Win, lose, or draw: trends in English football match results

Is the game getting more exciting?

Football (soccer) fans like to see exciting matches. Draws are boring but wins or losses are interesting; fans want to see teams give their all on the pitch. Which begs the question, is the game getting more or less thrilling over time? One way to answer this question is to look at fraction of matches in a league that end in a draw. The most boring extreme is every game is a draw (draw fraction = 1). The most engaging extreme is that every game ends in a win/loss (draw fraction = 0). How does the proportion of drawn games change over time?

(User:Aloba, Public domain, via Wikimedia Commons)

Draws by league and by season

From multiple sources, I put together a file containing all English national league games from the foundation of the league system in 1888 to the end of the 2024-2025 season. The different leagues started in different seasons, with the National League (tier 5) being the most recent. The top tier (tier 1) is currently called the Premier League, though, like the other leagues, it has undergone a number of confusing name changes.

From this data set, I calculated the fraction of all matches in a season and a league that ended in a draw. I also calculated the standard deviations so you can get a sense of the spread of the data. (Because the standard deviation values aren’t close to 0 or 1, I don’t need to use the Wilson Score Interval approach here, the “usual” way of calculating the standard deviation or standard error of a proportion is good enough.)

This chart shows the fraction of draws by league by season. The salmon-colored blocks are World War I and World War II. I’ll explain the blue lines later. The chart is interactive; click on the legend to turn the leagues off and on.

The standard deviation makes this chart hard to understand, so I’ve re-drawn the chart without it (below). Again, it’s interactive.

Let’s look at the top tier (tier 1 – currently, the Premier League). The fraction of draws started off low (around 0.167 in 1888) increasing up to the start of the First World War (0.274 in 1914). Things were more or less stable in the interwar period and the immediate post-war years. In 1968, the draw fraction shot up to 0.303, remained more or less steady, before starting a slow decline after 1993 (0.216 in 2023). The other leagues show a similar pattern, except they show no decline post 1993. How do we explain what’s going on?

Are rule changes the cause?

Let’s start by looking at significant rule changes in the game. The blue lines represent the first season significant rule changes were introduced:

• In 1968, clubs were allowed to make substitutions for any reason (rather than just replace injured players). 
• In 1981, the points system was changed from 2 for a win, 1 for a draw, 0 for a loss, to 3 for a win, 1 for a draw, and 0 for a loss. The thinking here was that this would encourage clubs to be more risk taking and go for a win when them might otherwise accept a draw. 
• Of course, in 1993, the Premier League was founded.

You can judge for yourself the impact of the 1968 and 1981 changes.

The foundation of the Premier League (the new top tier) marks the start of a decline in the draw fraction for the top tier only; none of the other leagues show a similar sustained drop. The question is, why?

More or less equal?

There are at least two reasons why the fraction of draws in a league might change: 

• The clubs are becoming more (less) equal. If all the teams in a league were equally skilled, we might expect every match to be a draw. 
• The style of play changes so that wins/losses become more prevalent than draws. The clubs could still be equally skilled though; in the case of equally matched clubs, they might go from drawing all games, to winning and losing in equal measure. 

For the Premier League case, we have two competing explanations for why the draw fraction changed: unequal clubs vs. playing style. Fortunately, there is a way of analyzing the data for evidence supporting one of these explanations.

If all clubs are equal, then the fraction of matches each team wins will be about the same for all clubs. If clubs are very unequal, some clubs will win way more matches than others, so the win fraction for the top performing clubs will be higher than the win fraction for the low performing clubs. In other words, equality means little variance in the win fraction and inequality means high variance in the win variance.

For each league tier and each season, I calculated the standard deviation of the win fraction (which is the square root of the variance). The chart below shows the results. Bear in mind, the lower the standard deviation win fraction, the more equal the teams are, the higher it is, the less equal the teams are.

Prior to 1914, tier 1 and tier 2 show the same trend, a decline in win fraction standard deviation, suggesting the leagues are becoming more equal. Post-World War II, tiers 2, 3, 4 and 5 show no change over time with a low win fraction standard deviation, again suggesting equally matched teams. The Premier League is different, it shows an increasing win fraction over time, implying this league is becoming less and less equal; the difference between the winners and losers is getting bigger. 

To be clear, these results suggest that the cause of the Premier League declining draw fraction is not due to a change in the style of play, it’s due to the league tier becoming more unequal.

Supporting evidence

If you’re a football fan, inequality in the Premier League comes as no surprise. There’s been a lot of discussion about the Premier League having a league-within-a-league of top clubs (currently Liverpool, Manchester City, Chelsea, and Arsenal). A few years ago, the top clubs in Europe talked about forming a breakaway European super-league which lends credence to the idea that top clubs really are different. On the flip side, it’s also true that most clubs that get promoted into the league get relegated soon after (for example, Luton).

Money might be the cause, but the picture is more complicated than it seems.

A Deloitte analysis [https://www.deloitte.com/uk/en/services/consulting/research/annual-review-of-football-finance-premier-league-clubs.html] for the 2023-2024 season shows Manchester City had a revenue of £719 million, compared to Luton's £132 million. Any guesses how these teams finished the season? The drop from most revenue to least isn't linear either.

For the same 2023-2024 season, the entire English Championship League (tier 2) was £958 million, which is less than the revenue from just the top two Premiership clubs. Championship club revenue is also not evenly distributed, showing the same kind "top heavy" pattern the Premier League shows (see [https://swissramble.substack.com/p/english-clubs-by-revenue-grouping] for a chart). If the Championship also has unequal revenue distributions, why doesn't it show the same win fraction standard deviation as the Premier League? I'm not sure, but I can offer a couple of ideas. Money buys talent, but maybe it isn't even. For example, a £100 million player maybe twice as good as a £50 million player, but a £10 million player might only be 1.5 times as good as a £5 million player (there are many more £5 million players than  there are £100 million players). This would mean lower leagues become more equitable because the money difference matters less as the amount of money goes down. 

The bottom line is, the Premier League is becoming more winner-takes-all while the lower leagues are more equitable.

What does this mean?

It may well be true that draws are boring, but having dominant teams is also boring. If you can reliably predict who will win a match, it’s not as interesting. If your team always wins, why do you care?

Inequality has implications for promotion and relegation. Inequality suggests there may be a revolving door of the same few clubs moving between the Premier League and the Championship. If newly promoted clubs mostly get relegated, that's pretty dull. If it's the "usual suspects" for promotion/relegation then things start to look the same season to season.

Given that the Premier League was founded over money sharing issues, it's hard to see any changes that would more equitably distribute money. 

For the foreseeable future, we may have a very unequal Premier League with much more equal lower leagues.

Other football posts:

• A mini-app that has highlights of all the blog posts - https://blog.engora.com/2025/08/english-league-football-mini-app.html
• Win, lose, or draw: trends in English football match results - https://blog.engora.com/2025/06/english-football-data.html
• G-O-A-L! Goals in English football - https://blog.engora.com/2025/07/g-o-l-goals-in-english-football.html
• Attendance at English football: tragedy and recovery - https://blog.engora.com/2025/07/attendance-at-english-football-tale-of.html
• Vanishing home field advantage in English football - https://blog.engora.com/2025/07/vanishing-home-field-advantage-in.html
• Visualizing multi-dimensional data: score distributions in English football - https://blog.engora.com/2025/07/visualizing-multi-dimensional-data.html
• COVID and soccer home team advantage - winning less often - https://blog.engora.com/2021/01/covid-and-soccer-home-team-advantage.html - an old post based on older data.

AI networking in the Boston area

A lot's happening in Boston - where should I find out more?

There's a lot of AI work going on in the Boston area covering the whole spectrum, from foundational model development, to new AI applications, to corporates developing new AI-powered apps, to entrepreneurs creating new businesses, to students building prototypes in 12 hours. Pretty much every night of the week you can go to a group where you can find out more; there are a ton of different groups out there. But not all of them are created equal. I've been to a lot of groups and here are my recommendations for the best ones that meet on a regular basis. The list is alphabetical.

(Google Gemini)

AI Tinkerers

What it is

Monthly meeting where participants show the AI projects they've been working on. Mostly, but not exclusively, presentations from the Sundai Club (Harvard and MIT weekly hackathons). Attendance is over 150.

Commentary

This is where I go to when I want to see what's possible and find out more about the cutting edge. It's where I found out what tools like Cursor could really do. There are a number of VCs in attendance watching for anything interesting.

How often it meets

Once a month at Microsoft NERD.

Positives

You get to see what the cutting edge is really like.

Negatives

I found networking at this event less useful than some of the other events.

How to join

https://boston.aitinkerers.org/

AI Woodstock

What it is 

A networking event for people interested in AI. It attracts practitioners,  some VCs, recruiters, academics, and entrepreneurs. Attendee numbers vary, but typically over 100.

Commentary

This is networking only, there are no presentations or speakers of any kind. You turn up to the venue and introduce yourself to other people, and get talking. I've met people who are starting companies, people who are working on side gigs, and people who are working in AI for large companies. 

The quality is high; I've learned a lot about what's going on and what companies in the Boston area are doing. 

The venue is both good and bad. It's held in a corner of the Time Out Market near Fenway Park. This is a large space with lots of food and drink vendors, it attracts the bright young things of the Boston area who go there to eat and drink after work. AI Woodstock doesn't take over the whole space or rope off a portion of it and AI Woodstock attendees are only identified by name badges. This means you're chatting away to someone about their AI enabled app while someone is walking by with their drink and app to meet their friends. The background noise level can be really high at times.

How often it meets 

Once a month at the Time Out Market near Fenway Park.

Positives

• Networking. This is one of the best places to meet people who are active in AI in Boston.
• Venue. It's nice to meet somewhere that's not Cambridge and the food and drink offerings are great.

Negatives

• Venue. The noise level can get high and it can get quite crowded. The mix of bright young things out to have a good time and AI people is a bit odd.

How to join

https://www.meetup.com/ai-woodstock/ - choose Boston

Boston Generative AI Meetup

What it is

This is a combination of networking and panel session. During the networking, I've met VCs, solo entrepreneurs, AI staff at large companies, academics, and more. Attendance varies, but typically over 200.

Commentary

This is held in Microsoft NERD in Cambridge and it's the only event in the space. This means it starts a bit later and has to finish on time. 

Quality is very high and I've met a lot of interesting people. I met someone who showed me an app they'd developed and told me how they'd done it, which was impressive and informative.

The panel sessions have been a mixed bag; it's interesting to see people speak, and I found out a lot of useful information, but the panel topics were just so-so for me. Frankly, what the panelists said was useful but the overall topic was not.

How often it meets

About once a month.

Positives

• Networking. 
• Venue.
• Information. The panels have mentioned things I found really useful.

Negatives

• Panel session topics were a bit blah.

How to join

https://www.meetup.com/boston-generative-ai-meetup/

PyData Boston

What it is

Presentations plus networking. This is almost all machine learning/data science/AI practitioners in the Boston area (no VCs, no business people, instead there are academics and engineers). The presentations are mostly on technical topics, e.g. JAX. Attendance varies, but usually 50-100.

Commentary

I've learned more technical content from this group than any other. The presentations are in-depth and not for people who don't have a goodish background in Python or data science.

How often it meets

Once a month, usually at the Moderna building in Cambridge.

Positives

• Best technical event. In-depth presentations have helped educate me and point out areas where I need to learn more. Conversations have been (technically) informative.
• Probably the friendliest group of all of them.

Negatives

• No entrepreneurs, no VCs, no executive management.

How to join

https://www.meetup.com/pydata-boston-cambridge/

Common problems

There's a refrain I've heard from almost all event organizers and that's the problem of no-shows. The no-show rate is typically 40% or so, which is hugely frustrating as there's often a a waiting list of attendees. Some of these events have instituted a sign-in policy, if you don't turn up and sign in, you can't attend future events, and I can see more events doing it in future. If you sign up, go.

One-off events

As well as these monthly events, there are also one-off events that happen sporadically. Obviously, I can't review them here, but I will say this, the quality is mostly very high but it is variable.

What's missing

I'm surprised by what I'm not hearing at these events. I'm not hearing implementation stories from existing ("mature") companies. Through private channels, I'm hearing that the failure rate for AI projects can be quite high, but by contrast I've been told that insurance companies are embracing AI for customer facing work and getting great results. I've met developers working on AI enabled apps for insurance companies and they tell me their projects have management buy-in and are being rolled out.

I'd love to hear someone from one of these large companies get up and speak about what they did to encourage success and the roadblocks on the way. In other words, I'd like to see something like "Strategies and tactics for successful AI projects" run by people who've done it.

Your thoughts

I've surely missed off groups from this list. If you know of a good group, please let me know either through LinkedIn or commenting on this post.

Logistic regression - a simple briefing

A briefing on logistic regression

I've been looking again at logistic regression and going over some of the theory behind it. In a previous blog post, I talked about how I used Manus to get a report on logistic regression and I showed what Manus gave me. I thought it was good, B+, but not great, and I had some criticisms of what Manus produced. The obvious challenge is, could I do better? This blog post is my attempt to explain logistic regression better than Manus.

What problems are we trying to solve?

There are a huge class of problems where we’re trying to predict a binary result, here are some examples:

• The results of a referendum, e.g., whether or not to remain in or leave the EU.
• Whether to give drug A or drug B to a patient with a condition.
• Which team will win the World Cup or Super Bowl or World Series.
• Is this transaction fraudulent?

Typically, we’ll have a bunch of different data we can use to base our prediction model on. For example, for a drug choice, we may have age, gender, weight, smoker or not, and so on. These are called features. Corresponding to this feature set, we’ll have a set of outcomes (also called labels), for example, for the drug case, it might be something like percentage survival (a% survived given drug A compared to b% for drug B). This makes logistic regression a supervised machine learning method.

In this blog post, I’ll show you how you can turn feature data into binary classification predictions using logistic regression. I’ll also show you how you can extend logistic regression beyond binary classification problems.

Before we dive into logistic regression, I need to define some concepts.

What are the odds?

Logistic regression relies on the odds or the odds ratio, so I’m going to define what it is using an example.

For two different drug treatments, we have different rates of survival. Here’s a table adapted from [1] that shows the probability of survival for fictitious study. 

	Standard treatment	New treatment	Totals
Died	152 (38%)	17 (17%)	169
Survived	248 (62%)	103 (83%)	351
Totals	400 (100%)	120 (100%)	520

Plainly, the new treatment is much better. But how much better?

In statistics, we define the odds as being the ratio of the probability of something happening to it not happening:

\[odds = \dfrac{p}{1 - p}\]

So, if there’s a 70% chance of something happening, the odds of it happening are 2.333. Probabilities can range from 0 to 1 (or 0% to 100%), whereas odds can range from 0 to infinity. Here’s the table above recast in terms of odds.

	Standard treatment	New treatment
Died	0.613	0.165
Survived	1.632	6.059

The odds ratio tells us how much more likely an outcome is. A couple of examples should make this clearer. 

The odds ratio for death with the standard treatment compared to the new is:

\[odds \: ratio = \dfrac{0.613}{0.165} = 3.71...\]

This means a patient is 3.71 times more likely to die if they’re given the standard treatment compared to the new.

More hopefully, the odds ratio for survival with the new treatment compared to the old is:

\[odds \: ratio = \dfrac{6.059}{1.632} = 3.71...\]

Unfortunately, most of the websites out there are a bit sloppy with their definitions. Many of them conflate “odds” and “odds ratio”. You should be aware that they’re two different things:

• The odds is the probability of something happening divided by the probability of it not happening.
• The odds ratio compares the odds of an event in one group to the odds of the same event in another group.

The odds are going to be important for logistic regression.

The sigmoid function

Our goal is to model probability (e.g. the probability that the best treatment is drug A), so mathematically, we want a modeling function that has a y-value that varies between 0 and 1. Because we’re going to use gradient methods to fit values, we need the derivative of the function, so our modeling function must be differentiable. We don’t want gaps or ‘kinks’ in the modeling function, so we want it to be continuous.

There are many functions that fit these requirements (for example, the error function). In practice, the choice is the sigmoid function for deep mathematical reasons; if you analyze a two-class distribution using Bayesian analysis, the sigmoid function appears as part of the posterior probability distribution [2].  That's beyond where I want to go for this blog post, so if you want to find out more, chase down the reference.

Mathematically, the sigmoid function is:

\[\sigma(x) = \dfrac{1}{1 + e^{-x}} \]

And graphically, it looks like this:

I’ve shown the sigmoid function in one dimension, as a function of \(x\). It’s important to realize that the sigmoid function can have multiple parameters (e.g. \(\sigma(x, y, z)\)), it’s just much, much harder to draw.

The sigmoid and the odds

We can write the odds as:

\[odds = \dfrac{1}{1-p}\]

Taking the natural log of both sides (this is called the logit function):

\[ln(odds) = ln \left( \dfrac{1}{1-p} \right)\]

In machine learning, we're building a prediction function from \(n\) features \(x\), so we can write:

\[\hat{y} = w_1 \cdot x_1 + w_2 \cdot x_2 \cdots + w_n \cdot x_n\]

For reasons I'll explain later, this is the log odds:

\[\hat{y} = w_1 \cdot x_1 + w_2 \cdot x_2 \cdots + w_n \cdot x_n = ln \left( \dfrac{1}{1-p} \right)\]

With a little tedious rearranging, this becomes:

\[p = \dfrac{1}{1 + e^{-(w_1 \cdot x_1 + w_2 \cdot x_2 \cdots + w_n \cdot x_n)}}\]

Which is exactly the sigmoid function I showed you earlier.

So the probability \(p\) is modeled by the sigmoid function.

This is the "derivation" provided in most courses and textbooks, but it ought to leave you unsatisfied. The key point is unexplained,  why is the log odds the function \(w_1 \cdot x_1 + w_2 \cdot x_2 \cdots + w_n \cdot x_n \)? 

The answer is complicated and relies on a Bayesian analysis [3]. Remember, logistic regression is taught before Bayesian analysis, so lecturers or authors have a choice; either divert into Bayesian analysis, or use a hand-waving derivation like the one I've used above. Neither choice is good. I'm not going to go into Bayes here, I'll just refer you to more advanced references if you're interested [4].

Sigmoid to classification

In the previous section, I told you that we calculate a probability value. How does that relate to classification? Let's take an example.

Imagine two teams, A and B playing a game. The probability of team A winning is \(p(A)\) and the probability of team B winning is \(p(B)\). From probability theory, we know that \(p(A) + p(B) = 1\), which we can rearrange as \(p(B) = 1 - p(A)\). Let's say we're running a simulation of this game with the probability \(p = p(A)\). So when p is "close" to 1, we say A will win and when p is close to 0, we say B will win. 

What do we mean by close? By "default", we might say that if \(p >= 0.5\) then we chose A and if \(p < 0.5\) we chose B. That seems sensible and it's the default choice of scikit-learn as we'll see, but it is possible to select other thresholds.

(Don't worry about the difference between  \(p >= 0.5\) and \(p < 0.5\) - that only becomes an issue under very specific circumstances.) 

Features and functions

Before we dive into an example of using logistic regression, it's worth a quick detour to talk about some of the properties of the sigmoid function. 

• The y axis varies from 0 to 1.
• The x axis varies from \(-\infty\) to  \(\infty\)
• The gradient changes rapidly around \(x=0\) but much more slowly as you move away from zero. In fact, once you go past \(x=5\) or \(x=-5\) the curve pretty much flattens. This can be a problem for some models.
• The "transition region" between \(y=0\) and \(y=1\) is quite narrow, meaning we "should" be able to assign probabilities away from \(p=0.5\) most of the time, in other words, we can make strong predictions about classification.

How logistic regression works

Calculating a cost function is key, however, it does involve some math that would take several pages and I don't want to turn this into a huge blog post. There are a number of blog posts online that delve into the details if you want more, checkout references [7, 8].

In linear regression, the method used to minimize the cost function is gradient descent (or a similar method like ADAM). That's not the case with logistic regression. Instead we use something called maximum likelihood estimation, and as its name suggests, this is based on maximizing the likelihood our model will predict the data we see. This approach relies on calculating a log likelihood function and using a gradient ascent method to maximize likelihood. This is an iterative process. You can read more in references [5, 6].

Some code

I'm not going to show you a full set of code, but I am going to show you the "edited highlights". I created an example for this blog post, but all the ancillary stuff got in the way of what I wanted to tell you, so I just pulled out the pieces I thought that would be most helpful. For context, my code generates some data and attempts to classify it.

There are multiple libraries on Python that have logistic regression, I'm going to focus on the one most people use to explore ideas, scikit-learn.

from sklearn.linear_model import LogisticRegression

from sklearn.model_selection import train_test_split

from sklearn.metrics import accuracy_score, classification_report

from sklearn.preprocessing import StandardScaler

train_test_split splits the data into a test set and training set. I'm not going to show how that works, it's pretty standard,

Machine learning algorithms tend to work better when the features are scaled. A lot of the time, this isn't an issue, but if the values of features range very, very differently, this can be an issue for the numeric algorithms. Here's an example: let's say feature 1 ranges from 0.001 to 0.002 and feature 2 ranges from 1,000,000 to 2,000,000, then we may have a problem. The solution is to scale the features over the same 0 to 1 range. Notably, scaling is also a problem for many curve fitting type algorithms too.  Here's the scaling code for my simple example:

scaler = StandardScaler()

features_scaled = scaler.fit_transform(features)

Fitting is simply calling the fit method on the LogisticRegression model, so:

# Create and train scikit-learn logistic regression model

model = LogisticRegression(

random_state=random_state,

max_iter=max_iterations,

solver='liblinear'

)

# Train the model on scaled features

model.fit(features_scaled, labels)

As you might expect, max_iter stops the fitting process from going on forever. random_state controls the random number generator used; it's only applicable to some solvers like the 'liblinear' one I've used here. The solver is the type of equation solver used. There's a choice of different solvers which have different properties and are therefore good for different sorts of data, I've chosen 'liblinear' because it's simple.

fit works exactly as you think it might.

Here's how we make predictions with the test and training data sets:

test_features_scaled = scaler.transform(test_features)

train_features_scaled = scaler.transform(train_features)

train_predictions = model.predict(train_features_scaled)

test_predictions = model.predict(test_features_scaled)

This is pretty straightforward, but I want to draw your attention to the scaling going on here. Remember, we scaled the features when we created the model, so we have to scale the features when we're making predictions. 

The predict method uses a 0.5 threshold as I explained earlier. If we'd wanted another threshold, say 0.7, we would have used the predict_proba method.

We can measure how good our model is with the  function accuracy_score.

train_accuracy = accuracy_score(train_labels, train_predictions)

test_accuracy = accuracy_score(test_labels, test_predictions)

This gives a simple number for the accuracy of the train and test set predictions. 

You can get a more detailed report using classification_report:

        classification_report(test_labels, test_predictions)

This gives a set of various "correctness" measures.

Here's a summary of the stages:

• Test/train split
• Scaling
• Fit the model
• Predict results
• Check the accuracy of the prediction.

Some issues with the sigmoid

Logistic regression is core to neural nets (it's all in the activation function), and as you know, neural nets have exploded in popularity. So any issues with logistic regression take on an outsize importance. 

Sigmoids suffers from the "vanishing gradient" problem I hinted at earlier. As \(x\) becomes more positive or negative, the \(y\) value gets closer to 0 or 1, so the gradient (first derivative) gets smaller and smaller. In turn, this can make training deep neural nets harder.

Sigmoids aren't zero centered, which can cause problems for modeling some systems.

Exponential calculations cost more computation time than other, simpler functions. If you have thousands, or evens millions of nets, that soon adds up.

Fortunately, sigmoids aren't the only game in town. There are a number of alternatives to the sigmoid, but I won't go into them here. You should just know they exist.

Beyond binary

In this post, I've talked about simple binary classification. The formula and examples I've given all revolve around simple binary splits. But what if you want to classify something into three or more buckets?  Logistic regression can be extended for more than two possible outputs and can be extended to the case where the outputs are ordered (ordinal).

In practice, we use more or less the same code we used for the binary classification case, but we make slightly different calls to the LogisticRegression function. The scikit-learn documentation has a really nice three-way classification demo you can see here: https://scikit-learn.org/stable/auto_examples/linear_model/plot_logistic_multinomial.html.

What did Manus say?

Previously, I asked Manus to give me a report on logistic regression. I thought it's results were OK, but I thought I could do better. Here's what Manus did: https://blog.engora.com/2025/05/the-importance-of-logistic-regression.html, and of course, you're reading my take. 

Manus got the main points of logistic regression, but over emphasized some areas and glossed over others. It was a B+ effort I thought. Digging into it, I can see Manus reported back on the consensus of the blogs and articles out there on the web. That's fine (the "wisdom of the crowd"), but it's limited. There's a lot of repetition and low-quality content out there, and Manus reflected that. It missed nuances because most of the stuff out there did too.

The code Manus generated was good and it's explanation of the code was good. It did miss explaining some things I thought were important, but on the whole I was happy with it.

Overall, I'm still very bullish on Manus. It's a great place to start and may even be enough of itself for many people, but if you really want to know what's going on, you have to do the work.

References

[1] Sperandei S. Understanding logistic regression analysis. Biochem Med (Zagreb). 2014 Feb 15;24(1):12-8. doi: 10.11613/BM.2014.003. PMID: 24627710; PMCID: PMC3936971.

[2] Bishop, C.M. and Nasrabadi, N.M., 2006. Pattern recognition and machine learning (Vol. 4, No. 4, p. 738). New York: springer.

[3] https://www.dailydoseofds.com/why-do-we-use-sigmoid-in-logistic-regression/

[4] Norton, E.C. and Dowd, B.E., 2018. Log odds and the interpretation of logit models. Health services research, 53(2), pp.859-878.

[5] https://www.geeksforgeeks.org/machine-learning/understanding-logistic-regression/

[6] https://www.countbayesie.com/blog/2019/6/12/logistic-regression-from-bayes-theorem

[7] https://medium.com/analytics-vidhya/derivative-of-log-loss-function-for-logistic-regression-9b832f025c2d

[8] https://medium.com/data-science/introduction-to-logistic-regression-66248243c148

Tell me on a Sundai.club – something novel in Boston?

At several events in the Boston area, I heard talk of something called the Sundai Club, a weekly AI hackathon for MIT and Harvard students. At the AI Tinkerers group, I saw some of their projects and I was impressed. This blog post is about the club and what I’ve observed from their presentations and from their code.

(Canva)

What impressed me

During the AI Tinkerers event, I saw several demos of “products” created by small teams of Sandai Club undergraduate students in 12 hours. Of course, all of the demos used AI, either to do processing in the background and/or for code generation. These demos were good enough to clearly demonstrate a value proposition.  

Let me repeat this because it’s important. A small group of undergraduate students are regularly building working prototypes in 12 hours. The impressive thing is the productivity and the quality coming from students.

Of course, the output is a prototype, but with AI, they’ve got a substantial productivity boost. All the UIs looked good and all the prototypes did something interesting.

I was impressed enough to dig deeper, hence this review.

How the club operates

This is a student club for MIT and Harvard students. It meets every Sunday from 10am to 10pm for a full day’s hacking. Not all the 12 hours is spent hacking, there’s a sunset run and presentations. Some of the sessions are sponsored by AI companies or companies in the adjacent space. Sponsorship often means providing free compute resources for example, computing power or hosting etc.

They have a website you can visit: https://www.sundai.club/ 

My review of their code

Most of the projects are posted on the website and of those, most have GitHub pages where you can view the code. I spent some time dissecting several projects to figure out what’s going on. Here are my thoughts.

Code quality is surprisingly good. It’s readable and well-structured. Is this because it’s at least partly AI generated? Probably. 

Code length is surprisingly short. You can read over all the code for one of these projects in less than 10 minutes.

Notably, they do use a lot of “new” services. This includes newer libraries and newer hosting services. This is a hidden benefit: their development speed isn’t just from AI, it’s from using the right (non-AI) tools.

LLM connections are simple. It’s just API calls and prompts. This was the surprise for me, I was expecting something more complicated.

Importantly, they use agentic AI IDEs. Cursor was the one I saw used the most, but I’ve heard of projects using Lovable and I’m sure there’s Windsurf usage too. In fact, a Sundai club presentation was the first time I saw people “vibe coding” using voice (via the Whisper add-on). Agentic IDEs seem to be key to the productivity gains I saw. 

Why is this so interesting

• They’re producing prototype “products” in less than 12 hours with a small team. This would have taken more than two or three weeks in the past.
• The quality of the code is high. It’s at least as good as some professional code.
• They’re using the latest libraries, IDEs, and tools. They really are on the cutting edge.

Next steps

The most obvious thing you can do is visit their website: https://www.sundai.club/ and view some of their projects.

If you’re in the Boston area, you can often catch Sundai Club presentations at the AI Tinkerers group, which is open to anyone: https://boston.aitinkerers.org/ 

Don’t stop till you get enough – sample size in machine learning

How many samples of labeled data do you need?

It turns out, finding out how many labeled samples you need to “correctly” build a supervised machine learning (ML) model is a hard question with no clear answer. In this blog post, I’m going to run through the issues and finish with some advice for people managing ML model building.

(Canva)

Why does it matter?

Sample size plays into two big related themes for ML models:

• Correctness. This means how correctly your model predicts results at a point in time.
• Reliability. This means how correctly your model works over time.

Small sample sizes tend to give models that have a lower correctness and that give worse performance over time. This is all tied up with variance and the “law of small numbers”.

Let’s say your manager comes to you and asks you to build a ML model on a data set. When do you express concern at the size of the data set? When it’s 10, 100, 1000, 10000, or 100000 samples? What happens if your manager asks you to justify your concern?  

For a correct, stable model, you typically need a “big enough” data set to train with, but how much is “big enough”? 

What does sample size mean?

Before I dive into this some more, I should define what I mean by sample size. I mean the size of the labeled data set used for training a supervised machine learning model excluding cross-validation and hold out data sets. For example, if you use 20% of your data for hold outs, and 80% of your cross-validation data is training, only 0.8*0.8 = 0.64 of your data counts towards sample size. 

Why is this a hard problem?

There’s very little in the literature, there’s almost nothing in the leading books on machine learning, and it’s only mentioned in passing on machine learning courses. It’s an area of active research, which means there’s nothing packaged for easy use.

I’ve spent hours searching for and reading papers on this topic but I’ve not found anything useful. What I did find is that the field that’s most advanced is medicine. Researchers are increasingly using ML models for clinical trials and they need to know how many patients to enroll in their trials. It seems that they’re mostly using statistical tests (see below) for sample size however, some researchers are trying to develop robust statistical methods to independently estimate sample size. However, as of June 2025, there’s no consensus on the best approach. 

What do other disciplines do?

In frequentist statistics, there’s a recipe for determining sample size given significance, power, and effect size for a single comparison test (formally, a null-hypothesis test). The code exists in R and Python libraries, so all you have to do is put the numbers into a formula and you get your minimum sample size. Everyone doing randomized control tests (RCTs, AKA A/B tests) works out sample size before running a test.

The nearest statistical equivalent to ML is multi-comparison null-hypothesis tests, which is really something different, but it does give you some idea of sample size. The math is more complex and most people use something called the Bonferroni correction to go from single comparison to multi-comparison testing. To give you an idea of numbers, the table below shows the minimum sample size for a proportion z-test with a significance level of 5%, a power of 85%, a baseline proportion of 5%, and a 5% effect size, with Bonferroni correction.

Comparisons	Sample size
1	272,944
2	409,416
3	545,888
4	682,360
5	818,832
...

Two things here: the sample size starts at 272,944 and it goes up for each test you add. 

Notably, the sample size for a null-hypothesis test depends on the effect size; a big effect leads to smaller tests. This is why most drug trials have sample sizes in the low hundreds, the effect they’re looking for is large. Conversely, in retail, effect sizes can be small leading to sample sizes in the high hundreds of thousands or even millions. This might be an important clue for ML sample sizes.

What rules of thumb are there?

The general consensus is, if you have n samples and f features, then n >> f. I’ve heard people talk about a 50x, 100x, or 1,000x ratio as being minimal. So, if you have 5 features, you need a minimum of 250-5000 samples. But even this crude figure might not be enough depending on the model.

What do people do in practice?

I’ve never come across a data scientist who estimates needed sample size before building a model. People use the cost function instead: if the cost function is “good enough” this suggests the sample size is good enough too. There are variations on this with people using confusion matrices, precision-recall, etc. etc. as “proxies”; if the metric is good enough the sample size is good enough.

But relying on the cost function or metrics alone isn’t enough. I’ve seen people develop models using under a hundred samples with over five features. The cost function results were OK, but as you might expect, the model wasn’t very robust and gave poor results some of the time.

Let me draw a comparison with an RCT to evaluate a new drug. All trials have an initial estimate of the sample size needed, but let’s say they didn’t and relied on metrics (e.g., fraction of patients cured). Do you think this would be OK, would you take the drug? Would you take the drug if the sample size was 10, 100, or 1000 patients? Or would you prefer there to be a robust estimate of the needed sample size?

My recommendations

The situation isn’t very satisfactory. Frequentists statistics suggests hundreds of thousands of samples which looks very different from the 50x-1,000x rule of thumb. Even the 50x-1,000x rule of thumb gives a huge range of answers. Using the cost function or metrics alone doesn’t feel very safe either.

I’m not in a position to give a robust statistical recipe to calculate sample size. All I can do is offer some advice. Take it for what it's worth.

1. Ideally, have a sample size of at least 100,000, but make sure you have at least 1,000x as much data as you have features. If you really have to model with less than 100,000 samples, recognize you're on very slippery ground.
2. Run a feature importance analysis. If you have many features each with a small contribution, that’s a warning sign; you should consider increasing your sample size.
3. Regularly performance check your model and have pre-determined thresholds for taking action.

Don't stop till you get enough

I was thinking of this song when I was writing this.

How to Thrive in an AI-Driven Future

On Wednesday, I went to a panel session in Boston on AI. I thought my notes might be useful to others, so here they are. The title of the panel was "How to Thrive in an AI-Driven Future", the thing thriving is the city of Boston and the surrounding areas.

What was the panel about?

The panel was about the current state of AI in the Boston area, focused on how Boston might become a hub for AI in the near term. It discussed the Boston areas' strengths and weaknesses, and along the way, it pointed out a number of great AI resources in the local area.

(King of Hearts, CC BY-SA 4.0 <https://creativecommons.org/licenses/by-sa/4.0>, via Wikimedia Commons)

It was held in the Boston Museum of Science on Wednesday June 4th, 2025.

Who was on the panel?

• Paul Baier, CEO GAI Insights - moderator 
• Sabrina Mansur, new Executive Director of Mass AI Hub
• Jay Ash, CEO of MACP, Massachusetts Competitive Partnership
• Chloe Fang, President of MIT Sloan AI Club

What did the panel talk about?

The panel was at its most comfortable talking about universities and students. There was a lot of chatter about Harvard and MIT (the event was sponsored by Harvard Business School Alumni) and the 435,000 students in the state. There was also mention of the state's great educational record. Jay brought up the term "the brain state" for Massachusetts.

Apparently, there are about 100 business incubators in MA housing 7,500 companies, with 20,000 employees. These are bigger numbers than I would have expected.

Several panel members mentioned the Massachusetts Green Higher Performance Computing Center in Holyoke. I didn't know about it and it's nice to hear about these kinds of initiatives, but no-one connected it to promoting or developing AI in the state.

Sabrina talked about the state releasing some data sets in the near future as data commons. The panel all agreed this would be a great step forward. It wasn't clear what data was going to be released, when, and how, but comments later on seemed to indicate this would be Massachusetts data only.

A good deal was made of the state's $100mn AI Hub initiative, but it seems like this money has been approved but not allocated and it's not clear what it will be spent on and when. There was a hint that there might be some focus on SMBs rather than large businesses.

Chloe talked about how AI and code gen has enabled new people players. She said that a few years ago, MBA students didn't have the technical skills to build demo products, but now, with the rise of code gen, they can. She talked about MBA hackathons, something that would have been impossible until recently.

The whole panel seemed to have a love affair with the MIT and Harvard Sundai Club. This is a student club that meets on a Sunday and produces complete apps in a 12 hour period, obviously focused on AI. (I agree, there's some very interesting things going on there.)

There was some discussion on making regulation in the state appropriate, but no discussion about what that might mean.

While there was a lot of discussion on problems, there were strikingly few ideas on how to resolve them. Two issues in particular came up:

• Funding
• Livability

The panel contrasted how "easy" it is to get funding in San Francisco compared to Boston, and that's both at the early stage and the growth stage. There were some comments that this view is overblown and that it's easier than people think to get funding in the Boston area. Frankly, there were no real suggestions on how to change things. One ideas was to tell students in Boston that it's possible to get funding here, but that's about the only suggestion that panel had.

There were a couple of questions around livability. An audience question pointed out that rents in the Boston area are high (though San Francisco and New York rents are probably higher), but the panel dodged the question. On the subject of "things to do for twenty-somethings", the panel deferred to the youngest panel member, but again, nothing substantive was said. The panelists did talk about Boston being an international city and how its downtown doesn't really live up to that right now; the view was, Boston city government needed to step up.

Boston AI Week, which is being held in the Fall, was heavily promoted. 

What were my take-aways?

While there was a lot of discussion on problems, there were strikingly few ideas on how to resolve them and I'm not sure then panel had thought the issues through.

MIT and Harvard (in that order), dominate the intellectual landscape and mind share. They certainly dominated the panel's thinking. In my view, this is fair and the other universities only have themselves to blame for being left behind. While they don't have the resources of Harvard and MIT, they could run the equivalent of the Sundai Club, and they could put people up for panel sessions like this. They could also organize events etc. Yes, it's harder for them, and yes MIT and Harvard have more resources, but they could still do a lot more.

I was left with the feeling that there's no real coordination behind Boston's AI groups. While there are individuals doing great things (Paul Baier being one), I don't get the sense of an overarching and coordinated strategy. 

(Two of the three trains I had to catch to get home.)

The International city thing struck a chord with me. My trip in was easy, I parked up and got one train right to the door of the Museum of Science. On the way back, things went wrong. I had to get three trains and a shuttle bus (almost two hours door-to-door, shocking). Nothing about my return trip said "international city".

Google's ADK for agentic AI development - and some general thoughts

Some observations on agent development

On Tuesday, June 3rd, 2025, I spent the day at Google's Cambridge, MA site at their "Build with AI" event. It was a hands-on tutorial to make agentic AI systems using Google's technology. The event crystallized a few things for me and helped sharpen my thinking. 

In this blog post, I'm going to review the workshop and talk about my general thoughts

(Kenneth C. Zirkel, CC BY 4.0 <https://creativecommons.org/licenses/by/4.0>, via Wikimedia Commons)

Workshop review

The goal of the workshop was to build a working agentic AI system using MCP, A2A, and the Google Agent Development Kit (ADK). Of course, this was all done using GCP.

The session started with an overview and some theory. Thankfully, this was done well; the team kept the introductions short and dived straight into the workshop. The theory was standard stuff, an introduction to the technologies used and some of the relevant history.

Something like 70% of the workshop was setting up various Google services, for example, a web server to serve the app, a server to serve the backend, and so on. Thankfully, this was all script based, but there were a lot of scripts. This really brought home to me the role of DevOps in AI. Someone asked about AI ops, and I agree with the question, it all felt like an outgrowth of DevOps.

The Python code we did use was pretty simple. Frankly, it was just a few API calls.  The focus was on the API call arguments, making sure we had the right arguments in place for what we were trying to do. I'm going to go as far as saying that the Python coding piece was trivial; there was nothing that would cause problems even for an entry-level programmer. It was made even easier by being cut and paste, we didn't even have to figure out the right arguments.

The presenter was keen to point out the message passing between servers and how we could debug it through the Google environment. This was my concern. I've tried to debug message passing between independent systems before, and it wasn't a good experience. Having Google provide a "trace" is very helpful and reduces my concerns quite a bit.

The workshop took about four hours and I managed to build the complete system a little while before the end.

Overall, I enjoyed it and got a lot out of it. Could I build their demo system from scratch by myself? No. The reason is, all the setup that needs to be done with the various servers. It's not at all clear to me the why behind some of the config scripts. But note the problem is not a data science one, or a software one, it's a DevOps problem. Do I feel I understand A2A and MCP better? Yes. Do I recommend the workshop? Yes.

The workshop is called "Build with AI" and it's going on the road soon.

General thoughts

Agentic systems are not the preserve of data scientists any more. In fact, it's hard to understand what benefits a data scientist would bring to the table.

Over the last year, the development of various abstractions, for example, A2A, MCP, LangChain, etc.  have made it much easier to build AI systems. We've got to the stage where these things are pretty much "off-the-shelf" APIs. With one glaring exception,  AI, and agentic AI in particular, now looks like a software engineering problem, so it feels like the preserve of software engineers.

Because frameworks like MCP and A2A are all about inter-system communication, message passing is now key. Frameworks all use some form of JSON message passing underneath. This make debugging much harder and means we need to see what messages have been passed between systems. To their credit, Google knows this and has produced software to let you trace messages. Debugging message passing is still new to many software engineers and I expect some problems, even with Google's tools.

AI systems are all about calls from one system to another. This obviously means permissioning, but it also means cost. A poor set up can cost a company a great deal of money and/or give a poor user experience. These kinds of problems are usually associated with DevOps. In fact, my overall impression of the Google workshop was it was mostly DevOps with some basic coding thrown in. 

In mid-2025, what skills do you need to develop agentic AI systems?

• Software engineering
• Devops.

There's no requirement for data science. In fact, you don't need to know how any of the LLMs work under the hood.

This is a brave new world.

Cursor for data science - a scorecard

What is this scorecard?

I've been investigating how to use Cursor for data science. This means using it on a real project and finding out its strengths and weaknesses. This blog post is a summary of my experiences and I'm posting it as a guide to others.

(Gemini)

Things in this space are changing quickly. This post is up to date as of June 2025. I may update this post in the future, but if you're reading this six months in the future and it hasn't been updated, please contact me if you want to hear more (https://www.linkedin.com/in/mikewoodward/).

Cursor scorecard

General

Area	Grade
Getting started	D
Usability	B
Debugging	C
Code generation	C
Code completion	A
Code commenting	A
Code tidying	D
PEP8 compliance	B
Documentation	A
GitHub integration	C
Error finding	B

Specific tasks

Area	Grade
Pandas dataframe manipulation	C
Web scraping	D
Data cleansing	C
Prototyping	A

Getting started with Cursor

Getting started is hard. This is very definitely an early adopter tool: 

• Product documentation is sparse. 
• There are very few online written tutorials. 
• There are a handful of courses, but only on Udemy. 
• Although there are many, many videos on YouTube, there are problems with them.

All of the YouTube videos I watched followed the same format, the development of a UI-based app. In all cases, the videos showed connections to LLMs to do some form of text processing, and in some cases, videos went through the process of connecting to databases, but none of the videos showed any significant (data science) computation in Python. On reflection, pretty much every Cursor demo I’ve seen has been focused on prototyping. That's fine if your application is a prototype, but not so great otherwise.

I got started by watching videos, talking to people at Meetup groups, and working on this project. That’s great for me, but it’s not scalable.

Although the Cursor free tier is useful, you very quickly exhaust your free tokens. To do any form of evaluation, you need a subscription. It’s cheap enough for that not to be a problem, but you should be aware you’ll need to spend some money.

Usability

The obvious problem is that Cursor isn’t a notebook. Given that most data scientists are addicted to notebooks (with good reason), it’s a major stumbling block any data science roll-out will have to deal with. In fact, it may well stop data science adoption dead in its tracks in some organizations.

Once you get round the notebook issue, usability is mostly good, but it’s a mixed bag. There are settings like rules which should be easier and more obvious to set up; the fact you an specify rules in “natural” English feels like a benefit, but I’d rather have something more restrictive that’s less open to interpretation. Rules have a bit of a voodoo flavor right now.

Debugging

Frankly, I found debugging harder than other environments. I missed having notebook-like features. There’s a variable explorer, but it’s weaker than in an IDE like Spyder. On the plus side, you can set breakpoints and step through the code.

Code generation

Very, very mixed results here.

Bottom line: code generation often can’t be trusted for anything technical and requires manual review. However for commodity tasks, it does very well.

Positives

It did outstandingly well at generating a UI in Streamlit. The code was a little old-fashioned and didn’t use the latest features, but it got me to a working solution astonishingly fast.

It produces ‘framework’ code really well and saved a lot of time. For example, I wanted to save results to a CSV and save intermediate results. It generated that code for me in seconds. Similarly, I wanted to create ‘commodity’ functions to do relatively simple tasks, and it generated them very quickly. It can automate much of the ‘boring’ coding work.

It also did well on some low-level and obscure tasks that would have otherwise requires some time on Stack Overflow, e.g. date conversion.

Negatives

Technical code generation is not a good story. With very careful prompting, it got me to an acceptable solution for statistics-oriented code. But I had to check the code carefully. Several times, it produced code that was either flat-out wrong or just a really bad implementation.

I found that code that required details instructions (e.g. specific dataframe joins) could be generated, but given how detailed the prompt needed to be, the cost savings for code generation were minimal.

On occasions, code generation gave overly-complex solutions to simple tasks, for example, its solution for changing the text “an example” to “An Example” was a function using a loop.

From a higher-level code structure perspective, code generation is not good. Persistently, it would create new functions rather than generalizing and re-using existing functions. For example, I had boiler-plate code to open a CSV and read it into a Pandas dataframe with error checking. Code generation created a new function to read in data rather than re-use the existing code. Once I told it to consolidate all the read functions, it did. Overall, it’s not good at generating well-structured code.

Although it’s a niche topic, it’s worth mentioning that code generation didn’t work at all well for web scraping.

Code completion

Excellent. Best I’ve come across.

There were several cases where code generation didn’t work very well, but code completion did. Code completion works well if the context is good, for example, if you create a clear comment, the system will offer code completion based on your comment, and almost all the time, it will do well.

I found code completion to be a very compelling feature.

Commenting code

This is definitely a Cursor superpower. It’s almost unbelievably good.

Code tidying

Some of the time, if you ask it to tidy your code, it will do the right thing. However, most of the time I found it introduces errors. 

PEP8 compliance

Surprisingly, generated code/completion code isn’t PEP8 ‘out of the box’, for example, it will happily give you code that’s way over 79 characters. Even asking the AI to make the code PEP8 compliant sometimes takes multiple attempts. I had set a rule for PEP8 compliance, but it still did didn’t fully comply.

Documentation

This means creating markdown files that explain what the code is doing. It did a really great job here.

GitHub integration

Setup was really easy. Usage was mostly OK, but I ran into a few issues where Cursor needlessly tied itself in knots. More seriously, it deleted a bunch of data files. 

Contrasting the usability of GitHub in Cursor with the GitHub desktop app, the GitHub desktop app has the edge right now. 

Github integration needs some work.

Error finding

In most cases, it did really well finding and correcting run-time errors, however I found a case where its error correction made the code much worse; this was processing a complex HTML table. Code generation couldn’t give me a correct answer, and asking the engine (Claude) to correct the error just produced worse code.

Pandas dataframe manipulation

This means the ability to manipulate Pandas dataframes in non-trivial ways, for example, using groupby correctly.

Cursor can do it quite well for basic manipulations, but it fails for even moderately complicated tasks. For example, I asked it to find cases where a club only appeared as an away team or a home team. The generated code looked as if it might be correct, but it wasn’t. In fact, the code didn’t work at all and I had to write it by hand. This was by no means a one-off, Cursor consistently failed to produce correct code for dataframe manipulations.

Code generation for scraping data

On the plus side, it managed to give me the URLs for the pages I wanted to scrape purely on a prompt, which frankly felt a bit supernatural.

On the negative side, it really can’t generate code that works for anything other than a simple scrape. Even asking it to correct its errors doesn’t work very well. The general code structure was OK, but a little too restrictive and I had to remove some of its generated functions. It’s marginal to me whether it’s really worth using code generation here. However, code completion was helpful.

Data cleansing

Cleaning data with code generation ran into the Pandas dataframe problem I've discussed above. Code completion was helpful, but once the manipulations become more complex, I had to hand write them.

Prototyping

By prototyping, I mean creating a UI-based application, for example, a Streamlit app, or even a standalone web app using react.js with a Python backend.

The results were outstanding.

You can generate apps in a fraction of the time it takes to do it by hand.

There are some downsides:

• Security is often not baked-in and has to be added later.
• The code often uses structures that are a little behind the latest thinking, e.g. not using new features of libraries.