Understanding the Cavs three-point variance

A closer look into what actually determines three-point shooting luck.

It’s that time of year again for the Cleveland Cavaliers: Playoff time! The time of hopes, dreams, and cliches. We all know them. We all say them.

“It’s a make-or-miss league.”

“They won’t shoot that good again.”

“We won’t shoot that poorly again!”

The truth is that they might. So, before we get started on the most hopeful playoff run since Lebron James wore the wine and gold, let’s take a journey into the wild world of statistics and get more grounded on what is, and is not, lucky.

Let’s try to understand variance.

Basketball is a complicated game, but for this exercise, we will start simple and focus on rules of thumb and reference charts. For now, we will ignore the real world, defenses, confidence, shot profiles, and rotations. We will treat shooting like flipping a weighted coin, where the result is purely random. We will also avoid most math jargon and formulas (at first).

Everybody understands field goal percentage and three-point percentage. We all know what making 40% of three-point attempts means. The difference between being a 40% shooter and a 36% shooter, as more and more shots go up, is when everything becomes less intuitive. It can be tricky to tell the difference between a bad night and a shooting slump. Thus, we introduce the bell curve and the 68 / 95 / 99.7 rule-of-thumb.

If you have a set of events, say shooting ten three-pointers, then you can calculate the mean and standard deviation of that set. The mean is the center point. It’s the average outcome of the event if you do it over and over again. It’s how often you make or miss the shot. That’s your field-goal percentage and the point where the curve is centered. Forty percent is in the figure above. The standard deviation is the measure of variance if you redo the whole set again. If you shoot ten more shots.

But what does that actually mean? This is where sample size comes in. Say you shoot 10 shots and you are a 40% shooter. On average, I expect you to make four shots. But we all know that won’t always happen every time you shoot 10 shots. Sometimes you’ll make four, sometimes six, sometimes two. If we make a simple random outcome assumption, the standard deviation on 10 shots for a 40% shooter is 1.55 makes.

Confusing? This is where the 68 / 95 / 99.7 rule comes in. Your average plus or minus one standard deviation happens about 68% of the time. Your average plus or minus two standard deviations happens about 95% of the time. So take our 10 shots at 40% example:

Sixty-eight percent of the time, you’ll make between 2.45 and 5.55 shots. That’s business as usual. It’s what usually happens. If you make three, four, or five shots, then there is nothing interesting to see here. That wasn’t lucky or unlucky.

Ninety-five percent of the time, you’ll make between 0.95 and 7.1 shots. So if you only make one shot, that’s really unlucky. You’d expect to do better 97.5% of the time (remember, the 5% of outcomes not included with 2 standard deviations include the very best and very worst outcomes). If you make seven shots, that’s very lucky. You expect to do worse 97.5% of the time. Anything outside one to seven makes, and you are in wild luck territory.

You can see how sample size is very important here. The more shots you take, the more the expectation is that you’ll make somewhere around your average number. More shots in the sample give a smaller standard deviation. Fewer shots give a larger standard deviation. Then let the 68 / 95 / 99.7 rule do the mental work for you.

So, how does standard deviation move with sample size?

These plots show how sample size changes how you think about luck. If you only shot five shots, the blue curve, it isn’t unreasonable to make it or miss them all. Those outcomes are still within the 95% rule. If you shoot 20 shots, it’s very unlikely you make or miss them all. The way it works is intuitive. The hard part is knowing how lucky a specific shooting outcome is.

Now that we understand all the ideas and the basics of the math, let’s make ourselves a reference chart for the playoffs.

Now we are ready. If Darius Garland, shooting 40% on the year from deep, goes 2 -9 from deep in one game, then we can look at our chart and see we’re just outside the bounds of the 65 rule. That’s a little unlucky but not crazy. If Malik Monk, a 32.5% shooter from three on the year, drops 4-7 on us on a semi-important Sunday? Annoying, yes. But it happens.

But what if these trends continue at that rate as the shot count rises? If Garland goes 4-18 over two games? Now we’re starting to press into very unlucky territory. Now we start asking if anything is wrong, wondering about injury or fatigue, and looking at what the defense is doing to his shot profile.

Bonus Section - Introducing the Z-Score!

Are all these rules-of-thumb and rough look-up reference charts not for you? Do you absolutely need a hard and fast number? Are you, like me, afflicted with the tragic burden of “doing too much, bro” in casual barroom conversation?

Well, have no fear. The Z-Score is here for you. Unfortunately, there is no simple, closed-form, calculator-ready way to calculate the percentile of the binomial approximation we’ve been doing thus far. Fortunately, we can just flip things around and talk in terms of a Z-Score to say how lucky or unlucky something is (if everything were purely random… which it obviously isn’t).

The Z-Score is simply taking our nice bell curve from the beginning and speaking in terms of how many standard deviations from the mean the result is.

A Z-Score of one is right at that “this outcome happens 65% of the time” boundary. A two is at the 95% of the time boundary. And you can calculate it pretty easily!

As an example, let’s look at Darius Garland from March 1 through April 10. He was 40.3% from deep on the year. In that time frame, he shot 50-143 from three-point range.

A Z-Score of -1.30. That’s well past the line of the outcome we expect 68% of the time. It’s an extended cold streak to be certain.

But what about our good friend and leader, Donovan Mitchell, over the same time frame? He’s 36.8% from deep on the year and has shot 31-122 from three-point range. That is a Z-Score of -2.61. If his shooting was due purely to random chance, then this run would be a run approaching extreme bad luck territory. This is where you start considering other factors like fatigue, injury, or the type of shots he is taking. This is where we hope that rest is all he needs.

It also fully explains the March slump. Sometimes, basketball really is that simple. The Cavs’ two most important, highest volume shooters have been ice cold for over a month. And the Cavs still went 14-7. This team is very, very good.

So here is to good fortune and high Z-Scores in the playoffs! Keep on counting, stat-heads.

And if you like this type of content and want deeper statistical looks at the Cavaliers' season, let us know in the comments section.