1. Mathematics SL
Margaux LOESCHE
Internal Assessment
POPULATION TRENDS IN CHINA
26/03/2012

2. Population Trends in China are common to observe since China has the largest population on Earth
since a number of years. We can observe its growth with the data collected from 1950 to 1995.
The variable for this data is the growing populations. The data collected is not completely
accuratethough since a lot of parameters influence this data in many ways. The health of the people in China
affects the data. The new medicine and technology discovered throughout the years affected a lot the
conditions of living in many countries such as China. The migration of the people throughout the world also
affects the set of data evidently. During this period of time there were a lot of geographical and natural
disasters that affected the population of China by killing many in an unexpected situation. China is also a
particular country because it set a one child policy in the country; this is a factor that affected a lot the
growth of the population.
Year 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995
Populations 554.8 609.0 657.5 729.2 830.7 927.8 998.9 1070.0 1155.3 1220.5
in Millions
Here is a chart with the data and a graph showing its growth throughout the years. The X is the years
in which the data was collected, the variable Y is the population that varies:
1400 Population in Millions
1300
1200
1100
1000
900
800
700
600
Year
500
1950 1960 1970 1980 1990 2000 2010
This is a clear graph obtained using Autograph by plotting in the data.
We can see on this graph that the population is constantly increasing. However we can also identify a
small part of a curve which could by identified as an exponential function from the year 1950 up to the year
1975. A quadratic or sine curve could be used to fit this data. The points following that year however follow a
simple linear function. So the data could fit a simple linear function. I tried a linear function to see how well it
could fit.

3. The equation of a linear function is y=mx+p where m is the gradient and p is the y intercept
M=
M=
M=
M=15.05
To find p you substitute the coordinates of a point into the equation. Here I used x=1985 y=1070
Y=15.05x+p
1070=15.05×1985+p
1070=29874.25+p
-P=29874.25-1070
-P=28804.25
P=-28804.25
Therefore the equation for the function would be:
Y=15.05x-28804.25
1500 Population in Millions
1400
1300
1200
1100
1000
900
800
700
600
Year
500
1950 1960 1970 1980 1990 2000
We can see that this model does not correctly fit the data because it does not go through all the
points.

4. Year Population Value of population in Difference
in millions millions produced by
the model (using
technology)
1950 554.8 543.25 554.8-543.25= 11.55
1955 609 618.5 609-618.5= -9.5
1960 657.5 693.75 657.5-693.75= -36.25
1965 729.2 729 729.5-729= .5
1970 830.7 844.25 830.7-844.25= -13.55
1975 927.8 919.5 927.8-919.5= 8.3
1980 998.9 994.75 998.9-994.75= 4.15
1985 1070 1070 1070-1070= 0
1990 1155.3 1145.25 1155.3-1145.25= 10.05
1995 1220.5 1220.5 1220.5-1220.5= 0
=
=-2.475
The difference between the model and the data is quite high since the sum of all the differences for
each point and then divided by the number of points is equal to -2.475. The closer this value is to zero the
better the model will fit the data.
Another model that could be used to fit this equation could be a polynomial. Using the calculator we
can find the model that fits the best and that would be following this equation:
y= + +ex+f
Where
a=0,0003161
b=-0,01663
c= 0,3948
d= -3,848
e=22,58
f=554,4
This gives us the following equation:
y=0,0003161x⁵-0,01663x⁴+0,3948x³-3,848x²+22,58x+554,4
When this equation is graphed on the data points we get a very accurate model that fits the
data.

5. 1500 Population in millions
1400
1300
1200
1100
1000
900
800
700
600
Years
500
0 10 20 30 40 50
Year Population Value of population in Difference
in millions millions produced by
the model (using
technology)
1950 554.8 554.4 554.8-554.4= .4
1955 609 611 609-611= -2
1960 657.5 653.3 657.5-653.3= 4.2
1965 729.2 732.3 729.5-732.3= -2.8
1970 830.7 832 830.7-832= -1.3
1975 927.8 924.5 927.8-924.5= 3.3
1980 998.9 999.9 998.9-999.9= -1
1985 1070 1071 1070-1071= -1
1990 1155.3 1154 1155.3-1154= 1.3
1995 1220.5 1221 1220.5-1221= -0.5
=
=0.06
By doing a difference table here we managed to see that this model fits almost perfectly to
the data since the sum of all the differences between points over the number of points is only equal
to 0.06 which is very close to 0.
A researcher suggests that the population, P at time t can be modeled by :
Where K, L and M are parameters.

6. P=Population
T=Time
K=carrying capacity
L=growth rate
M=rate of change in growth rate
Using the GDC, the logistic we can easily find the values of these unknowns.
K 1946
L 2.619
M 0.03332
P(t)=
1500 Population in Millions
1400
1300
1200
1100
1000
900
800
700
600
Year
500
0 10 20 30 40 50

7. Year Population Value of population in Difference
in millions millions produced by
the model (using
technology)
1950 554.8 537.7 554.8-537.7= 17.1
1955 609 604.9 609-604.9= 4.1
1960 657.5 676.4 657.5-676.4= -18.9
1965 729.2 751.7 729.5-751.7= -22.2
1970 830.7 829.9 830.7-829.9= 0.8
1975 927.8 909.9 927.8-909.9= 17.9
1980 998.9 990.9 998.9-990.9= 8
1985 1070 1072 1070-1072= -2
1990 1155.3 1151 1155.3-1151= 4.3
1995 1220.5 1228 1220.5-1228= -7.5
=
=0.16
Here is some additional data given:
Year 1983 1992 1997 2000 2003 2005 2008
Population in millions 1030.1 1171.1 1236.3 1267.4 1292.3 1307.6 1327.7
1500 Population in Millions
1400
1300
1200
1100
1000
900
800
700
600
Year
500
1950 1960 1970 1980 1990 2000 2010

8. 1500 Population in Millions
1400
1300
1200
1100
1000
900
800
700
600
Year
500
0 10 20 30 40 50 60
When we add the new data the researcher’s model does not fit as well as with the previous set of
data .we see that the points that were added do not correspond or are not close to points from the model
especially towards the end, those points were added points.
These result also go for the linear function used in the beginning:

9. By “fiddling” with the numbers of the linear equation I found a function that corresponded more to
the new set of data although not quit the perfect fit.
1500 Population in Millions
1400
1300
1200
1100
1000
900
800
700
600
Year
500
1950 1960 1970 1980 1990 2000 2010