What Strokes Gained Means In Disc Golf: The Math

A man in shorts pumps his fist as he walks on a grassy area in front of trees — Seppo Paju celebrates a big putt during a battle with Eagle McMahon at the 2019 Beaver State Fling. Original Paju photo credit: Alyssa Van Lanen

Strokes Gained is a concept borrowed from traditional golf that we are introducing to UDisc Live in 2021. If you know nothing about Strokes Gained, we suggest you give our article "Strokes Gained Stats Come To Disc Golf On UDisc Live" a read before continuing this one.

Below you'll find an in-depth explanation of the math behind Strokes Gained on UDisc Live with accompanying examples.

The Concept

The core idea behind Strokes Gained relies on the statistical notion of expected value.

In traditional golf, the PGA determines the number of strokes they expect it will take for a player to hole out from any position on a golf hole based on how Tour players have performed in the past from those positions. The main factors considered in the expected value computations are the distance to the pin, the relative difficulty of the hole, and whether the shot is taken from the fairway, the rough, or the green.

The general formula for the number of strokes gained on the field for a shot or a sequence of shots is this:

[SG] = [E_start] − [E_end] − [# Strokes]

In that formula, [SG] is the number of strokes gained on the field, [E_start] is the expected number of strokes to complete the hole from the starting position, [E_end] is the expected number of strokes to complete the hole from the ending position, and [# Strokes] is the number of strokes it took to get from the starting position to the ending position.

For clarity, let's apply that formula to a hypothetical shot.

A player is standing on the tee, and the expected number of strokes to complete the hole they're playing is 3.7. They drive, and from the spot they land in, it's expected that a player would complete the hole in 2.4 strokes. We now have our [E_start] and [E_end]: 3.7 and 2.4, respectively. How many strokes did it take for the player to get from [E_start] to [E_end]? Just one. That makes [# Strokes] equal to one.

Now, just plug everything into the formula to get Strokes Gained:

3.7 − 2.4 − 1 = 0.3

The player's above-average drive yielded 0.3 Strokes Gained, which then contributes to an appropriate Strokes Gained statistic.

So, where on the course will UDisc Live be tracking Strokes Gained, and how will we determine the expected number of strokes from each position? We answer both of those questions in the following sections.

Where Can Players Gain Strokes?

UDisc tracks three core Strokes Gained statistics:

Strokes Gained: Circle 1X [SG:C1X] measures a player's performance on putts from inside Circle 1X (over 11 feet/3.3 meters from the basket but within 33 feet/10 meters).
Strokes Gained: Circle 2 [SG:C2] measures a player's performance on putts from inside Circle 2 (over 33 feet/10 meters from the basket but within 66 feet/20 meters).
Strokes Gained: Tee → Green [SG:TG] measures a player's performance on all shots from outside Circle 2.

For all three of the core Strokes Gained statistics, a higher number represents a better performance. This can be slightly confusing for people who are used to looking at total score relative to par where lower numbers represent better performance.

All three of the core Strokes Gained statistics ignore penalty strokes. Instead, all penalty strokes are accumulated into a fourth statistic:

Strokes Lost: Out of Bounds [SL:OB]

We summarize a player's performance with two additional aggregate statistics:

Strokes Gained: Putting [SG:P] = [SG:C1X] + [SG:C2]
Strokes Gained: Total [SG:T] = [SG:C1X] + [SG:C2] + [SG:TG] - [SL:OB]

The player with the highest [SG:T] stat for a tournament is the winner of the tournament. In fact, [SG:T] measures the number of strokes by which a player exceeds the penalty-free scoring average. For example, if a player scores 49 during a round where the penalty-free scoring average is 53, then [SG:T] = 4.

Expected Values

So how do we actually compute the Strokes Gained statistics? As mentioned previously, this computation depends on the notion of expected value.

The expected number of throws from the Tee is just the penalty-free scoring average for the hole.
The expected number of throws from Circle 1X, Circle 2, and a Recovery Position (a position outside C2 occurring after a player has already been inside C2 or closer) are a more complex computation which we'll describe near the end of the article. The expected number of throws tends to be around 1.1–1.4 from C1X, 1.8–2.0 from C2, and 2.0–2.2 from Recovery Positions.
The expected number of throws from Parked (i.e., less than 11 feet/3.3 meters) is always 1.
As an important formality, the expected number of throws from inside the Basket is 0.

Throws from C1X and C2 contribute to the [SG:C1X] and [SG:C2] stats, respectively. Because of the way UDisc determines whether a player is Parked, every throw from Parked goes in the basket and therefore these throws don't contribute to any of the Strokes Gained statistics (if a player misses a short putt it counts as a missed C1X putt). All throws from outside of C2, including throws from the Tee and from Recovery Positions, contribute to the [SG:TG] stat.

One important subtlety is that we compute expected value from the Tee on a per-hole basis, while we compute expected value from C1X, C2, and Recovery Positions on a per-round basis. We do this to ensure that enough data points go into the calculation. If the field size is small, there might only be one or two – or even zero – C2 putts on a given hole.

Example 1: Beaver State Fling 2019

Now that we have the nuts and bolts, let's take a look at a full example. We'll set the scene a little before we get to the stats, though.

Going into the 450-foot/137-meter, par 3 hole 17 during the final round of the 2019 Beaver State Fling, Eagle McMahon had a two-stroke lead over Seppo Paju.

Paju was the first of the two to tee and threw a backhand that faded out a little too early, leaving him far off the fairway and deep in the rough. McMahon opted for a safe forehand play into a wide-open spot on the fairway, planning to park a 150-foot/46-meter upshot that should be routine for a player of his caliber.

Paju's next shot stayed in the woods, giving him a highly obstructed look from C2 to save his par. McMahon didn't manage to press his advantage, though. He leaked his upshot a little wide, and his par look was only just barely inside C1X.

Then, Paju did this:

And McMahon followed with this:

Now just one stroke apart, Paju and McMahon threw similar high backhand tee shots with overstable discs on the 18th hole. Paju's spiked down just outside of C1 and McMahon's caught some tree branches before landing about 50 feet/15 meters from the basket. McMahon decided to lay up his putt underneath the basket to force Paju to make his long putt. Paju failed to deliver, allowing McMahon to tap it in for the tournament victory.

We have a table below where you can see both McMahon's and Paju's Strokes Gained stats over the course of these two holes. We'll explain McMahon's in detail and you can follow along in the table. After that, see if you can follow Paju's table on your own.

The penalty-free scoring averages for the two holes were 3.17 and 2.83, respectively, and the C1X and C2 expected values for the round were 1.16 and 1.86.
On hole 17, McMahon took two strokes to get from an expected value of 3.17 (the Tee) to an expected value of 1.16 (C1X), so his Tee → Green stat was [SG:TG] = 3.17 − 1.16 − 2 = 0.01.
On his C1X putt, McMahon took one stroke to get from an expected value of 1.16 to an expected value of 1 (Parked), so his C1X stat was [SG:C1X] = 1.16 − 1 − 1 = −0.84.
For his bogey tap-in, he took one stroke to get from an expected value of 1 to an expected value of 0, gaining 1 − 1 − 0 = 0 strokes.
On hole 18, McMahon's tee shot into C2 gained him 2.83 − 1.86 − 1 = −0.03 strokes. That is, he lost a fraction of a stroke to the field.
His layup under the basket lost him 0.14 strokes to the field (1.86 − 1 − 1 = −0.14) though hindsight shows that he made the correct decision.

Eagle McMahon	Start Zone	E_start	End Zone	E_end	# Throws	Strokes Gained	Strokes Gained Statistic
Hole 17	Tee	3.17	Fairway
	Fairway		Circle 1X	1.16	2	0.01	Tee → Green
	Circle 1X	1.16	Parked	1.00	1	−0.84	Circle 1X
	Parked	1.00	Basket	0.00	1	0.00
Total	Tee	3.17	Basket	0.00	4	−0.83	Total
Hole 18	Tee	2.83	Circle 2	1.86	1	−0.03	Tee → Green
	Circle 2	1.86	Parked	1.00	1	−0.14	Circle 2
	Parked	1.00	Basket	0.00	1	0.00
Total	Tee	2.83	Basket	0.00	3	−0.17	Total
Seppo Paju	Start Zone	E_start	End Zone	E_end	# Throws	Strokes Gained	Strokes Gained Statistic
Hole 17	Tee	3.17	Off Fairway
	Off Fairway		Circle 2	1.86	2	−0.69	Tee → Green
	Circle 2	1.86	Basket	0.00	1	0.86	Circle 2
Total	Tee	3.17	Basket	0.00	4	−0.83	Total
Hole 18	Tee	2.83	Circle 2	1.86	1	−0.03	Tee → Green
	Circle 2	1.86	Parked	1.00	1	−0.14	Circle 2
	Parked	1.00	Basket	0.00	1	0.00
Total	Tee	2.83	Basket	0.00	3	−0.17	Total

Of course, even though the finish was exciting, McMahon and Paju played an entire tournament to get to where they were. Comparing McMahon's and Paju's Strokes Gained statistics for the entire tournament, we see that Paju outperformed McMahon on the putting green, but McMahon edged out Paju by throwing the disc. Both players rose high above the rest of the field in all areas.


	SG:TG	SG:P	SG:C1X	SG:C2	SL:OB	SG:T
Eagle McMahon	14.290	5.354	1.604	3.750	2	17.644
Seppo Paju	11.676	7.968	3.647	4.321	3	16.644
James Conrad	7.783	6.861	2.835	4.026	4	10.644
Jordan Castro	4.588	7.056	4.318	2.738	2	9.644
A.J. Risley	12.261	−0.616	0.372	−0.989	2	9.644
Simon Lizotte	12.208	−1.564	−1.866	0.303	1	9.644

Example 2: Waco Annual Charity Open 2018

The treacherous finishing holes of the final round of the 2018 Waco Annual Charity Open proved to be nail-biters for Paige Pierce and Sarah Hokom. Pierce had a two-stroke lead going into the 17th hole. Those two strokes were anything but comfortable on a windy day with OB everywhere. Hokom and Pierce threw solid tee shots to similar landing areas. Hokom's laser-accurate forehand proved to be the perfect tool to carry over the water and fade to a long C1X position. Facing the water and a headwind, Hokom left her putt just short of the basket and tapped in a 4 (par on the hole). With a two-stroke lead and a pair of large trees in her line, Pierce elected to throw a jump-putt layup to play for an easy approach. Unfortunately, the wind made its way beneath her putter and carried it out-of-bounds. She played a high forehand from an awkward position which left her well outside of C2. Getting a little aggressive on her approach, she ended up three-putting and walking to the tee of hole 18 with an 8, now two strokes behind Hokom.

Hokom, now with the two-stroke lead, decided to play a safe backhand to improve her position for the next shot. Her disc came out a bit too low and with too much hyzer and skipped onto an OB sidewalk. Pierce, seizing this window of opportunity, decided to go for the pin 492ft/150m away, carrying water the entire way. She executed the shot beautifully, leaving herself just outside of C2. Hokom, still slightly out of position, elected to lay up again — this time with the desired result. From there, she would need to finish the hole in two shots for a double-bogey to force a playoff at best (if Pierce were to miss her long putt). Hokom's approach to the basket nearly skipped in, but came to a halt about 20f/6m from the pin. Pierce made an excellent effort on her outside C2 jump-putt, hitting the cage and leaving her with a tap-in birdie. All the pressure was on Hokom to make her putt and go into extra holes, but it came up just short, putting an end to the turbulence.

The penalty-free scoring averages for the two holes were 4.87 and 4.27. The Circle 1X expectation was 1.51 on the windy day. You can see the computations of the Strokes Gained statistics shot-for-shot in the table below.

Paige Pierce	Start Zone	E_start	End Zone	E_end	# Throws	Strokes Gained	Strokes Gained Statistic
Hole 17	Tee	4.87	Fairway
	Fairway		OB → Off Fairway			−1.00	[SL:OB] = 1
	Off Fairway		Fairway
	Fairway		Circle 1X	1.51	4	−0.64	Tee → Green
	Circle 1X	1.51	Circle 1X	1.51	1	−1.00	Circle 1X
	Circle 1X	1.51	Parked	1.00	1	−0.49	Circle 1X
	Parked	1.00	Basket	0.00	1	0.00
Total	Tee	4.87	Basket	0.00	7	−3.13	Total
Hole 18	Tee	4.27	Fairway
	Fairway		Parked	1.00	2	1.27	Tee → Green
	Parked	1.00	Basket	0.00	1	0.00
Total	Tee	4.27	Basket	0.00	3	1.27	Total
Sarah Hokom	Start Zone	E_start	End Zone	E_end	# Throws	Strokes Gained	Strokes Gained Statistic
Hole 17	Tee	4.87	Fairway
	Fairway		Circle 1X	1.51	2	1.36	Tee → Green
	Circle 1X	1.51	Parked	1.00	1	−0.49	Circle 1X
	Parked	1.00	Basket	0.00	1	0.00
Total	Tee	4.87	Basket	0.00	4	0.87	Total
Hole 18	Tee	4.27	OB → Fairway			−1.00	[SL:OB] = 1
	Fairway		Fairway
	Fairway		Circle 1X	1.51	3	−0.24	Tee → Green
	Circle 1X	1.51	Parked	1.00	1	−0.49	Circle 1X
	Parked	1.00	Basket	0.00	1	0.00
Total	Tee	4.27	Basket	0.00	5	−1.73	Total

All in all, for the two holes Pierce gained 0.63 strokes Tee → Green, lost 1.49 strokes from Circle 1X putting, and lost a stroke due to OB. In total, she lost 1.86 strokes. Hokom, on the other hand, gained 1.12 strokes Tee → Green, lost 0.98 strokes from Circle 1X putting, and lost a stroke due to OB, In total, she lost 0.86 strokes, one fewer than Pierce. Reviewing the Strokes Gained statistics for the top finishers of the tournament, we see that Hokom edged out Pierce in the throwing game, but Pierce made up the strokes from putting — specifically from Circle 1X.

	SG:TG	SG:P	SG:C1X	SG:C2	SL:OB	SG:T
Paige Pierce	16.617	7.983	4.049	3.935	4	20.600
Sarah Hokom	21.879	2.721	−1.323	4.043	5	19.600
Lisa Fajkus	9.337	2.263	1.803	0.461	2	9.600
Rebecca Cox	12.397	−0.797	−0.318	−0.480	5	6.600
Ellen Widboom	7.393	2.207	4.346	−2.139	3	6.600

Markov Chains

Now that we understand how to compute the Strokes Gained statistics from expected value, let's back up and discuss how we determine expected value in the first place.

As mentioned above, expected value from the Tee is the penalty-free scoring average for the hole; expected value from Parked is one; and expected value from inside the basket is zero. To determine expected value from C1X, C2, and Recovery Positions (recall that these are positions outside of C2 after a player has already been inside C2 or better), we need the statistical notion of a Markov chain.

Roughly speaking, a Markov chain is a mathematical system that moves from state to state over time, with the defining property that the probability of moving to a particular state only depends on the current state (and not on what has happened in the past).

One popular example of a Markov chain is the board game Monopoly (played without chance or community chest cards). During your turn, the probability of landing on a given space depends only on which space you are currently on – it doesn't matter where you have been previously.

The Markov chain in our case consists of five states: C1X, C2, Recovery Positions, Parked, and the Basket. We replace the notion of probability with relative frequency: Of all the C1X putts during a round, what proportion of them went in the Basket? What proportion were Parked? What proportion landed in Circle 1X? Circle 2? Recovery Positions? At the end of the day, we get a matrix that looks like this:

	Ending Position
Starting Position		Recovery	Circle 2	Circle 1X	Parked	Basket
	Recovery	p_R,R	p_R,C2	p_R,C1X	p_R,P	p_R,B
	Circle 2	p_C2,R	p_C2,C2	p_C2,C1X	p_C2,P	p_C2,B
	Circle 1X	p_C1X,R	p_C1X,C2	p_C1X,C1X	p_C1X,P	p_C1X,B
	Parked	0	0	0	0	1
	Basket	0	0	0	0	1

For example, p_C2,C1X is the proportion of C2 putts that land in C1X. Note that from Parked, the next shot is guaranteed to go in the basket, and once you are in the Basket, you can't leave the Basket.

The quantities that we are looking for are called the mean hitting times of the Markov chain: If I am in a Recovery Position, how many throws will it take, on average, to get in the basket? Fortunately, there is a well-known – at least to mathematicians – theorem that reduces this question to solving a system of linear equations (you can find that theorem, for example, in these lecture notes). For us, that system looks like this:

(p_R,R − 1) E_R + p_R,C2 E_C2 + p_R,C1X E_C1X = −1 − p_R,P
p_C2,R E_R + (p_C2,C2 − 1) E_C2 + p_C2,C1X E_C1X = −1 − p_C2,P
p_C1X,R E_R + p_C1X,C2 E_C2 + (p_C1X,C1X − 1) E_C1X = −1 − p_C1X,P

where E_R, E_C2, and E_C1X are the (unknown) mean hitting times from Recovery, C2, and C1X, respectively. Note that there are three unknowns in the system and three equations.

It is possible to solve the system by hand, but we'd rather just ask a computer to do it for us. Once we have all the expected values, we can compute Strokes Gained statistics as described above.

Enough With the Math. Who Are the Best Putters and Throwers?

All statistics, including Strokes Gained statistics, should be thought of as small parts of the big picture. As such, we really can't use Strokes Gained statistics (or any statistic) to declare an undisputed "best putter" or "best thrower."

Moreover, since the number of strokes gained on the field average is highly dependent on how the field performs during a given round, the notion of "average Strokes Gained" over a series of events is almost meaningless. The best we can do is to compare total or average Strokes Gained statistics for a series of events that a group of players all have in common.

With that disclaimer out of the way, we can still use Strokes Gained statistics to highlight some standout performances by averaging each player's Strokes Gained statistics over all the holes played in a tournament (note that these are averages not totals like you see in the tables above).

Putting Superlatives: Open

Highest Average Strokes Gained From Total Putting: 4.88 by David Cox at the 2019 Idlewild Open
Highest Average Strokes Gained From C1X Putting: 4.26 by Drew Gibson at the 2017 Waco Annual Charity Open
Highest Average Strokes Gained From C2 Putting: 4.25 by Cox at the 2019 Idlewild Open

Putting Superlatives: Open Women

Highest Average Strokes Gained From Total Putting: 4.79 by Paige Pierce at the 2017 Idlewild Open
Highest Average Strokes Gained From C1X Putting: 3.66 by Ohn Scoggins at the 2020 Challenge at Goat Hill
Highest Average Strokes Gained From C2 Putting: 2.06 by Heather Young at the 2020 Ledgestone Insurance Open

Throwing Superlative: Open

Highest Average Strokes Gained From Throwing: 10.38 by Paul McBeth at the 2019 Throw Down the Mountain

Throwing Superlative: Open Women

Highest Average Strokes Gained From Throwing: 9.99 by Paige Pierce at the 2017 Idlewild Open

Find More Strokes Gained Stats Now on UDisc Live

The reason we have all these stats from past years is we've retroactively applied Strokes Gained to all previous UDisc Live tournaments. Just follow that link, click or tap on the year you see underneath the text "UDisc Live Schedule," and pick the year of the tournament you'd like to explore.

Though we're obviously thrilled to release Strokes Gained, it's not the only innovation we have in store. Keep an eye out for announcements about new additions to UDisc Live, the UDisc app, and UDisc online here on Release Point or make sure to never miss one by signing up for our twice-monthly newsletter.