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NicerTopology

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vital statistics population studies demography health statistics

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This document covers vital statistics, including methods of obtaining data, measurements of population, sex ratio, mortality, fertility, and population growth. It details various rates like crude death rates, age-specific death rates, and fertility rates. The document analyzes different methods for measuring and understanding population trends.

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# VITAL STATISTICS The events related to human life are known as vital events. **Eg:** Birth, Death, divorce, marriage, immigration, emigration. The statistics based on vital events is known as vital statistics. **Uses of vital statistics:** 1. To study the pop" trend. 2. Used in public admini...

# VITAL STATISTICS The events related to human life are known as vital events. **Eg:** Birth, Death, divorce, marriage, immigration, emigration. The statistics based on vital events is known as vital statistics. **Uses of vital statistics:** 1. To study the pop" trend. 2. Used in public administration 3. used in research and medical 4. Used in government and private agencies 5. Used in actuarial science. **Method of obtaining vital statistics** **OR** **Source of demographic data** The vital statistics data can be obtained by the following method 1. Registration method 2. Census method 3. Hospital Records 4. Adhoc survey. **Registration Method** :- The most important source of obtaining vital statistic data is the registration method. In registration method data related to birth, marriages, divorce, immigration, emigration is recorded continuously. These data are also legal documents. Registration of Birth provide information on place of Birth, sex, age, religion of Parents, occupation of parents etc. Registration of death provides information on place of death, age, sex, cause of death etc. Similar information can be obtained four marriage, divorce, emigration, immigration. In rural India many vital events (birth, death, marriage etc) are unregistered. Similarly data relating to cause of death, disease is not available in rural areas. The religions customs do not require compulsory registration of marriages in Hindu and Muslims. Hence we don’t get reliable data of marriages for the whole country. **Census method** :- In almost all the countries population census is conducted at the regular interval of time usually after 10 years. Census consist of complete enumeration of the population and data is collected regarding age, sex, marital status occupation, religion etc. The main drawback of census method is that vital statistic is available only for the census year. If data is required for all the year then information is not available. # **MEASUREMENT OF POPULATION** :- * **Pt:** Pop" at any time it. * **Po:** Pop" at last census. * **B:** The total number of births in a given region during the given period. * **D:** Total no. of deaths in a given region during the given period. * **I:** Total no. of immigrants in a given region during a given period. * **E:** Total no. of emigrants in a given region during a given period. **P t = P o + (B-D) + (I-E)** # **Sex Ratio** :- The sex ratio is defined as the total no. of females per thousand males. **Sex Ratio = female pop" x 1000/ male Pop"** **= fp x 1000 / mp** # **Measurement of mortality:** Crude death rate is defined as - **m = D x 1000/ P** * **m = C.D.R = deaths accured in a given region during the given time period.** * **D = No. of deaths accured in a given region during the given time period.** * **P = Total population in a given region during the given time period.** **CDR for male = male death x 1000 / male popn** **CDR for female = female death x 1000 / female popn** # **Merits of CDR** :- 1. It is easy to calculate and simple to follow 2. It is most widely used 3. It represents prob. of dying for a person in a given region during a given period. # **Demerits of CDR** :- 1. Ignores the age and sex distribution of the popn. 2. Ignores the fact that prob. of dying is not same for young and old or for males and females. 3. It's not suitable for comparing the mortality pattern of two places. # **Specific death rate (SDR)** :- **SDR = No. of deaths accured in a specific section of the pop" in a given region during a given period x 1000 / Total no. of person in a " " " " " "** **→Specification is being done w.r.t age and sex** # **Age specific death rate:** * **nDx = No. of death in the age gp (x, x+n)** * **nPx = Total pop" in the age gp (x, x+n)** * **ASDR for the age gp (x, x+n)** * **nmx = nDx / nPx** * **ASDR for the age gp (x, x+n) = nmx x 1000** **Taking n = 1** **Annual age specific death rate is** * **Mx = Dx / Px x 1000** * **ASDR for male = mMx = mDx / mPx x 1000** * **ASDR for female = fMx = fDx / fPx x 1000** # **Merits of ASDR** :- 1. It is better than CDR because age and sex distribution of popn is considered. 2. It helps in finding NRR. 3. It helps in construction of life table. # **Demerits** :- 1. It is not a better measure for comparing mortality pattern of two places. # **Infant mortality Rate** :- Infant mortality rate is defined as the probability of dying a newly born infant within a year of its life. **Do² = No. of death (excluding foetal death) of children in age gp. (0,1) in a given region during the given year Z** **Bo² = Total no. of live birth in a given region during the given year Z.** **The infant mortality rate is defined as -** **IMR = Im = Do²/ Bo² x 1000** # **Standardized death rate (STDR)** :- STDR is used to compare the mortality of two places or mortality of same places over two time period. The CDR for two regions A & B are given as - ** CDR for A = ma = Da x 1000 / Pa = Σ m aP a / Σ P a ** **CDR for B = mb = Db x 1000 / Pb = Σ m bPb / Σ Pb** * CDR defined above is weighted mean of age SDR. * Weights being equal to the corresponding pop". * CDR for two popn do not same since age dist. of two pop" in not same . This drawback is removed by fallowing two method: # **1) Direct method of standardisation:-** In direct method of standardisation 3rd population is taken as standard pop" denoted as Px **For example: if we have to compare mortality pattern of two region A and B** * **A = UP** * **B = Bihar** 3rd popn is taken as pop" of India denoted by Px. The weighted mean of age SDR for A is called standardised death rate for A. ** (STDR) A = Σ m aP a / Σ P a = Σ m bP a / Σ P a [No. person should be known for SDR]** ** (STDR) B = Σ m aP b / Σ P a = Σ m bP b / Σ P a** If popn is standardised with respect to age and sex both then - * **(STDR) A = ( Σ m aP a + Σ f aP a / Σ m sP a + Σ f sP a)** * **(STDR) B = ( Σ m aP b + Σ f aP b / Σ m sP b + Σ f sP b )** # **Merits** :- 1. It is easy to calculate and simple to follow 2. It use age SDR which gives the clear picture of the population. # **Demerits:-** 1. In calculation of STDR the main problem is choice of standard pop". # **2) Indirect method of standardisation :-** In calculation of STDR from direct method of standardisation, the no. of person and specific death rates for all segment of the popn should be known. If specific death rate is not known for any age gp, only total no. of deaths out CDR is known them indirect method of standardisation is used to find STDR. Let C be adjustment factor **STDR = (CDR) x C** **(STDR) A = (CDR for A)x C** **Σ m aP a / Σ P a = ( Σ m aP a / Σ P a ) x C [ Σ m aP a / Σ P a = Σ m aP a / Σ P a ) x C ]** **where C = Σ m aP a / Σ P a / Σ m aP b / Σ P b** Since m a is unknown hence it is replaced by m x to estimate the value of C where m x is age SDR for standard popn. **m x = Σ m aP a / Σ P a / Σ m aP b / Σ P b** **Hence C = Σ m aP a / Σ P a / Σ m aP b / Σ P b** ** (STDR) A = (CDR) A x C** # **Fertility** :- It measures the live births. **Measures of fertility:-** **Coude Birth rate (CBR):** CBR is defined as - **i' = B x 1000/ P** * **i' = CBR per 1000 of the pop"** * **B = No of live births occured in a given region during a given period** * **P = Total popn in a given region during a given period** **Merits and Demerits:-** 1. It is easy to calculate and simple to follow 2. CBR is not a better measure of fertility since it ignores the age and sex dist of the popn 3. CBR is not a probability ratio since the whole pop" is not exposed to risk 4. CBR can not be used to compare fertility pattern of two places # **General fertility rate (GFR):** GFR is defined as - **GFR = i = B / Px x 1000** * **i = GFR per 1000 females in the reproductive period** * **B = No. of live births accured in a given region during the given period** * **fPx = No. of females in reproductive aged in a given region during the given period** * **λι and λ2 = lower and upper limits of female reproductive period** * **Generally λ1 = 15, λ2 = 49** **Merits and Demerits:-** 1. GFR is a probability ratio since it takes into account the total female popn at risk. 2. GFR reflects how much popn is increased by female popn in the reproductive period. 3. GFR may not be used for comparing fertility pattern of the two places. # **Specific fertility rate (SFR):** It is defined as - **SFR = No. of births to the popn in the specific section of the pop" in a given region during the given period x 1000/ Total no. of females in the specific section of the pop" in a given region during the given period** # **Age specific fertility (ASFR):** ASFR for age group (x, x+n) is defined as- **nix = nBx / nPx x 1000** * **nBx = No. of live births to the female pop" in the age gp (x, x+n)** * **nPx = Total no. of females in the age gp (x, x+n)** **Taking n = 1, annual ASFR is defined as-** * **ix = Bx / fPx x 1000** Where, * **ix = annual age SFR per 1000** * **Bx = No. of live births to the female aged x in a given region during the given period** * **fPx = Total pop" of females aged x in a given region during the given period.** # **Total fertility rate (TFR):** TFR is obtained by adding the annual age SFR. Thus TFR is defined as- **TFR = Σ ix = Σ Bx/ fPx x 1000 ** Where, * **ix = annual age SFR per 1000** * **Bx = No. of live birth to the female aged x in a given region during the given period** * **fPx = Total population of females aged x in a given region during the given period** * **λ1, λ2 = lower and upper limit of female reproductive period** * **Generally λ1 = 15, λ2 = 49** TFR gives the no. of children born per 1000 females in the child bearing ages. It is approximately given by the formula. **TFR = Σ n(nix) [ie n=5]** If we deal with quinquennial age gp the TFR become **TFR = Σ 5(six) = 5 Σ (six)** # **Measurement of popn growth :-** Fertility rates are inadequate to give any idea about the rate of pop" growth since they ignore the mortality pattern. # **1) Crude rate of natural increase and Pearle’s vital Index :-** It is the simplest measure of pop" growth. **Crude rate of natural increase = CBR-CDR** **Pearle’s vital index = CBR x 100 / CDR** where, * **CBR = Crude birth rate per 1000** * **CDR = Crude death rate per 1000** Among the above two formula of popn growth Pearle’s vital index is more reliable. * If Pearle’s vital index < 100, * then there is no population growth it shows decay of the popn. * If Pearle’s vital index = 100. then it shows that population is constant ie, there is no increase or decrease in the population. * If Pearle’s vital index > 100, popn is increasing ie, popn has good medical cause. The main drawback of this method is that it does not tell us whether the popn has tendency to increase or decrease. # **2) Gross reproduction rate (GRR):** To obtain the rate of popn growth the sex of new born child is also taken into account ie, female births are taken into account in calculation of GRR. **GRR = ( Σ λ1, λ2 fBx / fPx x 1000 ) = ( Σ λ1, λ2 fBx / fPx x 1000 )** Where, * **fBx = no. of female births to the women of age x in a given region during the given period.** * **fPx = Total no. of females aged x in a given region during the given period.** * **λ1, λ2 = Lower and Upper limit of female reproductive period** * **fix = Annual age specific FR of females per 1000** GRR is a modified form of TFR. GRR tell us the vrato at which mothers would be replaced by daughters. GRR is approximately given by the formula. **GRR = Σ n(nix) = Σ (n fBx / fPx x 1000)** Where, * **nix = Female ASFR per 1000** * **nBx = No. of live births of females to the women in the age gp (x, x+n)** * **fPx = Total females pop" in the age gp (x, x+n)** If we deal with quinquenneal age gp ie n = 5 then GRR become. **GRR = Σ 5(six) = 5 Σ (six )** **GRR = No. of female births x TFR / Total no. of births** GRR is useful for comparing fertility pattern of two places or for comparing fertility pattern of same place for different time periods. * **If GRR < 1, then popn will decline.** * **If GRR = 1, then popn will remain constant.** * **If GRR > 1, then popn will increase.** Theoretically, OS GRR ≤ 5. The main drawback of GRR is, that it does not take into account the mortality pattern of female births. # **3) Net reproduction rate (NRR):** The main drawback of GRR is that it completely ignores the current mortality and takes into account only current fertility. GRR adjusted for the effects of mortality is known as NRR. As a measure of population growth NRR is defined as - **NRR = Σ n (fBx / fPx ). f n x x 1000** **= Σ n (mIx) . ( f n x) x 1000** **= 1000 Σ n. (Female age SFR) (Survival factor for females)** Where, * **fBx = No, of live births of females to the women in th age gp. (x, x+n)** * **f Px = Total popn of female in the age gp (x, x+n)** * **f nx = Survival factor** * **NRR ≤ GRR ; GRR is upper limit of NRR.** * **NRR = GRR; If a new born girl survives till the end of reproductive periodb.** Theoretically, OS NRR ≤ 5. * **NRR < 1, the popn is said to have a tendency to decrease** * **NRR = 1, the popn is said to have a tendency to remain constant** * **NRR > 1, the popn is said to have a tendency to increase.** # **Net Reproduction rate per women = Σ fBx x f nx / 1000** # **Mortality Table or Life table** The life table gives the life history of a hypothetical group or cohort. Life table is used for measuring the prob. of life and deaths of various age sectors. The data for the construction of life table are the census data and death registration data. # **Notation :-** * **lo = Cohort or radix of life table** * **l x = No. of persons living at any specified age x** * **dx = No. of persons among the l x persons who died before reaching the age x+1** * **dx = l x- l x+1 = (l x+1 - l x) = - (Δl x )** * **nPx = Probability that a person aged x survives upto age x+n** * **nPx = l x +n / l x for n>1; Px = l x+1 / l x** * **qx = 1-px = px** * **qx = Prob. that a person aged x will die within 1 year.** **qx = lx- l x+1 / l x** ** px = l x+1 / l x = px l x = (1-qx) l x = l x - qx l x** ** l x+1 = l x - dx** * **l x+1 = l x - dx** * **l x = The no. of years lived in aggrigate by the cohort of lo person b/w age x to x+1** ** L x = ∫l x+t dt** **l x = l x-1 - dx** **= l x -1 [l x - l x+1]** **= l x -1 - l x + l x+1** **= 1/2 ( l x + l x+1)** * **Tx = Total future life time of the persons who reached age x** **Tx: l x + l x+1 + l x+2 + - - - - - l w-1** * **where w is the highest age, of survival s.t lw = 0** **Tx = Σ (x+i) = l x + Tx+1** **(l x = Tx - Tx+1)** # **Expectation of life** The curate expectation of life ex. It is the avg no. of complete years of life lived by cohort of lo persons after age x. # **The complete expectation of life ex:-** It is the avg. no. of years a person aged x is expected to live under the prevailing mortality conditions. **e x = ( e x+1 t +1 / l x )** **e x = Tx / l x** **P x = e x +1 / 1 + e x+1** **qx = 1-px** # **Central mortality Rate (mx):-** It is defined as the ratio of no. of deaths with in ( x, x+1) to l x with in ( x, x+1) **mx = No of deaths within (x,x+1) / l x** **= dx / l x** **= dx / l x-1 - dx** **( dx / l x - dx ) = ( 1/2 dx )** **= ( qx l x / l x -1 -qx l x ) ** **= 2qx / 2 - 2qx** **= 2mx / 2 + mx** **d lx 0 ; q1 = 2mx / 2 + mx** # **Force of Mortality (µx):** The force of mortality at age x ( µx ) is defined as the ratio of rate of decrease in l x to the value of l x . It gives annual rate of mortality ie, the prob. of a person of age x exactly dying with in the year of risk of dying is same. **µx = -d ( 1/l x ) / dx** **= -1 / l x ( dx / dx)** **= -1 / l x ( dx / dx )** **= - d ( loge l x ) / dx** **µx+1 = mx** # **Assumptions of life table:-** 1. The cohourt is closed to immigration or emigration. 2. The mortality pattern of individual is predeterminec or fixed. 3. Deaths are uniformly distributed b/w 1st birthday and next birthday. 4. The cohort is usually taken as 10,000 or 1,00,000, and it originates from standard population. # **Discription of life table our construction of life table on columns of life table:** There are 11 columns in a life table. **(1)** (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) **l x** **d x** **dx** **qx** **px** **mx** **Lx** **Tx** **ex** **e x** **Steps for construction of life table are:-** * **For x = 0, 1, 2, - - - ** * **(1)** **dx = l x- l x+1** * **(2)** **qx = dx / l x** * **(3)** **px = 1-qx** * **(4)** **mx = 2qx / 2 -qx** * **(5)** **mx = 2mx / 2 + mx** * **(6)** **Lx = ( l x + l x+1) ** * **(7)** **Tx = Lx + Lx+1 + Lx+2 + - - - -** * **(8)** **e x = Tx / l x** * **(9)** **e x = e x-1 - 1/2** **The qx - column is called the pivotal column of life table. The above life table is called a complete life table.** # **Uses of life table:** 1. It gives a clear picture of age dist" of mortality for a given popn. 2. Used in demographic and public health studies. 3. To find rate of premium in insurance 4. For studying rate of population growth. 5. For determining rate of retirement benefits. 6. for comparing mortality pattern of two places. # **Stationary Population:** A popn is said to be stationary if it is of constant size, const. age and sex composition. In stationary popn. no. of births is assumed to be equal to no. of deaths. → Stationary pop" is not affected by immigration or emigration # **Stable population:** (A.J Lotka) A popn. is said to be stable if it has fixed age and sex dist", constant mortality and fertility rates. Stable pop" is closed to immigration or emigration. **Rate of overall change in pop" = 0; for stationary popn.** **Const; for stable popn.** Note: A stationary pop" is always stable but a stable pop" need not to be always stable. # **Lotka and Dublin’s Model for stable pop" :-** **Assumptions:-** 1. The fertility rates are indep, of time (t). 2. The mortality rates are indep. of time (t). 3. The age dist" b/w ages (x, x+dx ) is indep . of time (t). 4. The pop" is closed to migration. 5. The analysis is done w.r.t female pop" only. **Model**:- * **P(t) = Size of the pop" at any time t** * **l(x,t) = Proportion of the pop" in the age gp (x,x+dx) at time t** * **B(t) = Total no. of births at time t** * **p(x) = Prob. that a female child will survive upto age x under the given mortality condition.** * **l (x, ∞) = Prob. that a women will give birth to a female child when it is in the age gp (x, x+dx) .** **Then** **P(t) C(x,t) S x = B(t-x). p(x) S(x) = pop" in the age gp (x, x+dx) at time t** **P(t). C(x,t) B(t-x). p(x)** Multiplying both sides by 1 and integrating w.r.t x over (0, ∞). We get - **∫ P(t) C(x,t) l(x) dx = ∫ B(t-x) p(x) l(x) dx [1]** It is integral eqn with lag x. Lotka and Dublin suggested a Grail solution to this eqn given as- **B(t) = Σ Qn. eint [n ≥ 0]** * **where Q0, Q1, Q2….. are the size of the pop" and a0, a1, a2…. are growth rate of the popn. Putting this value in eqn ①** **Σ Qn. eint = Σ Qn. eint ∫ e-in(t-x) p(x) l(x) dx** **ie Σ Qn. eint = Σ Qn. eint ∫e-inx. p(x) l(x) dx** Which will be true iff **∫ e-inx p(x). l(x) dx = 1; for n = 0. 1. 2... , ∞** **⇒ a0, a1, a2...... are the root of the eqn.** **∫ e-ax. p(x). l(x) dx = 1** **This is known as "Lotka’s integral eqn."** **∫ e-ax dx / ∫ e-a x Ф(x) dx = 1 ** **[where Ф(x) = p(x) l(x) = net maternity b "** **Differentiating w.r.t a'** **dy / da = ∫e-ax (- x) Ф(x) dx = ( ∫∞ to ∞ -x e- ax Ф(x) dx ) / (∫∞ to ∞ e-ax Ф(x) dx ) ** **= -A(α).y** **where A(α ) = ∫∞ to ∞ e-ax Ф(x) dx / ∫∞ to ∞ e-ax Ф(x) dx ** **A(α) = ∫∞ to ∞ ( 1-α x + α²x²/2! +….. ) Ф(x) dx / ∫∞ to ∞ e-ax Ф(x) dx ** **= ∫∞ to ∞ ( 1 - α x + α²x²/2! +….. ) Ф(x) dx / ∫∞ to ∞ e-ax Ф(x) dx ** ** = ∫∞ to ∞ [ Σ ( -1)i αi xi / i! ] Ф(x) dx** **= ∫∞ to ∞ [ Σ ( -1)i αi xi / i! ] Ф(x) dx** **= 1/ 1! ∫∞ to ∞ Ф(x) dx - α / 2! ∫∞ to ∞ x Ф(x) dx + α² / 3! ∫∞ to ∞ x² Ф(x) dx - - - - - ** **= (1/ 1!) Σ( -1)i / i! {Ri + 1} ; where Ri = ∫∞ to ∞ xi Ф(x) dx ** **= (1/ 1!) Σ( -1)i / i! {Ri} )** A(α) = R1 -α R2 + α²/2! R3 - - - - - **= R1 -α R2 + α²/2! R3 - - - - - / R0- α R1 + α²/2! R2 - - - - -** **= (R1 (1-αR1 + (1-α) R1 ) / ( R0 (1-αR1 + (1-α) R1 )** **= ((R1 / R0 )(1-αR1 / R0 ) / ( 1-αR1 / R0 ) ** **A(α) = (R1 / R0 )(1-αR1 / R0 ) / ( 1-αR1 / R0 ) ** **=(R1 / R0 )(1+αR1 /R0) / (1 + αR1 /R0)** **=(R1 / R0 ) + α( (R1 /R0 )² - ( R1 /R0 ) ) / (1 + αR1 / R0 )** **=(R1 / R0) + a ( (R1 / R0 )² - ( R1 /R0 ) ) / ( 1 + αR1 / R0 )** **= a + β / 1 + αp** **Integrating w.r.t a** **∫A(α) da = α + β∫ da / 1 + αp** **P0 + 2cα - 2log R0 = 0** **a = R1 / R0 ** **β = ( (R1 / R0 )² - ( R1 / R0 ) ) / ( 1 + αR1 / R0 ) ** Where, * **R0, R1 and R2 are estimated as -** * **R 0 = Σ Фi(x) = NRR for females** * **R 1 = Σ x Фi(x)** * **R 2 = Σ x² Фi(x)** * **R0 = Σ Фi(x)** * **R1 = Σ x Фi(x)** * **R 2 = Σ x² Фi(x)** Putting values of R0, R1 and R2 in egn @ we get value of α or β. The value of α give growth parameter of stable popn. # **Abridged Life table:-** In complete life table the values of life table for ex, dx, bx, qx, - - - etc are given for the year. In Abridged life table the values of life table for de - - - etc are not given for all the years ie, they are given at a distance of 5-years or 10 years. The methods of construction of Abridged life table are- 1. Reed- Merrel Method (q.d, P, m) 2. Greville’s Method 3. King’s Method 4. Chicing’s Method # **Construction of Abridged life table** There are 7 columns of Abridged life table **(1)** (2) (3) (4) (5) (6) (7) **l x** **nLx** **ndx** **nlx** **Tx** **ex** **Rnown** * **For x = x0 , x0 + n, x0 + 2n, - - - - -** * **(1) nqx = 1- np x = ndx / l x** * **(2) nqx = ( ndx / l x ) = ( ndx / l x - dx ) = nqx. l x** * **(3) nLx = ∫ l x+t dx = ½ (l x+ l x+n)** * **(4) e x = Tx / l x** # **Reed Merrel Method :-** **nqx = The prob. that a person who is in the age gp. (x, x+n) will die un calendler year (z)** **nqx = ndx / l x** **ndx = No. of deaths in the age gp (x, x+n) in the calendler year z** **ndx = l x - l x+n** **nPx = The prub. that a person who is in the age gp (x, x+n) will survive in the calendler year z.** **nmx = Central mortality rate for the age gp (x, x+n) in the calendler year z.** **then ngx = 2n(nmx²) / 2+n(nmx²) ** **where nmx = ndx / nPx** **Proof:** * **dx = Ex² - 1 / 2 ( lx - l x+n )** Assuming that deaths are uniformly distributed **nPx = ∫ l x+t. dt = 1/2 ( l x+ l x+n )** * **nPx = 1/2 ( l x+ l x+n ) using ndx = l x - l x+n** **nPx = 1/2 ( 2lx - ndx ) ** **2 nPx = 2 lx - ndx** **2nPx = 2l x - ndx²** **2nPx / n = 2l x / n - ndx² / n** **∫ (2l x / n - ndx² / n) dx = ∫ l x - l x+n dx** **1/2 [ 2

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