How to Understand Statistics with Sports
A practical and easy guide to understanding the most valuable statistical concepts
For a long time, the core ideas of statistics escaped me. Probabilities? Things either happened or they didn’t. Like many, I believed if there was an 85% chance of something, it might as well have been 100%. If the opposite outcome occurred, it was because statisticians were wrong and stupid. Unfortunately, I was the one who was wrong and stupid. I hadn’t yet internalized the key ideas that made statistics valuable. Without truly understanding these ideas, I was a man using the handle of a hammer to hang pictures. What I was doing didn’t make any sense.
I was blaming the tools because I didn’t understand how the tools worked.
By watching the sports I love, I slowly began to understand statistical tools through practical examples. Over time, I began to understand the value of probabilistic thinking. Despite seeming elusive, it was my love of sports that began to illuminate statistical concepts. For example, when a basketball player who typically shoots 33% starts a game shooting 3/3, does that mean he’s now a 100% shooter? Well, not really. We haven’t seen him shoot enough shots to have a representative sample. What’s more likely is that if he keeps shooting, by the end of the game his make percentage will be much closer to his typical shot percentage.
Over time I began to realize that most people understand stats. This simple example shows how you already understand many statistical concepts. It contains core concepts like variance, the effect of small sample sizes, and averages. What many people didn't have was a simple vocabulary to describe what they were seeing. For something to stick in my brain, I need a practical example of the concept. Not a lofty academic definition. The guide below contains simple, concrete examples of statistical concepts that you can use and reapply to every area of your life. With applications in sports and business, you will be able to wield powerful statistical tools without utilizing significant mental horsepower.
Helps us form reasonable expectations
Consistent way to change expectations when conditions change
Scientifically measure and analyze performance of new tactics
We can visualize and experience the concepts with something we’ve know instead of charts and graphs
Sports are extremely popular all across the world
Easy to analyze and measure because of clear game rules
Valuable (You can make money!)
Visible (Almost everyone works for a business!)
I like it (so the examples are easy for me to come up with)
How it Works
Here’s a guide to navigating this blog. Each section will cover a statistical concept and include a sports and business example. It’s easiest if you read it in order but not required.
Here are the statistical tools:
Here are strategies using the tools:
Shots on Goal
Disclaimer: This guide is about practical, not theoretical stats. That means some of the terms used are my own terms to help you understand the idea behind the concept better.
Expected value tells us what value to expect! It sounds crazy, but it's just that simple.
If James Harden takes 10 three-pointers at tomorrow’s game and he typically makes 40% of his 3’s, we can expect him to earn 12 points from those shots (.4 * 10 shots * 3 points per shot).
Why this matters: We can make expectations on how many points we can or should have scored based on who shot what shots, where, and how many times.
Expected Value in Business: Predicting Sales
If we have a business and we want to close 10 customers and we have a 10% close rate, we need to reach out to 100 customers. Anything less than that then we can infer we got lucky, we are dealing with a different population, or our process improved. Expected value gives us a framework to form reasonable expectations about future events.
Sometimes you're hot, sometimes you're cold, but over time you converge on your average. Just because you’re a 33% shooter, doesn’t mean you always make one shot and miss the next two.
Why this matters: If someone makes 3 shots in a row, you shouldn’t expect that result to continue forever into the future. Their performance varies over time converging on their average production.
Variance in Business: Stock Volatility
In investing, the stock price varies nearly every second of every day. If stats help us form reasonable expectations, when observing a stock that moves up 1% today, can we expect a stock to grow at 1% every day? Once again, the obvious answer is no. An infinite number of factors could've driven this price change. What explains this non-linear phenomenon? Variance (or in business parlance, volatility). Volatility basically means that we can expect that even though a stock might grow 15% a year, we cannot expect this to be a smooth, linear ride to y*1.15. While this isn't the exact academic definition of volatility, it's close enough to help us form reasonable expectations.
The key point with volatility, expect the unexpected.
Taking the difference between two values may not reflect the true impact. Sometimes you need to take the difference between the two options to understand what is going on. Net Net is when we take the difference of a difference.
If Draymond Green is a 30% 3 point shooter and Steph Curry is a 40% 3 point shooter, Steph is not a 10% better shooter but actually a 33% better shooter! (Net Net)
Why this matters: This means that over 100 shots, Draymond will earn 90 points, and Steph will earn 120 points (a 33% gain!) because he is a 33% percent better shooter Net Net.
Net Net in Business: Deciding where to invest
Today, many active investors must convince LP's to give them their money by outperforming a set benchmark. Let's call this benchmark the S&P 500. Let's say the hurdle rate (rate at which investors must outperform to see a result over the S&P hint: it's got net net built in) is 7%/year. In this example, active investors must not only make 6% per year, they must also net something greater than that to get an investor to invest in them! This means if you pitch a 9% yearly return, you're not pitching a 3% improvement but a 50% increase in their return! How?
If I give you $100 and it returns 6% that leaves us with $106. If I give you $100 and it returns 9%, that leaves us with $109. (9-6)/6 = 3/6 = 50%!
This concept may be counter intuitive but it's very powerful in games that are very competitive (like basketball and finance). When percentages are small, small changes can mean huge net net gains!
Shots on Goal
Shots on Goal comes from the world of soccer. It’s a strategy to employ when our shooting percentage is predictable and has low variance. When you employ this strategy, you expect to win by simply taking more shots.
Why this matters: If a certain shot does not decrease in expected value over time or frequency of use, we want to maximize our “shots on goal”. This means sometimes to have the best shot at winning, you have to shoot your best shot over and over and over again. You’re likely to have a greater total of missed shots per game, but this is okay.
Shots on Goal in Business: Increasing Sales
Shots on goal tightly relates to the framework of expected value. If we have low variance on our close-rate, we can reasonably expect that increasing the number of reps should create outcomes in lockstep. For example, if we close at 1%, and currently outreach 100 prospects per week, we close 1 customer per week. If we're highly confident about this 1% figure, doubling to 200 outreached prospects per week means we can expect 2 customers per week. Now the catch is these are small values and there IS variance in most systems like this one but this is simply to let us know over a large time frame if we're on track or if something is wrong.
Sometimes combining two extremes is better than playing the average.
The Houston Rockets are a perfect example of a barbell strategy. Instead of trying to be able to shoot every shot on the floor, they are highly selective about which shots they take. They want to maximize the number of open 3's, layups, and free-throws. Why? Because these shots have the highest expected value. They barbell high value shots (3's) with high percentage shots (layups and free throws). Together, it makes for one of the most efficient offenses of all time.
Why this matters: Going head to head with an opponent and matching strategies works for dominant teams. For underdogs, it may make sense to try to go to the extremes of the distributions to find out-performance.
Barbell Strategies in Business: Google’s Moonshots
Google executes on a barbell strategy. They have on extremely consistent and profitable product, search ads. Then they take large chunks of their money and distribute it to "moonshots". These moonshots have an extremely low probability of success, but the nature of these projects are that they must have the potential to become the next search ads if successful. Why would companies do this? Because once you reach a certain size, a million dollar return on a project does nothing. The hurdle rate to continue growing is so high it nearly necessitates taking moonshots.
Once we understand variance, expected value, and net net calculations, we can make intentional trade-offs on what we can and can’t accept.
Take a look here to see one of the best sports game plans of all time. As a defense, we can determine which shots and areas of the floor we want to protect. We cannot be everywhere all the time, and that means sometimes people are open.
Why this matters: We can strategically select which shots from which areas from which players are below a certain threshold (kind of like an inverted hurdle rate) and intentionally allow the offense to shoot these shots!
Trade-Offs in Business: Where to spend our marketing dollars
Trade-offs inform us of what we’re giving up when we employ a particular strategy. Using expected value, we can attempt to quantify what strategy is best. For example, if we have $10,000 to spend on marketing this month, how should we distribute it over the following 3 strategies?
We spend all $10,000 on Facebook ads that have a 99% chance of returning us 1000 sales of $20 product.
We spend $10,000 on a print ad that is expected to return us 500 sales on a $50 product, but has a 20% chance of returning 2000 sales.
We spend $10,000 on a billboard that is expected to generate 200 sales on a $20 product but has a 1% chance of being placed next to a valuable property that would generate 50,000 sales.
With the first strategy, we are giving up a potentially greater outcome for the near certainty of performance. The second strategy is high variance but still can generate a significant return on our investment. The third strategy is essentially a moonshot. More often than not you lose your invested money. Will that 1% make it worth it? How certain that it’s actually 1% and not 0%? How long can you stay in the game and not run out of money? These are all trade-offs you must think about when considering where to spend your marketing dollars.
Now these examples are a bit far fetched, often you can distribute your dollars to many different channels. What’s important when deciding this is understanding what you give up by not choosing a strategy.
Putting it all together
Most importantly, when we understand these tools, we can build strong strategies. Integrate these tools into your everyday thinking and have a profound impact on your outcomes.
If we take the expected value of every player, we can compare their production levels net net! Using this, we can make trade-offs on who to defend and where on the court based on their shot selection. If certain players are outperforming in a game then it’s likely that they’re having a high-variance game, and/or an adjustment needs to be made to combat this.
We use the same tools to model and measure our sales team’s performance, what is going on in the stock market, and where to invest our money. Using examples from different domains, we begin to find patterns everywhere.
Learn the tools once, apply them everywhere. Go use these tools to make great decisions!