chhattisgarh | 16 | 0.089 | 0.073 | 0.005 | 69.14 | 0.056 | 0.116 | 0.008 | 69.13
Delhi | 9 | 0.052 | 0.029 | 0.002 | 69.27 | 0.164 | 0.148 | 0.009 | 68.67
Gujarat | 25 | 0.022 | 0.118 | 0.002 | 69.19 | 0.186 | 0.227 | 0.004 | 68.27
Haryana | 19 | 0.063 | 0.036 | 0.001 | 69.25 | 0.041 | 0.445 | 0.007 | 68.02
Himachal Pradesh | 12 | 0.134 | 0.082 | 0.005 | 69.00 | 0.166 | 0.054 | 0.002 | 68.97
Jammu Kashmir | 14 | 0.112 | 0.081 | 0.003 | 69.07 | 0.140 | 0.446 | 0.012 | 67.37
Jharkhand | 18 | 0.097 | 0.099 | 0.002 | 69.07 | 0.041 | 0.389 | 0.014 | 68.27
Karnataka | 27 | 0.104 | 0.064 | 0.002 | 69.13 | 0.034 | 0.196 | 0.004 | 69.00
Kerala | 14 | 0.022 | 0.047 | 0.001 | 69.28 | 0.037 | 0.374 | 0.006 | 68.32
Madhya Pradesh | 45 | 0.055 | 0.098 | 0.002 | 69.18 | 0.070 | 0.280 | 0.006 | 68.66
Maharashtra | 35 | 0.067 | 0.047 | 0.001 | 69.23 | -0.022 | 0.381 | 0.012 | 68.61
Manipur | 9 | 0.043 | 0.057 | 0.002 | 69.25 | 0.051 | 0.076 | 0.005 | 69.21
Meghalaya | 7 | 0.142 | 0.063 | 0.003 | 69.03 | -0.010 | 0.208 | 0.002 | 69.05
Mizoram | 8 | 0.006 | 0.118 | 0.002 | 69.21 | 0.096 | 0.023 | 0.003 | 69.21
Nagaland | 8 | 0.065 | 0.053 | 0.002 | 69.22 | -0.172 | 0.480 | 0.020 | 68.61
Odisha | 30 | 0.057 | 0.148 | 0.007 | 69.06 | 0.029 | 0.719 | 0.023 | 66.90
Punjab | 17 | 0.038 | 0.037 | 0.001 | 69.28 | 0.172 | 0.170 | 0.005 | 68.56
Rajasthan | 32 | 0.095 | 0.073 | 0.006 | 69.14 | 0.072 | 0.190 | 0.003 | 68.91
Tamilnadu | 30 | 0.068 | 0.036 | 0.002 | 69.24 | 0.027 | 0.323 | 0.005 | 68.63
Uttarakhand | 13 | 0.034 | 0.134 | 0.003 | 69.13 | -0.185 | 1.054 | 0.020 | 69.13
Uttar Pradesh | 70 | 0.061 | 0.101 | 0.002 | 69.17 | 0.098 | 0.578 | 0.022 | 67.45
West Bengal | 18 | -0.015 | 0.161 | 0.004 | 69.17 | 0.181 | 0.354 | 0.008 | 67.64

We apply our proposed analysis methodology for these two variables, depicting
the gender based inequalities in LR and WPR, and the resulting parameter
estimates and the UP values are presented in Table 2. For better understanding
of the change in distributional uncertainty from the year 2001 to 2011, the UP
values for both SR-LR and SR-WPR are also presented graphically in Figure 3
along with the linear fits. It can be seen that, on an average, there are
sublinear growth of the distributional uncertainty (UP values) for both SR-LR
and SR-WPR between years 2001 and 2011, indicating reduced gender-
discriminations in literacy rate as well as in work-participation rate. For
SR-LR, all states indeed have increased UP value in 2011, except for Haryana
and Mizoram. The lowest UP value (most discrimination) is observed in
Arunachal Pradesh for SR-LR and in Uttarakhand for SR-WPR. In general, the
distributional uncertainty for SR-WPR remains mostly stable in the states
having higher UP values, with slight uniform variations.

##  IV Conclusion

In this paper we have quantified the distributional uncertainty present in the
various socio-economic indicators of all the Indian states, by studying the
distribution of the districts within each individual states. The analysis is
based on the DGB distribution, a typical rank-order distribution, which fits
very well across many disciplines of Arts and Science. Despite the immense
diversity of human settlements and extraordinary geographic variability, we
have shown that all states obey DGB distribution not only for population size
but also for other socioeconomic factors like literacy rate and work
participation rate. The two model parameters of the distribution for different
states construct a cluster. With the evolution of time there is no noted
change in this cluster structure. Our primary analytical focus here is
concerned with the entropy of the distribution. Although, the value of the
entropy for a particular state for all factors are different, the Uncertainty
Percentage (UP), a measure defined for the uncertainty, is almost the same for
all the states of different size, culture and economic and social conditions.
This is one of the most intriguing outcome of this analysis. The results also
indicate that this value of UP is also nearly equal to the value of the UP
when it is studied with all the districts across the states. Remarkably, the
study for different years also indicate a similar result.

It is important to note that the proposed UP measure of distributional
uncertainty is quite a general formulation which can further be applied to
analyze the distributions of different important socio-economic variables of
any country having stratified administrative structure. Beside such
applications, it would also be important to investigate the detailed
theoretical properties of the proposed UP measure as possible future research
works.

Acknowledgment:
OM is thankful to Indian Statistical Institute, Kolkata, India for allowing
her to work as a Visiting Student at the Physics and Applied Mathematics Unit
of the Institute. SC thanks Science and Engineering Research Board (SERB),
Government of India for financial support through Junior Research Fellowship
(Grant No.CRG/2019/001461). The research of AG and BB is partially supported
by a research grant (No. CRG/2019/001461) from the Science and Engineering
Research Board (SERB), Government of India.

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##  Appendix A Supplementary Tables and Figures

Table S1: Population (Pop), Literacy Rate (LR) and Work Participation Rate (WPR), along with the number of districts ($N$), for every states of India | 2011 | 2001
---|---|---
State | $N$ | Pop | LR | WPR | $N$ | Pop | LR | WPR
Andaman Nicobar (UT) | 3 | 380581 | 77.32 | 40.08 | 2 | 356152 | 71.07 | 38.26
Andhra Pradesh | 23 | 84580777 | 59.77 | 46.61 | 23 | 76210007 | 52.40 | 45.79
Arunachal Pradesh | 16 | 1383727 | 55.36 | 42.47 | 13 | 1097968 | 44.15 | 43.98
Assam | 27 | 31205576 | 61.46 | 38.36 | 23 | 26655528 | 52.58 | 35.78
Bihar | 38 | 104099452 | 50.44 | 33.36 | 37 | 82998509 | 37.48 | 33.70
Chandigarh (UT) | 1 | 1055450 | 76.31 | 38.29 | 1 | 900635 | 71.42 | 37.80
Chattisgarh | 18 | 25545198 | 60.21 | 47.68 | 16 | 20833803 | 53.63 | 46.46
Dadra - Nagar Haveli (UT) | 1 | 343709 | 64.95 | 45.73 | 1 | 220490 | 47.12 | 51.76
Daman - Diu (UT) | 2 | 243247 | 77.45 | 49.86 | 2 | 158204 | 68.01 | 46.01
Delhi (UT) | 9 | 16787941 | 75.87 | 33.28 | 9 | 13850507 | 69.78 | 32.82
Goa | 2 | 1458545 | 79.91 | 39.58 | 2 | 1347668 | 73.13 | 38.80
Gujarat | 26 | 60439692 | 67.99 | 40.98 | 25 | 50671017 | 58.87 | 41.95
Haryana | 21 | 25351462 | 65.48 | 35.17 | 19 | 21144564 | 57.20 | 39.62
Himachal Pradesh | 12 | 6864602 | 73.42 | 51.85 | 12 | 6077900 | 66.50 | 49.24
Jammu Kashmir | 22 | 12541302 | 56.35 | 34.47 | 14 | 10143700 | 47.39 | 37.01
Jharkhand | 24 | 32988134 | 55.56 | 39.71 | 18 | 26945829 | 43.71 | 37.52
Karnataka | 30 | 61095297 | 66.53 | 45.62 | 27 | 52850562 | 57.59 | 44.53
Kerala | 14 | 33406061 | 84.22 | 34.78 | 14 | 31841374 | 80.04 | 32.30
Lakshadweep (UT) | 1 | 64473 | 81.51 | 29.09 | 1 | 60650 | 73.67 | 25.32
Madhya Pradesh | 50 | 72626809 | 59.00 | 43.47 | 50 | 60348023 | 52.35 | 42.74
Maharashtra | 35 | 112374333 | 72.57 | 43.99 | 35 | 96878627 | 66.03 | 42.50
Manipur | 9 | 2855794 | 66.83 | 45.68 | 9 | 2293896 | 57.13 | 41.21
Meghalaya | 7 | 2966889 | 60.16 | 39.96 | 7 | 2318822 | 49.93 | 41.84
Mizoram | 8 | 1097206 | 77.30 | 44.36 | 8 | 888573 | 74.44 | 52.57
Nagaland | 11 | 1978502 | 67.85 | 49.24 | 8 | 1990036 | 56.90 | 42.60
Odisha | 30 | 41974218 | 63.71 | 41.79 | 30 | 36804660 | 53.90 | 38.79
Puducherry (UT) | 4 | 1247953 | 76.71 | 35.66 | 4 | 974345 | 71.47 | 35.17
Punjab | 20 | 27743338 | 67.43 | 35.67 | 17 | 24358999 | 60.58 | 37.47
Rajasthan | 33 | 68548437 | 55.84 | 43.60 | 32 | 56507188 | 49.02 | 42.06
Sikkim | 4 | 610577 | 72.87 | 50.47 | 4 | 540851 | 58.86 | 48.64
Tamilnadu | 32 | 72147030 | 71.85 | 45.58 | 30 | 62405679 | 64.94 | 44.67
Tripura | 4 | 3673917 | 76.34 | 40.00 | 4 | 3199203 | 63.21 | 36.25
Uttar Pradesh | 71 | 199812341 | 57.25 | 32.94 | 70 | 166197921 | 45.56 | 32.48
Uttarakhand | 13 | 10086292 | 68.22 | 38.39 | 13 | 8489349 | 60.14 | 36.92
West Bengal | 19 | 91276115 | 67.42 | 38.08 | 18 | 80176197 | 58.87 | 36.77

(a) Year 2011

(b) Year 2011

(c) Year 2011

(d) Year 2001

(e) Year 2001

(f) Year 2001

Figure S1: Spearman Rank Correlation of the estimated parameters and UP values
for Populations, LR and WPR in 2011 and 2001