Pop China Final

Laura Cristina L’Hoeste Roldan000564-020

Population trends in China

The objective of this task is to investigate different functions that best model the population of China from 1950 to 1995.

Year 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995Population in millions

554.8 609.0 657.

5 729.2 830.7 927.8 998.9 1070.

0 1155.3 1220.5

In order to graph these values we will place the years in the x-axis since these are independent values, the years are set and do not depend on any other factor to occur. In order to graph these values more accurately we will assign a scale representing these values. Subsequently we are going to assign the scale to the values of years.

Year Equivalent in scale

1950 51955 101960 151965 201970 251975 301980 351985 401990 451995 50

Note that we have begun our scale at 5 instead of 0 because the graph would be drawn inaccurately if we introduce the first year in our calculations as year 0.

On the y-axis, then we will place the population values, since this is the dependent variable as we are determining how the population varies per year. We will graph the values as appear in the chart in small decimals showing millions because there is no need to convert the values to full million numbers, this would be irrelevant as the relationship between the values, which we use to observe the function, will be exactly the same.

So if we graph these values we can observe there is a notorious increase yearly, the population does not present any apparent decrease.

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5 10 15 20 25 30 35 40 45 500.0

200.0400.0600.0800.0

1000.01200.01400.0

Variation of population in China per year

Since the graph shows a constant growth, in a population, we could interpret this as a population with very stable conditions, low mortality rates, no significant pandemics or natural disasters which could have reduced the population at some point. From this graph however, we cannot conclude this immediately for China because on one hand, we require a further study of the population of China and its lifestyle, and on the other hand, the fact that we only have values for every 5 years is a limitation which doesn’t allow us to observe the exact behavior of the population, still, since the growth of the population is continuous, from this graph we can surely make a very accurate study of the population trends in China.

Consequently, we will examine several types of functions, which might fit the trend of increase for this population; this will help us to make a more accurate study of the population in china between 1950 and 1995, and further more, this could serve to understand this population and the factors affecting it, and moreover, we may project this trend to future and past years to make a much better understanding of the population. So to begin with, the most evident trend in this data is a linear trend. The following graph shows the line of best fit assuming that the increase of the population in China is linear.

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In this graph we can observe visually that there is an evident correspondence between all the values of population plotted and the line drawn. Stating that the growth of the population in China has a linear growth, entails that yearly, the population grows exactly by the same factor. We can observe in this line of best fit an r2value of 0.99466. This r2 value is known in statistics as the coefficient of determination. This coefficient of determination is a values that “gives the proportion of the variance (fluctuation) of one variable that is predictable from the other variable. It is a measure that allows us to determine how certain one can be in making predictions from a certain model/graph.”1 Consequently, we understand that this is a value ranging from 0≤r2≤1, and the closer this value is to 1, this means that the certainty of the line in comparison to the plotted values, is the highest. The higher the coefficient of determination, the higher the correspondence of the drawn line. In this case, then, we can observe that the coefficient of determination is established as 0.99466, which is extremely close to one, evidencing that the linear function fits quite adequately the values of the population of China between the year of 1950 and 1995.

These results, could be conceived to an extent for this specific population due to the social and political situation in China. We know that China has implemented several politics as the one-child policy which was elaborated as a result to the exuberant growth of the population which is “projected to have

1 MATHBITS.COM. Correlation Coefficient. (Internet link). (24 March of 2012) http://mathbits.com/mathbits/tisection/statistics2/correlation.htm

Population (in Millions)

Years

Linear increase of population in China

3

http://mathbits.com/mathbits/tisection/statistics2/correlation.htm


1.39 billion citizens by 2015”2. Such elevated values in the population are only approximated my the population in India in the world, and this overpopulation has much more implication than giving a special value to life; this instead brings serious economical, social, cultural and political complications to the Chinese population in general. So evidently, we understand that after comprehending the gravity of this overpopulation and its effect in the increase of unemployment, severe water shortages, availability of food and booming of the economy; the Chinese government has developed methods to control the increase of the population, which could account for a linear increase in the population. Nevertheless, we must notice that these policies have been reinforced recently, and the graph presents the population trend between 1950 and 1995, years, which do not exactly comply these mentioned policies.

Since the population in China was already the greatest in the world 150 years ago, the Chinese have been effective in producing birth control methods, however, this has not always been like this. In 1949 (as we can see first value corresponds to 1950), when Mao took power, he “condemned birth control and banned the import of contraceptives”3. This action should represent an increase in the rate of the population development, until in 1970 he was pressured to begin a policy to suggest people that they should have only two children or less (Which should now appear as a decrease in the rate of population). So evidently, if we observe on the first drawn graph, in which the dots have been just joined (without drawing a line of best fit) from year 5 to 25 which correspond exactly to years 1950 to 1970, the dots show a small slope which due to its shape determines an increase in rate, and then 1970 appears to be an inflection point to the graph because after this the slope changes revealing increase in the population but at a slower rate. These varying slopes reveal that even though a linear function appears to be highly qualified and valid to determine the trend for this population, if we contextualize it with China’s current social situation; nevertheless, for the specific sample of the population selected, the linear function doesn’t seem to be the best adjustment.

Additionally, the linear function proves to be invalid for this population trend if we extrapolate the line drawn, to observe the behavior of the graph at future and former years.

2 HAYS, Jeffrey. Population of China. (Internet link). (24 March of 2012) http://factsanddetails.com/china.php?itemid=129&catid=4&subcatid=15 3 Ibid. (24 March of 2012)

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http://factsanddetails.com/china.php?itemid=129&catid=4&subcatid=15


On this graph we have marked using dotted lines, the year in which, according to the linear function of best fit, would be when the population reaches 1.39 billion people (1390 million people), the value we have previously determined to be the expected value of the population for 2015. From this we could obtain that this population would be reached approximately in the year 60, using the relationship table established at the beginning for the graphs, we may conclude that according to the graph the population in China would have reached 1.39 billions in 2005. We know already that by 2005, the population was much smaller than this value, which shows how this linear function cannot be really employed accurately for other values, aside from the plotted.Moreover, we can observe in this extrapolation that since the line extends completely straight on both directions, so we can observe that it reaches a point, approximately year -30, which would be exactly 1915, in which, according to the graph, the population of China would have been 0 and before this year the population would have been negative. This is absolutely impossible; in consequence, we can eliminate the linear function for the possibilities for the trends of the population of China between 1950 and 1995.

Continuing with this investigation we have the exponential function, we can observe the line of best fit in the plotted values, if we assume that the population in China grows exponentially.


Years

Extrapolated Linear increase of population in China

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For this type of function, visually we can observe that there is not much concordance between the plotted values and the exponential line. Just a couple of dots appear to overlap with the line. Nevertheless, the graph specified a coefficient of determination of 0.9916. Even though this value is smaller than the previous value obtained for the linear function, it is still equally coincidental; it is still extremely close to 1. This means then that the section of values known from the population of China, coincides with a section of an exponential function. Moreover, taking in account the historical information previously established on China, we know that from 1950 to 1070 the population presented an accelerated increase, due to Mao’s policies, fact that could account for n exponential increase. However, as we have mentioned also, after 1970, the rate of increase of the population slowed down due to the incorporation of Mao’s knew policy to suggest people to have maximum two children, and in later years of the one-child policy (which is a consequence of Mao’s own policy), so this should all reveal a smaller increase in the population after 1970, nevertheless, in an exponential function, the increase of the population would produce elevated increase in population after every year. Presenting a flaw in the trend for an exponential function.

Furthermore, by observing the extrapolation of the exponential line, we are able to observe the overall behavior of the function.


Years

Exponential increase of population in China

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Now, we may

observe that while drawing the relationship of 1.39 billion population and years, we obtain that for an exponential function the expected year for this population to be reached is about 55 years in this graph, which corresponds to the year 2000. The respective year established with the exponential function is much lower, and therefore more erroneous, than the year established by the linear function.

Furthermore, analyzing the extrapolation of the line itself show one favorable detail that makes it adequate for measuring population. We observe that the exponential graph does not come under zero, this function has an horizontal asymptote which in this case appears to be close to y=150million population. This is a positive detail since it is appropriate to determine populations as a population can never come under cero and this functions serves for this purpose.

On the other hand, we may observe in the graph also that even though it has a horizontal asymptote which limits population values and regulates it to adequate values; the exponential function lacks of vertical function which means that the population will continue to grow every time at a higher factor, approaching to infinite with no limit or barrier to stop it from growing. Still, due to these same facts we observe why the exponential equation turns out to be improper since no population ever could grow exponentially to infinite, since population in general is limited by several factors such as space availability, food sources, health problems among others, there is always a limiting value for all populations. Consequently, exponential functions become improper to graphing trends for the population being analyzed.


Years

Extrapolation Exponential increase of population in China

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Next we have studied the quadratic equation and we have plotted the line of best fit which best satisfy the trend established by the values. Following this we can observe its corresponding graph. This graph looks very close to the straight line analyzed at the begging. Several values coincide with the trending line. This graph presents a coefficient of determination of 0.9955, which evidently tells us that it serves very well the group of given data. Since this section of the function appears to be straight, then I cannot comment on

apparent shapes and they meaning with the overall population. From this we me have then a further analysis from the extrapolation of the graph.

From this we can observe a similar shape tu the exponential function, revealing an horizontal assymptote, never the less this value is about -900 million people, so in reality becomes completely useless as we have established previously this entails that at son point in the past years the population of China was cero and before that it was a negative value; which is something completely unconceivable.

On the other hand, vertically we observe that also similar to the exponential function, the quadratic equation increases undefinedly. Which doesn’t contribute to the decreasing rate in the increase of Chinas population. This fact about Chinas population such as ia its current rate the future population would be very much reduced. As discussed, firstly due to the reasons previously established about all populations and their limiting factor but moreover duw to the graying population in China caused by the one-child policy which produces a significant deterioration of the population’s average age. This consequently reduces the amount of future adults which have more kids, so reducticion of population becomes a vicious cycle which will deteriorate all economical and social aspects of the Chinese. So taking this projected reduction in account we must establish that the population of China


Years

Quadratic increase of population in China


Extrapolation Quadratic increase of population in

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would not fit an infinite increase in its population as occurs in the quadratic function as well as in the exponential previously.

Following this we fin another model which fits the values even more accurately. The cubic function and its line of best fit according to the plotted values of China population are showed as follows.

This value is perhaps the one that best fits the set of data, as it follows the curvatory we have previously established, accelerated increase from 1950 to 1970 and then followed by increase at a slower rate from 1970 and on. This function presents a coefficient of determination of 0.99862; this is the highest coefficient among the trends we have studied in this investigation. This reveals the high

coincidence we have observed visually. If we extrapolate the graph however, any coincidence established around the plotted values vanishes completely.


Years

Cubic increase of population in China


Years

Extrapolation Cubic increase of population in China

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The shape of the function lacks of any correspondence with the trends of any population. It centers itself in a decreasing graph, which at its minimum increases for a small amount and then continues to decrease above negative values. This trend is completely unconceivable for any developing country with no apparent threat or factor that would decrease its population. Consequently, even if this model fits perfectly the section of the values plotted for the population in China between 1950 and 1995, when taking in account its global shape and behavior, this is the most inconceivable of all the models we have studied in this investigation.

So now, even though we have previously discarded all four of the studied models, I will analytically develop the one which taking in account its flaws could be adapted the most, to the population we are studying. So for instance, we have the exponential function, it has the least troubles from all functions at the time of representing the population in China, however, I believe that this trend is not appropriate for China taking in account its historical context as we have established along this investigation. Therefore, that is why, in my opinion, for this population specifically, the function which best fits the population is the linear function. I must clarify that a linear function is not appropriate to describe a population in development since every time the birth rate is increasing. This happens because every time there are more parents which produce more offspring and this happens consecutively. However, for this population specifically in which China has implemented so many tactics to control the growth of the population, a linear trend can be conceivable and even the most valid.

So, in order to calculate the linear regression (or the equation of the linear trend line), we must take in account that y=ax+b for a linear function.

I will express one more time a table showing the relationship between the two variables we are studying, years (In the equivalence form) and population.

Year Population in millions

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5 554.810 609.015 657.520 729.225 830.730 927.835 998.940 1070.045 1155.350 1220.5

Scatterplot of the values of the population of China. The scatterplot allows us to observe whether the relationship between the variables is linear, so we can use a linear regression technique or perhaps if it is another type of function, then we use other non-linear regression techniques. Even though we have already determined that the relationship is linear, I still include the scatterplot diagram since this is a general step used in regression and will eventually help us evaluate the veracity of our model.

5 10 15 20 25 30 35 40 45 500.0

200.0400.0600.0800.0

1000.01200.01400.0

Scatterplot of population in China per year

Scatterplot still evidences a linear relationship, so in order to solve for the desired equation of the line of best fit we will use linear regression.

Now, given that (x1 , y1 ) , (x2, y2 ) , ( x3 , y3 ) , (x4 , y4 ) , (x5 , y5) , (x6 , y6 ) , (x7 , y7 ) , (x8 , y8 ) , (x9 , y9 )∧( x10 , y10)=

(5 ,554.8 ) , (10 ,609.0 ) , (15 ,657.5 ) , (20 ,729.2 ) , (25 ,830.7 ) , (30 ,927.8 ) , (35 ,998.9 ) ,(40 ,1070.0 ) , (45 ,1155.3 )∧(50 ,1220.5 )

We are going to calculate y=ax+b

First, the slope (a) of a line is calculated using the formula

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a=n∑i=1

n

xi y i−∑i=1

n

x i∑i=1

n

y i

n∑i=1

n

x12−(∑

i=1

n

x i)2

So using the values of x and y correspondent to the years and the population, respectively; I will calculate the slope for this data.

n x y xy x2

1 5 554.8 2774.0 252 10 609.0 6090.0 1003 15 657.5 9862.5 2254 20 729.2 14584.0 4005 25 830.7 20767.5 6256 30 927.8 27834.0 9007 35 998.9 34961.5 12258 40 1070.0 42800.0 16009 45 1155.3 51988.5 202510 50 1220.5 61025.0 2500

Σ 275 8753.7 272687 9625

Now if we replace the variables from the equations with these variables.

a=n∑i=1

n

xi y i−∑i=1

n

x i∑i=1

n

y i

n∑i=1

n

x12−(∑

i=1

n

x i)2

a=10∑

i=1

10

x i y i−∑i=1

10

x i∑i=1

10

yi

10∑i=1

10

x12−(∑

i=1

10

xi)2

a=10 (272687 )−(275 ) (8753.7 )10 (9625 )−(275 )2

a=2726870−2407267.596250−75625

a=319602.520625

a=15.49587879≈15.495879

The y-intercept (b) is calculated using the following formula:b= y−a x

Average y

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y=554.8+609.0+657.5+729.2+830.7+927.8+998.9+1070.0+1155.3+1220.510

y=8753.710

=875.37

Average x

x=5+10+15+20+25+30+35+40+45+5010

x=27510

=27.5

Sob= y−a x

b=(875.37)−(15.495879)(27.5)b=(875.37 )−(426.136667)

b=449.233333

Finally we obtain the formula for the line of best fit, which would be y= (15.495879 ) x+449.233333

We can now plot this formula in order to observe its behavior in relation to the scattered diagram we have previously done.

For this we will use the same x values we have already plot, corresponding to the years, against the calculated value for y. Such as

x y

5 526.71272810 604.19212315 681.67151820 759.15091325 836.63030830 914.10970335 991.58909840 1069.06849345 1146.54788850 1224.027283

Which plotted with the scattered diagram would produce a linear equation as line of best fit.

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5 10 15 20 25 30 35 40 45 500.0

200.0

400.0

600.0

800.0

1000.0

1200.0

1400.0

Line of best fit of population in China per year

We can observe that the line fits very perfectly around the values of the population of China, proving the statements made previously that even though a linear function is not adequate to express the trend of a population, since it implies that every year the population increases exactly by the same factor and we have already established that this constant increase is not possible due to several economical, social and even political problems. Still, taking in account the background of China, we may conceive in these case a linear growth due to the various policies established to control growth.

Nevertheless, we must no forget the limitations of this function in order to be able to make predictions and analysis not only of the data stated, but also of previous and future years. Since a linear trend would produce that in some year in the past the population would be zero, which is impossible; as we have mentioned before.

Still in my opinion, this is the best trend from the functions we have investigated, to represent the growth of China’s population.

On the other hand, we find that a researcher suggests that the population P at time t can be modeled by

P (t )= K1+Le−Mt

Where K , L and M are parameters or limitations to the function.

We identify this function as the logistic function. The logistic function combines two characteristics of exponential growth, the exponential growth, like the one we have investigated previously, and bounded exponential growth which is a decaying exponential subtracted from a fixed boundary, this models

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growth that is limited by some fixed capacity. So by combining these, the logistic function models resource limited exponential growth. 4

Now, as we determine, K, L and M are parameters that determine the logistic function, so in order to determine what are their values in the logarithmic function we are investigating for the population of China, we will need to know what does each parameter represent. If we analyze the equation, we will assume that the variables represent positive constants. This is because we are studying growth of the population; consequently, growth would always signify a positive value. So now, as we progress in time, or t increase (which corresponds to the normal course of life), t increase, the term –Mt (exponent) would become a larger negative number. Consequently, e−Mt becomes smaller, since raising e to a negative number gives a small positive answer. Consequently, the term Le−Mt also becomes smaller. Therefore, the entire denominator 1+Le−Mt is always a number larger than 1 and decreases to 1 as t gets larger. Finally then, the value of y, which equals K divided by this denominator quantity, will always be a number smaller than K and increasing to K. It follows therefore that the parameter K represents the limiting value of the output. Following this same analysis based of a mathematics paper5, we can find that L represents the number of times that the initial population must grow to reach K and finally, M is much harder to interpret, so it is enough to establish that if it is positive, the logistic function will always increase, and is negative, it will decrease.

Now, in order to obtain the values for these variables I have plotted the values in the program I have been using throughout the investigation (Two Variable Analysis) and then display a logistic curve of best fit.

4 WMUELLER. Logistic Functions. (Internet link). (26 March of 2012) http://www.wmueller.com/precalculus/families/1_80.html 5 MATH 120. Elementary Functions. (Internet link). 926 March of 2012) http://cerebro.xu.edu/math/math120/01f/logistic.pdf


Extrapolated Logistic increase of population in China

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http://cerebro.xu.edu/math/math120/01f/logistic.pdf

http://www.wmueller.com/precalculus/families/1_80.html


It is now evident that this function is destined to specifically model population, since it supplies for all the problems we experienced previously with the investigated functions. Since we have an increase but it is not infinite, due to the carrying capacity, and we do not have any zero or negative values for previous years. We can see from the drawn line that according to the graph, the population would be 13.9 billions in 2005, as we have established it is the expected value for 2015, nevertheless, it is just a small difference in comparison to all the validity of the function.

From this graph we have been able to obtain that K=1946.2, which is the carrying capacity, of limiting value; L=3.0938 and M=0.033321.

Which produces that the researcher’s model for this population is

P (t )= 1946.21+3.0938 e−0.033321 t

Now we can see how the researcher’s model fits the original data in the following graph.


Popul

Logistic increase of population in China

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From a close distance we can now observe how does the logistic function fit original data, and it is evident that it complies all the parameters we had previously established for the population in China. We can se that the line is slightly curved in an accelerated shape from 1950 to about 1970, which shows the relative increase in the population due to Mao’s policies, and then at about 1970 we can observe the inflexion point of the curve, when restriction policies begin. The coefficient of determination for this function is of 0.99672, showing the validity of the function is very high, and that therefore this appears to be the correct function to represent the population in China.

An additional data of population was obtained from 2008 World Economic Outlook published by International Monetary Fund (IMF).

Year 1983 1992 1997 2000 2003 2005 2008Population in millions

1030.1

1171.1

1236.3

1267.4

1292.3

1307.6

1327.7

We will plot using the following scale conversion in accordance to the previous scale.

Year Equivalent in scale

1983 381992 471997 52

Popul

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2000 552003 582005 602008 63

So If I plot my model against these values we will have values such as

x y

38 1038.076735

47 1177.539646

52 1255.019041

55 1301.506678

58 1347.994315

60 1378.986073

63 1425.473710

While if we use the researcher’s model we will have values such as

x y

3 512.22669712 633.08698817 706.21127920 751.76380123 798.38846925 829.94503328 877.841729

I will plot both models together with the values in order to observe and compare the coincidence of both models.

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38 47 52 55 58 60 630.0

200.0

400.0

600.0

800.0

1000.0

1200.0

1400.0

1600.0

Fitting models with data

My ModelResearcher's

Axis Title

We can see that in this data both model’s fit very good, how ever we can observe the researcher’s model a little more in relation to the shape of the dotted values which correspond to the scattered diagram due of the values established for the population of China.

So now and finally I will include all the collected data and modify the logistic equation in order to determine the correct formula that describes this population.


Years


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All the values fit very well and we find that it has a coefficient of determinant of 0.9986, higher than any other of the coefficients we calculated to all the other functions in this experiment. This proves the validity of the function in describing the population of China

So to conclude, The equation for this overall population would be

P (t )= 20251+3.2261 e−0.032112t

Consequently, we can state that the population in China is better modeled by the logistic equation, and this is supported due to the context of the population that we have established throughout the entire investigation. Taking in account the policies of birth control established by the government of China, the growth among the period we are analyzing (which corresponds to this period in which the policies have been established), appear to be a linear trend at first sight. This is understandable because the control policies limit the reproduction of the families to just one child, which maintains the increase constant throughout the years. However, we must realize that a population as


Years


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resources and space which limit the increase of the population which in this case is 2.025 billion people, so this eventually decrease the rate of reproduction in any population, and moreover, it is necessary to realize that the population could have never been less than cero people, even in past years, only much time ago; so taking in account all these realizations, is why it is easy to notice that the logistic trend defines this population as best as possible.

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Pop China Final

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