Using Math to Check Rhetoric

Below is an analysis I posted that came out of a discussion in the comments in another post. I thought I would share.




Hey Sam,


Thanks for writing this well thought out and articulated post. It will help us both understand our thoughts better (I had never previously done the analysis below).


When you say that 200,000 people will be coming to Waterloo in the next 20 years (which I actually think will be low), I agree that they probably won’t live in the country (at least that of course should be the goal). I also agree with the goal of getting dense enough development around each station, because this is what will allow us to get cars off our road, our true goal.


My concern is that the LRT is going to have 16 nodes. Let’s assume that we can dictate that 100% of all future residential and office development will be placed directly beside one of these nodes (unlikely, but a good place to start). Then I believe that means your odds of living and working at the same node will be 6.25%* (although it has been a while since I did math! *edit: note, my math was indeed off originally, I have now fixed it) So that means there is a 93.75% that you will be living and working at different nodes. Now you will have to consider whether you have a car, and if so, whether you’ll take LRT. If you don’t have a car, you’ll be someone that is already taking the bus today, and will take the LRT tomorrow, so no reduction in cars on the road. But most likely you are someone that has a car, so that you can get to places outside of Waterloo like other cities, the beaches on Lake Huron, or for parts of your job. If you were one of these people, assuming both your office and work are right beside their closest node, would you LRT or drive? Well, if you LRT you have to walk to the station (0.5 min), wait for the train (3.5 min average during peak hours), ride the train (X min), and walk to work (0.5 min). So 4.5 + X min during peak hours. If you drive, you have to walk to your underground parking (0.5 min), drive X min or less (almost never traffic in Waterloo today, and most likely not for the next 10 years), and 0.5 min to park. So 1 + X min to drive. If we say X is 5 it will be 9.5 minutes instead of 6. So not only will driving take you less time, but you also won’t have to walk/wait in the hot/cold/rain/snow, and if after work you need to go somewhere off node, you won’t have to go home first to pick up your car. Which will you do? I’d give you at least a 90% chance of driving, but let’s put it at 50% to be safe.


So now we can tally it up. So for every 1000 people living on the corridor, where they both live and work right beside an LRT station, 531 people will take the LRT or walk, and 469 people will drive. Of course, this is best conditions, as many places to live or work will have a significant walks to the station first, increasing the odds that you’ll drive.


But now imagine that you could change the rules in the region, and to stop people from driving you said that you we’re only going to open 5 stations to start, so that development would focus there. Because using the same math then there would be a 20% chance you would live and work at the same node (1/5), resulting in only 400 people taking cars, a savings of 69 cars per 1000. Not bad but not great.


But now what if you took it to the extreme? What if you said you were only going to open a single station for the next year? That you were going to spend 800 million on it, and that it is going to be the most beautiful station anyone has ever seen. That it would make Grand Central Station in New York look like a toy. Do you think any condo developer build such that they couldn’t advertise as being right beside it? I don’t think so. Which would mean that now the chance of living and working at the same node would be 100%, and no one would take cars. A savings of another 400 cars per 1000, with the gracious assumption that 50% of people would take the LRT instead of drive when it would take longer to get there, ask you to go outside in the cold/hot/rain/snow, and eliminate your flexibility should you want to go somewhere off node after work.


I agree that we should be spending money to concentrate development so that we can take cars off the road. But if this is really our goal, we should start with one node and then move out from there. Because then everyone could walk, and no one would drive.


But of course you make a fantastic point. We are made up of 3 cities. So instead of deciding to pick a single node and get an extra 470 cars per 1000 people off the road, we have compromised, resulting in almost twice as many roads, twice as many road crews, and twice as many cars.


This is the problem with rhetoric. That it breaks down once you put it through actual analysis. This is why that instead of saying “we need to add mass transit to be able to handle growth”, we need to break that down into clear goals, and then analyze how the LRT will meet them. Because if we say the goal is to “spend the minimum amount of money to remove the maximum number of cars from the roads per 1000 people we grow” we will quickly see that there are much better options.



21 thoughts on “Using Math to Check Rhetoric

  1. Michael Druker

    This is the problem with rhetoric. That it breaks down once you put it through actual analysis.

    You have some serious gall writing that rhetoric.

    Then I believe that means your odds of living and working at the same node will be 6.25%* … So that means there is a 93.75% that you will be living and working at different nodes.

    No, cities don’t work that way. People and their jobs are not evenly distributed within the boundaries of an arbitrarily defined geographic region. People try to live where they have better commutes, or they do a trade-off between commute quality (for multiple family members) and other factors of interest (schools, leafiness of their street, what have you). One of the crucial factors in these kinds of decisions is transportation infrastructure, which is what connects disparate places.

    The thing that a good highway or a good transit line does is it brings places closer together, making it possible to travel to more places (or a bigger proportion of places) within the same span of time. That doesn’t mean distance is irrelevant. What it means for a transit line is that you get walkable, urban live/work/play nodes around stations (not all of them the same size or function) that strongly facilitate trips on foot. But if people do need to travel further, it’s more likely that they are going from one walkable node to another, and that they can do so with a transportation mode that extends their range on foot.

    1. Michael Druker

      P.S. Putting 200,000 people at a single node defined as a standard 800m / 10 minute walk radius is a density roughly four times that of Manhattan.

      1. Ted Post author

        Thanks for pointing this out Mike. What it we put 50,000 people at a node first (Manhattan density), and then grow the node (both size and population) to 200,000?

        1. Michael Druker

          I don’t really see the point in entertaining a “what if” that has no mechanism. You have not proposed any mechanism that has any evidence at achieving intensely high density in an area that is not dense to begin with.

          1. Ted Post author

            Yes, coming up with specific solutions would come next. But first we need to define the problem, and understand the variables that will drive it.

            I live at Bauer, easily arguable as the closest Condo to uptown Waterloo (this is why I chose it), which is easily arguable as the most dense area in KW today in which to “play”. And yet I never walk.

            Compare this to King and Spadina in Toronto, where I rent out a room that I have been going to on weekends for the past year. There everything is close, and I from Friday to Sunday I never drive. And it is amazing. What I miss most about Waterloo.

            By putting in 16 stations, we are almost guaranteeing we get 16 Bauers, instead of 16 King and Spadinas. I believe we should try to come up with something that will create one King and Spadina first, instead of trying to build 16 all at once. Not only will it make Waterloo a much better place to live, but it will also actually solve our problem by reducing the % of citizens that drive in the region. As it did for me in Toronto.

            1. Michael Druker

              Honestly, the main thing I take away from your comment is that you live in a very walkable area in Waterloo and choose not to walk anywhere, while doing something different in a very walkable area in Toronto. That really isn’t enough to generalize to behaviour of others, nor really is “stuff within walking distance” necessarily the only explanation. E.g. traffic and parking ought to be pretty big factors as well.

              By putting in 16 stations, we are almost guaranteeing we get 16 Bauers, instead of 16 King and Spadinas.

              You do realize that not all of the station areas are intended as high-intensity growth nodes, right? Plus, the process is organic – while part of it is attractive transit infrastructure and growth-supportive zoning, there is no mechanism by which a particular density can be willed to happen by a municipality.

              Realistically, the biggest growth areas in the short term will be downtown Kitchener through uptown Waterloo, as they already are doing both separately and in anticipation of LRT. There’s plenty of room for those to grow with demand. There’s also room for other areas to grow in different ways around stations.

              By putting in 16 stations along the Region’s most well-used transit corridor, we are guaranteeing to be getting useful transit infrastructure that helps people get where they want to go, and thus helps to both handle the real and growing demand and concentrate development.

              Walkability is a great goal, and one that is helped by an LRT-oriented urban form. But the overarching goal of the Region of Waterloo is growth management – limiting sprawl, limiting the amount of new infrastructure required to serve growth, and guiding development to happen as infill and reurbanization.

              1. Ted Post author

                Hi Mike,

                It is hard to debate with you when you talk in general terms. I laid out a specific, mathematical analysis of the LRT above that attempts to model how the LRT will affect the % of Waterloo residents that take cars.

                If you could show specifically where you think my math is wrong, and how it should be corrected, or if you could provide some mathematical analysis of your own, that would help move this debate forward.

                1. Michael Druker

                  Your math is not even wrong, to use a popular phrase. It’s the assumptions that are the issue. Such as that shortening the line would result in a better effect; whereas overwhelmingly the experience in other cities is that shorter lines are less useful and therefore don’t get used and don’t attract development. Or the assumption that it is possible to coerce dense development to happen in a very small geographic area and only there.

                  If you want modelling of ridership, you can find it on this site:

                  Keep in mind – there’s limits on how much driving can be done. The roads are fixed and with growth primarily being attracted to the central corridor, there’s not going to be widenings to go along with it. So new development can come with parking, but it can’t come with nearly as much driving as current development has – there’s just not the capacity to double or triple the number of cars as the urban population doubles or triples. For the development to happen at all, instead of congestion driving it away, there has to be mobility. The only way to provide mobility for substantially more people without knocking down tons of city to build roads is with transit.

                  1. Ted Post author

                    Hey Mike,

                    Re: assumptions. I am not saying shortening the line will make it more popular. I am saying we should spend the money to attract all the development to a single node first Obviously a train that goes nowhere won’t do this, it would have to be something else. This is how Dubai did it: Not saying we should do this, only that I would be shocked if there wasn’t a way. If I gave you $800mm to spend on whatever you wanted, do you think it would be impossible to attract all future development in Waterloo to one place?

    2. Ted Post author

      Mike, could you take me through how your different assumptions would mathematically change the logic I presented above? What numbers do you think would be different? Or what numbers/ calculations do you think we are missing, and how would you incorporate them?

      1. Michael Druker

        Sorry, I don’t think your numbers are meaningful, because I just don’t see much connection between them and reality. Unless you are master planning a completely new and totally isolated city, the present reality (which is the jumping-off point for change) is found largely in current trip patterns, current population patterns, and current urban form. That means existing nodes such as the universities, the downtowns, major commercial areas, major office and industrial parks, existing residential neighbourhoods; and existing transportation corridors.

        1. Ted Post author

          Mike, it seems like your argument is “you’re not a city planner, how could you know!” I can win that argument, but I think it is an attitude that will result in Waterloo being mediocre.

          Which numbers do you not think are meaningful? I would live to try to change them if you give me a chance.

          1. Michael Druker

            That’s not my argument. My argument is “you can’t predict much about how a city will work with figures pulled from thin air, but you can with figures rooted in how it currently works and why”. Non-incremental urban change is possible and often desirable, but it needs to nevertheless relate to existing patterns of urban use. In this day and age, the only way to get a clean slate for city-building is to build a completely new city.

            See this earlier comment about the figures.

            1. Ted Post author

              Driving around Toronto for 5 minutes and then driving around Waterloo for 5 minutes – we basically have a clean slate. There are literally corn fields a 5 minute walk from UW.

          2. Cory Albrecht

            Actually, the “You’re not an urban planner, so how can you know” is a legitimate point.

            Do you know what a person with an urban planning degree knows? Why should I just assume that you do, or that you know better than a person who has been educated and is working in that field? How much is there that you simply do not know about the field and therefore it doesn’t even cross your mind to add into these supposedly mathematical models you’ve come with?

            Every field has it’s “counter-intuitives”, like Goedel’s incompleteness theorems or the Monty Hall problem, where what the non-field-trained people get the answer wrong. How many of the intuitive assumptions you made in your models about how the urban environment will change based on the decisions people will make as the live and move here are simply wrong because they happen to be the counter-intuitives for urban planning?

            Would you ask a baker fix faulty electrical wiring in your home?

            1. Ted Post author

              Hey Cory,

              Follow the logic for yourself, and then find and bring forth the flaws you find, and I will do my best to incorporate them

              Even experts make mistake. In fact, here are a couple of good quotes from the white paper on the Ottawa LRT I posted (

              Citing research on rail estimates carried out by the US Department of Transportation, the
              authors observe that “for virtually every [rail transport] project the divergence between
              forecast and actual ridership was wider than the entire range of the critical decision
              variables. Actual ridership was 28 to 85 percent [average 65 percent] lower than forecast
              ridership, meaning that forecasts overshot actual development by 38 to 578 percent
              [average 257 percent].75 The authors conclude that the accuracy (or rather the degree of
              inaccuracy) of demand forecasting is a major source of uncertainty and risk in the
              appraisal of major projects.


              As the new mayor prepares to hand over Ottawa’s transit planning task to community
              volunteers in the new transit task force, it is difficult not to agree with Denley’s verdict
              …our highly paid experts and elected councillors have made a terrible mess
              of it. They poured millions of dollars and untold thousands of hours of staff
              time into a north-south rail plan that the public ultimately did not endorse.
              They’ve also spiked all the studies of the east-west rail plans. In truth, they
              have no plan, except a plan to spend.

              What are your thoughts on these quotes? Obviously Ottawa had experts too. Was it right not to question them as well?

              1. Cory Albrecht

                I never said that the experts were never wrong, I asked why should I believe you more than them? Again, would you ask a baker to fix the faulty electrical wiring in your house? Who is more likely to be right, who am I more likely to get accurate results from trusting when I do not know the field myself? The field-trained experts or some guy just spouting off who makes ignorant claims about things that are simply not true (e.g. iXpress getting canned)?

                Why should I believe those little models you’ve cooked up when people here in the comments on this website have shown you problems with your assumptions? Why should I believe yours rather than the models the urban planners involved with the LRT have come up with? How do you know that stuff you DO NOT KNOW about urban planning is unimportant and does not affect your models and why should I accept your claims? Like non-mathematicians and failing to solve the Monty Hall problem because of bad assumptions, how many of your assumptions are wrong and you just don’t know it?

                Yeah, Ottawa failed, but can you show me that the same problems that happened in Ottawa are explicitly happening here in Waterloo Region? Can you show me that the ridership models presented here have the flaws mentioned in the Ottawa review?

                You are making these claims – you need to to do the background work

                1. Ted Post author

                  Hey Cory,

                  You don’t have to believe me – I have tried to present it in a way that anyone could follow along and judge for themselves.

                  So far there have been two good flaws brought up about my analysis, one that one day driving will get much slower and then the LRT will make sense (which I agreed with, but that it was a depressing thing to wait for) and one that said parking will one day get expensive enough for the LRT to make sense (which I somewhat agreed with, but again today that certainly is not the case).

                  But you can help me. What situation are you in that you are so passionate about the LRT? Do you bus today? Where do you go? Help us move the discussion forward.

  2. Pingback: Off For the Weekend | An Attempt to Mathematically Prove the LRT Is Right

  3. Alex

    You are still ignoring that people do not choose their place of residence independently of their place of work. I’ve lived in eleven different places over the last twenty years and in each case an important parameter was ease of access to public transit (and I’ve owned a car for all that time). In the end in four of said locations the other variables outweighed bad access to public transit so I ended up taking a place with poor transit. In all others cases I rejected at least one otherwise reasonable location because of their lack of access to public transit.

    In fact you might not even think of it, but you yourself are a regular user of public transit. And why do you use it regularly? because it is the most convenient option, all variables considered.

    By now I’m sure you are saying “what the hell is he talking about, I don’t use public transit”. But you see, you do all the time, when you fly somewhere. And why do you choose flying like that? because is the most convenient option considering the alternatives (driving to Vancouver, swimming across the Atlantic, renting a private jet at a cost of $2K-10K an hour).

    You are building a “mathematical argument” around a false premise. I hope you are smart enough to see that.


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