Gender Equality in STEM Programs

Recently I’ve been interested in investigating how gender equality – or equivalently, inequality – has evolved over time in Canada. Using the University of Waterloo’s public Cognos cubes, specifically those for undergraduate enrollment, I have found some pretty interesting results. Below, I will detail a brief summary of these findings.

To begin, let’s talk a bit more about our population of study and the data that is used in our analysis. The target population that I’m examining here is set of all undergraduate university students and the sample population is the set of all undergraduate students who have enrolled at the University of Waterloo. The sample chosen in the analysis is the sample population restricted to students who have enrolled between 1996 to 2013 where terms are only selected in the sample if there are at least 20 distinct programs with at least 1 student in them. We make no distinction between students enrolled in or not enrolled in a co-operative program.

All analysis and visualization is done in the free academic version of Revolution R, Version 7.

For each date, at the Program (e.g. Life Sciences) and Faculty (e.g. Science) level, a total is computed for each of the female and male genders and the percentage of females is then calculated as |Females|/(|Males| + |Females|). A rendered ordered bar chart at each date, with the Program on the x-axis and percentage female on the y-axis, is then generated using the ggplot2 R package and a GIF animation of these charts is produced to study the time evolution as seen below.


Bar colors are dependent on the Faculty of the Program. Click the image above to properly view the animation.

The abbreviations for each Program can be clarified here. A quick scan over the image shows that there does not appear to be any noticeable change in the overall shape other than a -very- slight flattening of the center bars and slight increase in the slopes near the extreme ends during later years.

Using this data, I use the following method as a crude estimate for Faculty-wide, time dependent gender bias, where I define this as how gender bias a Faculty is relative to past or future states or enrollments of the university. Suppose that for a fixed date we have n programs and P=\{P_{1,F1}, P_{2,F2}, ..., P_{n,Fn}\} is a set of ordered values of percentages of females in n different programs, ordered by least to greatest percentage of females in the first index, and where the second index is representative of the Faculty in which the program falls under. Let P_{F}=\{P_{k,Fk} \in P: Fk = F\} and n_{F} = | P_{F} |. Then for each Faculty F, we denote the (female-dominated) gender bias as

G(F)= (P_{n_{F},Fn_{F}}+P_{n_{F-1},Fn_{F-1}}+P_{n_{F-2},Fn_{F-2}}) / 3n

Which we can think of as a three term average [1] of the quantile of the three most female dominated programs. A value close to 100% (less biased) is generally preferred.

Taking only the STEM Faculties into consideration (SCI, ENG, MATH), we plot out this measure over time using the lattice R package below:

Gender Bias over Time

The blue circles indicate points in time, the red lines are LOESS curves and the green lines are smoothing splines. The science faculty seems to follow a rather sinusoidal trend, the engineering faculty a mostly linear trend, except for the sudden rise in the 2003-2005 date range, and the maths faculty being the most sporadic of the three. There is an apparent outlier near the 2008 year in the maths faculty, although this may be explained by the increased interest in the new FARM program and other finance related programs in light of the latest U.S. recession.

A least squares regression with slope and intercept interaction factors is also done in R for computing long term trends and is shown below:

R Regression

Here, Idx is just a normalized Date variable. From the results, we can see that the long-run growth in the MATH and SCI Faculties are not significantly different from one another and we can expect a long-term growth of female gender bias of approximately 0.23% every term in these faculties in the near future, while for engineering, this is closer to 0.03%.

With this in mind, it looks like we won’t be seeing fair gender equality for at least 2 decades for the sciences and several times that amount for the mathematics and engineering faculties.

To replicate these results, as well as see the charts above in higher resolution and examine the source data, you can check out the relevant Skydrive directory here.

If you have any comments or suggestions for future statistical projects, let me know in the comments section below.

[1] An average is done here in order to smooth out any outliers, which from the data we can see a few, particularly in the architecture program.


Spring 2013 Course Notes

Hi everybody,

As I move into the edge of third year as an undergraduate student, the time that I have to actively contribute to this blog lessen more and more. However, with my new investment into a new tablet/laptop (that can last more than 1 hour) in the form of the latest Microsoft Surface Pro, I hope to make it up to everyone out there by offering not one, not two, but SIX sets of course notes this term.

Specifically I will be covering PMATH 450, PMATH 352, ACTSC 372, ACTSC 445, STAT 371 and STAT 330. I have already posted the most up to date versions of these notes in the Course Notes section of my blog and will continue to update them throughout the term, along with review notes for midterms and finals. Note that currently I am emphasizing the content of these notes rather than the aesthetics, so some areas such as the index and abstract are still under construction.

Hopefully I will have some time to present something of interest from my studies as the term goes on, but for now, all of you who are dying to know more about the details the non-measurability of transforms of \mathbb R \backslash \mathbb Q will have to make do with my notes.

Until next time,

Stochastic Seeker

Fall 2012 Exam Notes

Final exam review sheets have been posted for STAT 333 and PMATH 351 in the Course Notes section of this blog. I will update this post when more notes become available.

Update 1: The review sheet for ACTSC 371 is now available.

Update 2: The review sheet for CS 338 is now available.

Some end of term work

Just about one week after my exams, I went to work on tidying up course notes and making sure that errata such as final exam study notes were polished enough to put on the blog. They’re now ready in the Course Notes section of this blog along with a bonus set of notes based on the multivariable calculus video series here.

Taught by Prof. Auroux, I decided to focus on only the vector calculus component of the lectures as the other components are well covered by my MATH 247 notes (also found on the same blog page). Specifically these notes cover the following:

In addition to these notes, I also completed one of my little side projects which is a basic proof on the irrationality of Euler’s Number, e. Check it out in the Resources section.

Over the next couple months or so, I will also be preparing to write the Putnam exam in December, so stay tuned for future interesting problems.

For undergraduates who are taking the summer off, I hope you all have a nice vacation and a good rest for the upcoming fall term.

For co-op students like me who are back to work next week, stay diligent and strong. Your efforts will appreciate someday.

Small Beginnings

Finally, for all the people at Waterloo taking statistics and advanced calculus III, this term, I bring you the beginnings of my typesetted course notes! Check them out in the “Course Notes” section of this blog.

For the other people in some of my other classes, I’ve refrained from typesetting those due to how the lectures were structured:

  • CS330: Powerpoint slides were provided and classes were just discussions… exactly like a business course.
  • CS371: There are some awesome notes already online at this link.
  • ECON102: I do have notes on OneNote but these are not very clear unless one has attended L. Smith’s lectures.
  • ECON201: See CS330 for the same reasons.

I’ll continue to update the ones that I do have as the term goes on (without official posts), so be sure to spread the word to all of your friends taking the course. Hopefully in the coming weeks I will be able to find some more interesting content to blog about, but for now stay tuned for that Black-Scholes primer document!

High school life, research topics, and advice for the young aspiring mathematician. (Part 3 END)


This will be the final post in my three part saga about mathematics and high school. You can check out parts one and two here and here respectively if you haven’t already read them. In this post I want to end off by offering the final part that was mentioned in the title, research topics (I have decided to write about advice for the upcoming mathematics undergraduate in a different series some time far in the future).

If you have been reading any of my previous posts, you might remember me talking about how a critical part of learning mathematics is knowing how to explore, be creative, and think about problems outside of the classroom. It is better to sow your tiny seeds interests early than later, because as many of you find out from learning maths, many of the areas  interconnect beautifully if you just take a while at the subtleties.

Thus, below, I have provided some topics of interest that can be researched at the high school level but delve deeply into undergraduate topics; all they need is a bit of perseverance, time, and creativity to get started. You can probably even get your teacher in on this. Ask him/her if instead of completing droll textbook problems or monotonous exercises, you could start a research topic instead! No more spending one hour every day copying down exactly what the book asks you to do or sketching and drawing precise margins (at the university level, precisely drawn graphs is one of the lesser important aspects of mathematics). Instead, mathematics turns out to be a continuous process that you can do every day, without the stress.

... unless of course, your teacher wants you know the critical skills of copying and pasting. Then, it is advisable to keep at those 'exercises'. {MS Clipart}

So without further ado, here they are (with difficulty rankings relative to high school content):

Population Dynamics

Difficulty: 7/10
Mathematics Topics: Differential Equations, Chaos Theory
About: Students can examine the different  population models such as Malthusian, logistic, and island biogeography. Investigations can be about the limitations of each model, the strengths and the ideal conditions for the model to work.
Bonus: Students can talk about how current day population models are more similar to chaotic systems than deterministic ones

 Poker, Probability, and Game Theory

Difficulty: 6/10
Mathematics Topics: Game Theory, Probability, Statistics, Simulations
About: Students can investigate the best course of action to take with various poker hands, categorized by winning strategy. Approaches can be from a game theoretical, probabilistic or statistics side or all three.
Bonus: Create a computer simulation to accept or deny your theory through a null hypothesis and Monte Carlo simulations.

Rings, Fields, Groups, Vector Spaces, and Algebras

Difficulty: 9.5/10
Mathematics Topics:
Abstract Algebra, Linear Algebra
About: Students can discuss the differences and similarities between the above algebraic objects and how to construct them. Investigations can be made about various isomorphisms between algebraic objects and about interesting objects such as the power set of the real numbers being an algebra under the union and intersection operators.
Bonus: Construct your own simple algebraic object (maybe the set of all watermelons that taste like oranges; something silly) and see if you can go from a magma all the way to an algebra.

Fractals and Iteration

Difficulty: 9/10
Mathematics topics: Fractals, Chaos Theory
About: Students could take a look at the definition of a fractal and what exactly is fractal dimension. Investigations can be made about the construction of fractals under various domains, their applications (e.g. coastlines), and their self-similarity properties. Students could try to generate fractals through freeware found online.
Bonus: Students could investigate what higher dimensional fractals would look like if projected on to the 3D or 2D plane. An example of a 3D projection into the 2D plane would be a contour map.

Methods for Computing Mathematical Constants

Difficulty: 8/10
Mathematics Topics: Computer Science, Simulations, Calculus
About: Students can investigate the different methods for computing mathematical constants such as \pi, e, and \phi the golden ratio. Some investigation could be made about the complexity of the algorithms and how fast they converge to the constants.
Bonus: Come up with your own unique constant and an algorithm used to compute it. Talk about complexity and rate of convergence as well as if it can be represented by current day mathematical constants

Cardinality of Various Sets

Difficulty: 8.5/10
Mathematics Topics: Set Theory, Ordinals
About: Students can investigate the various cardinalities contained in sets such as the natural numbers, integers, irrational numbers, rational numbers, real numbers, and algebraic numbers. Students could also define their own set and try to determine the cardinality of that set.
Bonus: Students can devise a model for classifying the cardinalities of different sets based  on certain characteristics contained in each set.

If you have read Part 2, you might recall that I did write a research paper on a topic of my choice. From the above topics, my essay was actually about the last one, cardinalities, which you can access a copy, here. Some time in the future, I will be restoring that essay to a cleaner LaTeX typeset document. You may use it as a guide, but I highly encourage that readers take the time to learn about the above topics themselves. After all, to quote Georg Cantor:


English: Georg Cantor

"The essence of mathematics is in its freedom." {Image via Wikipedia}



I hope you all enjoyed my little mini-series rant about my high school experiences and if you did, I would love to hear about your experiences in the comments below. In the upcoming months, stay posted for even more content as I begin to dive straight into some serious mathematics this term and start to bring you my quality typsetted notes.

Complete LyX Templates Overhaul.

Those of you who have been following my blog from the start may recall that I created a couple of LyX templates to be used for educational and practical purposes. Well now I have added two more templates to the collection as well as some major changes to existing ones to better suit the needs of you, the readers.

University of Waterloo math & comp sci building

Image via Wikipedia

Mathematics Majors in co-op at the University of Waterloo may want to take advantage of the Mathematics Work Template as I have specifically made it to adhere to the work report structure specified here.

You can get a copy in the Resources section of my blog and included in the section is a link to the old version of the templates for anyone who doesn’t prefer the current ones.

To see  a full list of changes, please see the changelog below:

Changes from V1.0.0 to V2.0.0:

  • Added some “Lorem Ipsum”
  • Changed the default font to Computer Modern Sans Bright; hfbright package is required
  • “Contents” is now labeled “Table of Contents”
  • Added Roman numerals to the front matter without the need of the book document class
  • Two new Templates
    • Book Template
    • Mathematics Work Report Template (Uses Latin Modern font)
  • Added the captions package and modified the figure labeling styles
  • Removed the pesky URL boxes that appear when the PDF is compiled
  • Created custom colours for URL and other linking text via the hyperref package
  • Section Numbering in the Work Report now has a decimal added to it (i.e. Section 1 becomes Section 1.0)
  • Added a .bib file so the bibliographies in the templates will reference site
  • Added Appendices and List of Figures where applicable
  • Improved the appearance of the front and back matter
  • Made the “empty” fields more generic (e.g. Removed “Waterloo” from all of them)
  • All pages (except the title pages) use the “fancy” pagestyle; they also take advantage of the fancy headings features
  • Fixed a bug in the Table of Contents which caused some misalignment
  • All packages will use the AMS theorems modules
  • APA Styling is now used in the Bibliographies (except for the Work Report Template, which uses IEEE)