Summer of Math Exposition

The storytelling was great and it very cool to see how data from so long ago can be used. However, the structure and conclusion of the video left something to be desired. Only till the very end did it become clear where you were going with this (demography in fictional worlds). The conclusion was a bit weak and I still don’t really know what the main takeaway from watching this video should be.

Not bad at all but a bit weird video for my taste tbh. Started off well with mathematics, but got boring during the details of the mideval Doomsday book analysis. You lost me there, sorry, and was not able to get re-engaged after that. I do love your style, clear well paced explanation and simple but sufficient graphics.

nit: voiceover audio could use a de-clicker. mouse sounds are unpleasant and very distracting throughout the video.

Nice visuals. I like the “stand in boxes according to height” thing. makes the meaning of the pdf bar height intuitive. However, it may have been clearer if you started with this, had labels like 100cm-102cm for every bin, etc., and then shown that turning into the bars, and then animate the sample size and thin-ness of the bins increasing, then show that turning into the continuous pdf.

also, the transition into talking about goblin height distribution is pretty abrupt, unmotivated, and the distribution of it is kind of handed down from on high. It might have been better to start out from the very beginning with the goal of generating realistic goblin heights, rather than talking about pdf and bins first. Motivate concepts before explaining them.

transition to cdf is abrupt and not well-explained. could have gone a little slower there. ask the question “how many goblins are below a certain height” and explain it discretely, noting the pattern, and then show how that generalizes to continuous, or just leave continuous out for pdf and cdf.

nit: 50% point of cdf is median, not mean

in general, many of the things you show are unmotivated and not explained, and there are many jumps that someone unfamiliar with the subject would not be able to follow easily. I find that the best videos I’ve seen have a clear flow of establishing motivation for something first, starting simple and concrete, and then generalizing and becoming more complex through a series of clear, intuitive steps. Lots of concepts are peppered in but not motivated or well-explained like the CDF, inverting a CDF, etc. and you go back and forth between discrete and continuous without much explanation.

You say you want to randomly generate realistic goblin height distribution, which could be used as motivation for talking about all of this, but you don’t explain how to actually do that other than handing down an algorithm from on high in a (confusing) flowchart at the end without really talking about it much. I would’ve liked to see a more detailed explanation of how to actually use this information to generate statistically accurate goblin heights. You could have structured this section like:

1. let’s generate realistic goblin heights for a game
2. how are real human heights distributed? (show real data with a histogram/pdf)
3. show histogram turning into continuous pdf with larger sample size, pdf is idealized histogram, the “true” distribution
4. how could we use this to make an algorithm to generate statistically accurate heights? it’s easy to generate uniform random, but not our pdf.
5. explain how/why if only we had (something like inverse cdf but not saying inverse cdf), we could generate heights according to our pdf.
6. explain what that thing is and how it’s cdf. again, start discrete and get continuous. use an example.
7. put it all together in an algorithm and show the step by step run-through of it, and animate it filling up a histogram sample-by-sample for the big payoff

nit: You say “I want 1 bin to have 5% of goblins”, but that doesn’t really make any sense (which bin should have 5%?) and it’s unclear why you’d want to do that.

the quadratic on the city graph is completely pulled out of nowhere, and i’m pretty sure it’s incorrect. this should be a pareto distribution, which is not necessarily quadratic. try doing a log-log plot and it should end up being linear iirc, indicating a power law. watch this great vsauce video about it https://www.youtube.com/watch?v=fCn8zs912OE

I think this analysis is totally incorrect and missing out on a very interesting phenomenon discussed in that video.

nit: you don’t need to un-log your scale to make a probability distribution. log scale is just for graphing. you’re conflating things.

the critical step, which answers the question of the video title, is also totally wrong. You make lots of assumptions that things scale linearly even though they don’t, you eyeball the modified distributions for both regions, and also don’t rescale your distributions according to the different total population sizes as far as I can tell.

The whole video lacks focus. Almost the entire video is about PDF -> RNG in two different settings, which is not the title of the video, and then at the end you shove in this thing with the city estimation and it’s rushed and inaccurate. The 2 PDF -> RNG sections are redundant, the city section didn’t get the attention it deserved, and the video as a whole wasn’t very cohesive. However, the city RNG -> city estimation transition/buildup was done better than the goblin RNG -> city RNG transition since going from city RNG to estimation followed pretty naturally.

I love the core idea of using the historical data and the pattern to answer the question. that part’s great. I think the video would’ve benefitted from focusing on that more, establishing motivation at the very beginning, and building up to it with clear explanations with intuitive steps. Here is how the video could’ve been structured to achieve this:

1. establish the goal: how many cities are there in middle earth?
2. here is some historical data from domesday book. it follows this interesting distribution
3. it turns out this pattern shows up all over the place (watch https://www.youtube.com/watch?v=fCn8zs912OE). explain why (snowball effect)
4. this is a very general power law, and most countries follow this pattern
5. let’s assume middle earth follows this pattern
6. use this all to answer the question

I also think the idea of generating random numbers following a PDF is interesting and could benefit from an explanation like this if cleaned up, but it distracts from the whole number of cities thing and could be a completely separate video.

But good visuals, good concepts, and good application of a cool idea from math for having fun with fiction.

The main crux of this video is taking something that has been explained a million times on YouTube but applying it to an attention grabbing example. There are some big leaps to take though and as a viewer I kept getting taken out of it, wondering if this was even a worthwhile endeavor to compare against the fictional Middle Earth. As a teaching tool, I think it cuts both ways. On the one hand “yay engagement!”, but on the other “well actually.” Why not use the doomsday book to predict the data it was studying, Midieval England? Couldn’t there have been something interesting there, maybe before the black death or after one of England’s many wars. There could have been a lot of neat history to discuss, but instead, goblins.

I enjoyed the creative context of trying to sample city populations for the context of designing better fantasy worlds. I thought the discussion of inverse sampling was very well done.

This was certainly an interesting and novel idea. I’ve never really seen mathematics applied to fictional worlds. The depth of research you put into this is admirable. I would suggest however that you you introduce the main problem that you want to solve towards the beginning of the video, since for the beginning I was left wondering where you would take this or what direction you would go. I didn’t rate this video very high since statistics doesn’t appeal to me too much, but that’s just personal preference. Good shot nonetheless!

Sorry I did not get the point of the video. I think the introduction should present the objectives and stakes, and oh the results.

That was really nice!

Strong points:

• Well recorded audio; reasonably good speaking voice (if maybe a bit monotone). I had no problem listening to this for the length of the video, which isn’t true for a lot of youtube ;)
• Fun angle on PDF / CDF.
• I love the final pitch on “you should do models / look at data / be a data scientist”.
• Kinda like how meandering the video was with going on tangents about history, and how that plays into the previous point.

Weak points:

• I’m not entirely sold on any these models being all that appropriate for the things your trying to do. That’s beside the main point of the video, I’m sure, but it still might have been a bit better if the final models might have been better? Not sure.

This video was not a great math exposition. I learned more than I wanted about medieval settlements. The goal seems to have been to use the medieval settlements as a model to apply basic concepts in probability and statistics. Unfortunately, I don’t think most viewers would learn much math in the 25 minutes. The narration was good.