Maths Port. Type II

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Portfolio Task 2

Mathematics SL

Population trend in China

Name of the Candidate: Vikram Rathore

Candidate Number: dhc444

Session Number: 003528-001

Name of the School: Johnson Grammar School, I.C.S.E

School code: 003528

Examination session: May, 2012

Name of the instructor: Mrs. Shantha


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Introduction:

China has the world largest population. As such it is a chief objective of study for many

demographers.In order to sudy the population of this country effectively, its essential to know

the different trends the population takes on and the various forms it may take in the future.Envisaging the nature of the a population is very important for the country specially in the

fiels like economy or business and also its implication on the environment.

In order to study the above it is needed to device a mode to signify the population as a

function of time. Therefore a conjucture should be developed which could seem to fit the

given set of records.

Development of conjucture:

For the development of conjecture, the most important details that have to be considered as

follows;

The family of the function The variables concerned. The variable parameters which are concerned in the function.

Table 1: The data table below shows the data provided for the population of china.

Year Population (in million

1950 554.8

1955 609.0

1960 657.5

1965 729.2

1970 830.7

1975 927.8

1980 998.9

1985 1070.0

1990 1155.3

1995 1220.5


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Graph 1:

The graph is plotted using logger pro 3.6.1.

From table 1 and graph 1, it can be seen that the population of China is incresasing with the years ,therefore in order to define the incresing trend, first, the linear model is considered.

Linear model:

In the linear method first the value of a, t and b is calculated from the equation obtained as

, where t is number of years

i) Substituting t = 0 in eq (1), then555.5 = 0*t + b

b = 554.8

ii) Substituting t = 5and b = 554.8in eq (1), then609 = a(5) + b

609 = 5a + 554.8

Therefore, a = 10.84

Substituting a and b is the eq(1) for different years, population is calculated.

y = a*t + b (1)


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The table 2 below shows the % of error in population calculated and the given population

The table is drawn using MS excel2007.

Year Given

population

Found

population

Difference % Error

1950 554.8 554.8 0 0

1955 609.0 609 0 0

1960 657.5 663.2 5.7 0.87

1965 729.2 717.4 11.8 1.62

1970 830.7 771.6 59.1 7.11

1975 927.8 825.8 102 11.0

1980 998.9 880 118.9 11.9

1985 1070.0 934.2 135.8 12.7

1990 1155.3 988.4 1666.9 14.4

1995 1220.3 1042.6 177.7 14.6

The Average error obtained is 8.24%

Though the population trend shown by the linear model is increasing like the trend shown by

the given population, the average error obtained is 8.24%, which is quite high. The difference

found between the given population and the calculated population is different and it is not

fixed throughout, consequently, the linear fit model may not fit into all the data points.

Therefore, the linear fit model is not appropriate to show the population trend. This can also

be seen from the linear fit graph given below:


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Graph 2: The graph is plotted using logger pro 1.3.6.

Quadratic model:

After liner model the next polynomial is a quadratic which is used to model the data using the

equation: p(t) = at2+ bt + c

When t = 0, the population is 554.8 i.e., in the year 1950.

In the quadratic method, first the value of a, t and b is calculated from the equation obtained as

i) Substituting t = 0 in eq (2), thenP(0) = a(0)

2+ b(0) + c

554.8 = a(0)2

+ b(0) + c

Therefore, c = 554.8

p(t) = at2+ bt + c .. (2)


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Now, two more data points were substituted in the equation (2).

ii) Substituting t = 10 in eq (2), then657.5 = a(10)

2+ b(10) + 554.8 (3)

729.2 = a(15)2+ b(15) + 554.8 (4)

Solving the two equation i.e., the equation 3 and equation 4 simultaneously

a = 0.197

b = 6.745, are obtained.

P(t) = 0.197(t)2

+ 6.745(15) +554.8

= 700.3

P(15) = 0.197(15)2

+ 6.745(15) +554.8

= 768.5

From the equation, It is found that, when t = 15 the population is 700.3, but the actual

population is 729.2. As a result of it, the difference obtained is (729.2-700.3) = 28.9, and

hence the error percentage is calculated which turned out to be 3.96%.

P20) = 0.197(20)2

+ 6.745(20) +554.8

=768.5

The given population of China at 1970 is 830.7 million, but the population found by

substituting the calculated value of a and b in the equation is 768.5 million, thus the

difference obtained is 7.45 million.

Similarly, the difference in the weight and the percentage of error is calculated for different

years and presented in the column below-


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Table 3: percentage of error is calculated for different years

Year Given

population

Found

population

Difference % Error

1950 554.8

554.8

0 0

1955 609.0

593.45

5 0.83

1960 657.5

641.95

0 0

1965 729.2

700.3

28.9 3.96

1970 830.7

768.5

62.2 7.45

1975 927.8

846.55

73.5 7.92

1980 998.9

934.45

85.6 6.77

1985 1070.0

1032.2

37.8 3.53

1990 1155.3

1139.8

52.9 4.58

1995 1220.5

1257.25

23.8 1.96

Average error =

3.47%

Graph3: Population (in million) vs. years

The graph below shows difference in the trend of calculated population from the given population

0

200

400

600

800

1000

1200

1400

0 5 10 15 20 25 30 35 40 45

p

o

p

u

l

a

t

i

o

n

years

given population

calculated population


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Cubic model:

If is found that generally, cubic logarithmic or exponential functions are used to model the population

growth. A cubic function may be fit well for the data points. This function for the above data can be

modeled by an equation.

When we substitute the value t = 0 in the above equation

P(0) = a(t)3

+ b(t)2

+ c(t) + d

P(554.8) = a(03

+ b(0)2

+ c(0) + d

d = 554.8

Substituting two more data points in the equation (5) gives;

a =0.0358

b = -0.692

c = 0.00385

Table 4;

year Given population

(in millions)

Found

population

(in millions)

Error %

1950 554.8 554.8 0

1955 609.0 593.45 1.23E-05

1960 657.5 641.95 6.08E-05

1965 729.2 700.3 0.000168

1970830.7 768.5 2.684146

1975 927.8 846.55 14.01106

1980 998.9 934.45 37.40022

1985 1070.0 1032.2 70.65285

1990 1155.3 1139.8 111.8219

1995 1220.5 1257.25 167.5108

Mean = 40.40812

P(t) = a(t)3

+ b(t)2+ c(t) + d .. (5)


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It can be seen that the mean % of error obtained is 40.40812. This could be because of the

fact that many points doesnt seems to have co-relation. The obtained percentage is very high

and consequently the cubic model cannot be used.

Graph 4: Population (in million) vs. years

The graph below shows how well the cubic model fit the given data.

In the above graph the calculated data points doesnt seems to fit the trend from year 20 to 45, and

hence, the proposed model cannot be used to model the given data points.

P (t) =

where K, L and m are parameters. Where m>0 and t and m cannot be a

negative number. This is because if m takes a negative number then t cannot tend to 0. If t is

positive value the over value tend to K.

P (t) =

()-1 =

Substituting t= 0 in eq. (6)

P (0) = 554.8

Let k be integral multiple of 554.8

Let k= 3 X 554.8

0

200

400

600

800

1000

1200

1400

0 5 10 15 20 25 30 35 40 45

p

o

p

u

l

a

t

i

o

n

years

given population

found population

P (t) =

. (6)


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K = 1664.4

P (t) =

which is given

= = ()

= =

()- 1

=()

()

L =()

() (7)

Substituted k= 1664.4, t = 0 in eq(7)

L =

L = 2

Putting these values in eq (6), the following function is obtained.

P (t) =

The value of M is varied to get the significance of M in the graphical representation is

deduced.


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Table 5:

Year m = 0.1 m =0.2 m =0.3

0 554.8 554.8 554.8

5 752.2106 959.1458 1151.167

10 959.1458 1310.212 1513.937

15 1151.167 1513.937 1628.3325

20 1310.212 1605.729 1656.22

25 1430.006 1642.338 1662.57

30 1513.937 1656.22 1663.992

35 1569.802 1661.383 1664.309

40 1605.729 1663.29 1664.38

45 1628.325 1663.992 1664.395

Graph 5:Population (in million) vs. years

The graph is plotted by substituting m= 0.1 in eq (6)

0

200

400

600

800

1000

1200

1400

1600

1800

0 5 10 15 20 25 30 35 40 45

P

o

p

u

l

a

t

i

o

n

years

given population

found population


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From the graph 3 it can be seen that the when m = 0.3 the trend found is very much different

from the trend which the given population shows.


The graph is plotted by substituting m= 0.2 in eq (6)

When m = 0.2 the trend shown by the given population is unlike from the trend of found

population.

0

200

400

600

800

1000

1200

1400

1600

1800

0 5 10 15 20 25 30 35 40 45

P

o

p

u

l

a

t

i

o

n

Years

given population

found population


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The graph is plotted by substituting m= 0.3 in eq(6)

Again at m = 0.3 also the trend shown by the found population is different from the given

population.But it can also seen that when m = 0.1 the function seems to be more appropriate than

others.

After substituting more value of m, it is deduced that, when m = 0.037 the graph is more appropriate

and demonstrates the actual population to a great extent.

Table 6

year Given

population

(in millions)

m = 0.037 % Error

1950 554.8 554.8 0

1955 609.0 625.2408 2.67

1960 657.5 698.9912 6.31

1965 729.2 774.9534 6.27

1970 830.7 851.8863 2.55

1975 927.8 928.4841 0.07

1980 998.9 1003.463 0.46

1985 1070.0 1075.647 0.53

1990 1155.3 1144.036 0.97

1995 1220.3 1207.852 0.01

Average error percentage = 1.98 %

The average error obtained is 1.98% which is quite less and very much acceptable.

0

200

400

600

800

1000

1200

1400

1600

1800

0 5 10 15 20 25 30 35 40 45

Po

p

u

l

a

t

i

o

n

years

given population

found population


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The graph is plotted by substituting m= 0.037 in eq(6)

The graph of a proposed function co-relates well with nearly all the points excluding in the

range of 10-20. In table 6 the maximum percentage obtained is 6.27 and the average

percentage of error obtained is 1.98% which is very less relative to the previous model.

Therefore, this can be used to illustrate the trends in population of china logically.

The graph below shows the trend the graph will for the next 150 years. The predicted

population (in million) shows that the population trends toward a certain constant and will

never exceed it. This is accordance with what was hypothesized about the parameters which

states that K is maximum limit that the population can take.

0

200

400

600

800

1000

1200

1400

0 5 10 15 20 25 30 35 40 45

Po

p

u

l

a

t

i

o

n

years

Given population

found population


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Graph9: Population (in million) vs. years

The graph is plotted by substituting m= 0.037 in eq (6) for 150 years.

The function doesnt represent the population (in million) precisely since the random values

are to be assumed as K. if the maximum value which a population can take was provided it

would have helped considerably in both modeling the data provided and for the predicting its

trends in the future. The graph above provides only the quantitative analysis of the population

trend and not a quantitative one as no discrete pieces of evidences are used to define the

parameters for the values K and L.

Supplementary data published by the International Monetary Fund regarding Population

trends in China gives the following data:

Table 7

year Population ( in millions)

1983 1030.1

1992 1171.7

1997 1236.3

2000 1267.4

2003 1292.3

2005 1307.6

2008 1327.7

0

200

400

600

800

1000

1200

1400

1600

1800

0 25 50 75 100 125 150

P

o

p

u

l

a

t

i

o

n

years

Graph obtained from modelled function

found population


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Graph 10: represents the trend shown by the data given in table 8

By assigning the specific values for K, L and M in the researchers model, the function which

fittest the population was devised. Thought in the investigation it is found that most of data is

flitting the population of China for the higher values of the calculated population seems to be

higher beyond the year 2003 when compared to actual population.

By using technology and trying for the best fitting graph, it is found that the Gaussian

function of the form given below satisfies all the data points.

Therefore the population trend can be modeled by using Gaussian function as

P (t) =

()

+d ..(8)


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Conclusion:Therefore, the above Gaussian function defined for all the parameters mentioned acts as a

suitable model for which the data points fit within the respectable error boundaries.

Software used for the portfolio:

Logger pro 3.6.1. Excel spreadsheet

P(t) =

()

+d

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Maths Port. Type II