Exploration into Modelling a Population Pyramid (19/20)

Introduction

Constructing the Population Pyramid

Method: Constructing the Function

The Female Piecewise Function: f(x)

The Male Piecewise Function: m(x)

 Analyzing Development with f(x) and m(x)

 Limitations

 Conclusions

On this ideological spectrum, I fall completely in the humanities section, often feeling out of place and confused in my math classroom. On the other hand, in my HL Global Politics class, I’m entirely in the loop and understand everything happening. I wanted to make math more appealing but had no idea how to make something so word based into something numbered-based. Then it came to me; the population pyramid. The population pyramid is a beautiful geographical figure that can quickly help categorize countries in aspects such as infant mortality, life span, gender inequality, and immigration rates. A population pyramid is generally measured in 5-year increments, split between the percent of females, and males, representing the individual age groups. I wanted to model the population of a more economically developed country (MEDC) and compare this function to the plotted points of other countries, including Less Economically Developed Countries (LEDCs), to see how they compare, based on development, and gauge if population pyramids are a good

The baseline readings from the population pyramids can show the general development of a country. Furthermore, suppose Country A’s population pyramid was modeled. In that case, it should match Country B’s population pyramid if the development (measured in the UN unit of the Human Development Index) is similar. So, if I take my home country, Canada, and create my population pyramid, model the pyramid, and then place my piecewise function on the population pyramid of a country with a similar political background, HDI, and geographic location, like the United States of America, there should be a close fit. Even more, my parents emigrated to Canada from Ecuador due to the hindered development, so if I placed my piecewise function on Ecuador’s population pyramid, it should be completely wrong.

The first challenge I had to undertake was converting a table of numbers from the Canadian 2016 Census[1] into a population pyramid. To separate the populations by gender, the male population is generally converted into negative numbers, while the female remains positive.

Traditional population pyramid pyramids have the x-axis being the percent of the population and the y-axis being the age group. This creates a couple of problems; typically, in mathematics and science, the independent variable is presented upon the x-axis, while the dependent variable counters it upon the y-axis. However, in the case of the traditional population pyramid, the age is on the y-axis, despite it being a measurement of the independent variable, while the percentage of the population is on the x-axis, despite it being the dependent variable. The other problem is that I planned to model the population pyramid with a piecewise function, and the shape that the traditional population pyramid forms are not a function due to the population bubbles.

When constructing my piecewise function, I started by dividing the seemingly random points into manageable sections. I started at the end-of-life part of the x-axis to make manipulating the end of the life span easier. The beginning of the life span is simple, as the population simply appears. Contrarily at the end of life, people must slowly fade away into the x-axis of death. Any part of the modeling piecewise function following the original segment is slightly restricted in that the beginning must intersect with one point of the first segment in a way that is not too abrupt; furthermore, it’s easier, to begin with, the more finicky part to ensure maximum control.

After formulating the first segment of the piecewise function, I would create the next segment to the left by trying to match the slope of the new position to the first one. If I could reach the hill closely enough without it looking odd, then I would shift the function into place and adjust the horizontal and vertical stretches until the line fit with as many points as possible. In my piecewise functions, I used lines, exponential functions, logarithmic functions, parabolas, ellipse segments, and cosine waves. I especially liked the use of the cosine wave, as I found it easier to match the trough or crest of waves to the coinciding trough or crest for smooth and natural transitions. I would shift and stretch each function from various forms, which I found there be prime manipulability.

•  For linear equations, I used the slope-intercept form.

\(f(x)\ =\ mx\ +\ b\)

•  For exponential functions, I used the general transformative form.

\(\ f(x)\ =\ Ab^{B(x-C)}+D\)

•  For logarithmic functions, I used the general transformative form.

\(f(x)\ =\ Alog_B(x-C)+D \)

•  For parabolas, I used the vertex form. 

\(f(x)\ =\ A\left[B(x-C)]^2\right.+D\)

• For ellipses, I used the general transformative form.

\(\ f(x)\ =\ \frac{(x\ -\ h)^2}{a^2}+\frac{(y-k)^2}{b^2} \)​​​​​​​

I knew there were some limits to using an ellipse since an ellipse is not a function and breaks the rules of creating a piecewise function. However, I decided that the benefits outweighed the drawbacks. I figured that the ellipse fit the shape of the position quite well, as well as I knew that if I placed restrictions on the ellipse, it would be a function that could quickly work well with the rest of the piecewise function. Additionally, I wanted to explore various ways of creating a piecewise function, including methods I wasn’t accustomed to.

After tinkering with all of the form variables to create functions (and, in one case, an ellipse) that fit the patterns set by the plotted points, I would calculate the intersections using GeoGebra. Then I would restrict the domain (and the range in the case of the ellipse) to limit the individual functions from being too chaotic and create one smooth line for each gender surveyed by Census Canada.

Some of the equations and limitations ended up with numerous significant figures, but I chose to limit my answers to 3 influential figures for simplicity and not to be overly redundant. All calculations utilized the entire trail of decimals to maximize accuracy.

As I mentioned earlier, I began both of my functions from the elderly side of the x-axis, just to the left of (20,0); as to scale, this would equate to over 100 years old, and very few Canadians fall under this category.

Immediately there seemed to be an exponential decay pattern with points from (13, 5.5) spanning to (18, 1.1). It quickly became clear that the exponential curve was too drastic, and the corner approaching the horizontal asymptote was too extreme. So, I transitioned from the extreme exponential function to its mundane inverse brother, a logarithmic function. There I simply reflected the function across the x-axis via changing the A value to negative. If I had made the A value more than 2, the curve would have been too steep, so I tinkered with values between 1.01 and 2. I shifted upwards and leftwards by subtracting the difference between the final point on the diagram (18,1.1) and a point on the logarithmic curve that appeared similar to the curve as populations waver off (7.5,-11.05), which I shifted by\(\ \binom{10.5}{12.15}\). When I implemented this shift into my equation, I realized that my first calculations were slightly flawed. So I looked approximately where I wanted the function to intersect with the X axis and adjusted my estimates for this, resulting in a somewhat different final shift. My final equation for the first section of my piecewise function is\(f_1 (x)=-log_{1.20} (x-9.38)+12.75\)

From my newly gained experience with modelling, I was able to utilized the same techniques just adjusting the numbers slightly to better fit the slightly altered male population curve. Like the female function, I started at the elderly end of the population, with a logarithmic function doing the same method resulting in the equation\(m_1(x)\ =\ -\ log_{1.33}(x\ -\ 11.3)\ +\ 7.18\)​​​​​​​.

Instead of transitioning with an exponential function, I chose to use the edge of a parabola because I intended to use a cosine function after. I figured it would be far easier to transition from a parabola to a cosine wave than an exponential function to a cosine wave. I wanted the vertex to be at the already plotted point (12, 6.3) and then reflected the function to make it downward facing by changing the A value to harmful to result.\(m_2(x)\ =\ -\ \ (x\ -\ 12)^2\ +\ 6.3\)​​​​​​ .

I decided to first put both functions without the points on the same cartesian graph. Then I reflected the m(x) over the x-axis to create a clear difference between female and male populations, which can help underline gender inequality. In the case of Canada, the male and female populations are pretty similar.

Canada and the US have the same HDI of 0.920, Which is calculated with the equation \(HDI\ =\sqrt[3]{LEI-\ EI-\ II}\)​​​​​​​With LEI being the Life Expectancy Index, EI being the Education Index, and II being the Income index.

Then I gathered data from the US Census Bureau, and calculated the age distribution via excel. Theoretically, if the population pyramid is a proper indicator of development, the US’s points should align well with Canada’s. So I put the data points into GeoGebra, which resulted in a strikingly similar pattern.

The population for the United States, Bosnia and Herzegovina, and Ecuador were only estimates as no census was conducted in those countries in 2016. Therefore, the data was less reliable than the data I got from the Canadian Government. The sources used to get the other countries were still reliable as they came from organizations such as the UN and the US Census Bureau.

Another aspect is that I couldn’t find a way to quantitatively gauge how well my piecewise function fit. I suppose that I could attempt to measure the distance between the points and the lines and then get the mean distance, but that’s incredibly tedious.

Word-based ideas can sometimes be challenging to convert into numbers. What I tried to do is convert theories of development that I’ve learned in class into quantitative data and create a visual representation. I think that I was successful in modeling the Canadian population pyramid. Still, I was only left with a more vague understanding of what it means for a country to be developed or not.

During the Cold War, countries were separated into First, Second, and Third World Countries. That concept is now seen as outdated. We have now transitioned into using the terms More Economically Developed Countries (MEDCs) and Less Economically Developed Countries (LEDCs), but how long until this concept is out of date? Maybe math can help create a solution to the biased system of identifying what country is developed and what country is not while taking other aspects into account, such as minority groups. This information can be used to make decisions such as deciding whether a country receives aid from the World Bank and what kind of conditions should be placed on loans like the IMF. A mathematical solution to gauging development can prevent unnecessary international intervention and domestic crises caused by international intervention. The UN’s HDI is a step towards this but still has flaws.

Modelling population pyramids is not a solution to this problem but perhaps can be used in a step toward a better global community.