Here is what I have figured out for interpreting Difficulty of Schedule Scores:

- For rikishi who are out of the rank, we are going to see DoS scores in excess of 40.
- For rikishi who are well ranked, we are going to see DoS scores from 30 to 40.
- For Ozeki who are earning their rank, we are hope to see DoS scores from 20 to 30.
- For Ozeki who are nearing the threshold for promotion to Yokozuna we hope to see sub-20 scores.

There is one more cool thing you can do with Expected Wins – read below.

With these Expected Wins included in the chart, you can figure out how well your rikishi performed compared to how their DoS indicated they would perform. All you have to do is divide their actual wins in the tournament by their expected wins. (Obviously, the DoS won’t usually be exactly one of the numbers listed in the chart, so you’ll just have to round to the nearest 5: 32 is rounded to 30 and 33 is rounded to 35)

That’s all you need to know to use DoS. If you want to read a long winded explanation of how I figured out the chart above, read on. If not, enjoy the basho!

Apart from this long, long explanation below, this concludes the Sumo as Science series.

**Explanation**

**Goals of Calculating DoS**

The goal behind calculating Difficulty of Schedule scores for rikishi is to give a little bit more context to their win/loss record each basho. To figure a DoS score for a rikishi, you look at his performance against his opponents during the basho.

**Method of Calculating DoS**

If you look at the total win/loss record between two rikishi, it can sometimes be a gargantuan list of 20 or more bouts that date back years and years and years. When figuring a rikishi’s Difficulty of Schedule, it doesn’t seem right to say that a losing streak to an opponent 8 years ago is still relevant today. This is especially true that rikishi got better and better over time and has beaten his opponent every bout for the past 7 years.

So, with this in mind, we don’t look at the total win/loss record between two rikishi. Instead, we just look at the most recent 5 bouts they have fought together. If we are calculating the Difficulty of Schedule for Rikishi A and Rikishi A’s opponent is Rikishi B. We look at the last five bouts they have fought together and see that Rikishi A has lost 4 of those 5 bouts. That means Rikishi A’s DoS against Rikishi B is 4. On the other hand, if we are calculating the Difficulty of Schedule for Rikishi B and Rikishi B’s opponent is Rikishi A. We look at those same five bouts and see that Rikishi B has lost 1 of those 5 bouts. That means Rikishi B’s DoS against Rikishi A is 1.

The reason why we give a rikishi points for losing to their opponents is because a high Difficulty of Schedule score means a rikishi is likely to have trouble beating his opponents (the more you lose, the higher your score).

**What DoS tells us**

Now imagine a rikishi who lost 100% of the time (during the past five bouts fought) to 100% of his opponents during a basho. That means the rikishi’s DoS score against each rikishi he faces is 5. The rikishi faces 15 opponents in a basho, so the highest DoS score a rikishi can get in a basho is 75 (5 * 15).

Imagine the opposite case as well. If a rikishi wins 100% of the time (during the past five bouts fought) to 100% of his opponents during a basho, his DoS score against each of those rikishi is going to be 0. The rikishi faces 15 opponents in a basho, so the lowest DoS score a rikishi can get in a basho is 0 (0 * 75)

Finally, imagine a rikishi who loses 3 out 5 times to 7 of his opponents and 2 out of 5 time to 8 of his opponents. That means his DoS score for the basho is 37. It also means that out of the 75 bouts his has fought with the opponents he faced this basho, he won 37 times. In other words he won 49.3% of the time.

Roughly speaking then, since the opponents are the same, we can expect a rikishi who has lost 100% of the time against all of his opponents in the past to lose 100% of the bouts in the current basho. We can also expect a rikishi who has won 100% of the time against all of his opponents in the past to win 100% of the bouts in the current basho. By the same logic, we can expect a rikishi who has lost on average 49.3% against all his opponents to lose roughly 49.3% of the bouts in the current basho.

This is how we can use the DoS score to make a prediction about how our rikishi will perform in the current basho. If nothing changes, then a rikishi who has won against the opponents he will face 100% of the time will win roughly 100% of the bouts, a rikishi who has won against the opponents he will face 75% of the time will win roughly 75% of the bouts, a rikishi who has won 25% of the time will win rough 25% of the bouts in the current basho.

**DoS in terms of percentage of wins over the 75 bouts considered.**

I may not have explained this clearly enough before, but to calculate a cumulative DoS over the course of a tournament for a rikishi, you end up considering 75 total bouts. You consider the previous five bouts the rikishi has fought against his opponent. When you do this for fifteen opponents, that is 15*5 or 75 bouts.

The DoS socre is the number of those 75 bouts a rikishi has lost. That means if the DoS is zero, the rikishi has lost 0% of those 75 bouts. If the DoS is 75, the rikishi has lost 100% of those 75 bouts.

Here is the breakdown in increments of 5% of how many of those 75 bouts a rikishi has lost

- Lost 5% of the 75 bouts would be a DoS of 3.75
- Lost 10% of the 75 bouts would be a DoS of 7.5
- Lost 15% of the 75 bouts would be a DoS of 11.25
- Lost 20% of the 75 bouts would be a DoS of 15
- Lost 25% of the 75 bouts would be a DoS of 18.75
- Lost 30% of the 75 bouts would be a DoS of 22.5
- Lost 35% of the 75 bouts would be a DoS of 26.25
- Lost 40% of the 75 bouts would be a DoS of 30
- Lost 45% of the 75 bouts would be a DoS of 33.75
- Lost 50% of the 75 bouts would be a DoS of 37.5
- Lost 55% of the 75 bouts would be a DoS of 41.25
- Lost 60% of the 75 bouts would be a DoS of 45
- Lost 65% of the 75 bouts would be a DoS of 48.75
- Lost 70% of the 75 bouts would be a DoS of 52.5
- Lost 75% of the 75 bouts would be a DoS of 56.25
- Lost 80% of the 75 bouts would be a DoS of 60
- Lost 85% of the 75 bouts would be a DoS of 63.75
- Lost 90% of the 75 bouts would be a DoS of 67.5
- Lost 95% of the 75 bouts would be a DoS of 71.25
- Lost 100% of the 75 bouts would be a DoS of 75

**What DoS does and doesn’t do**

So the idea is that past performance will give us a snapshot of where a rikishi is coming from. Of course, the rikishi might have gotten better, in which case, even though they won 60% of the 75 DoS score bashos, they will win 75% of the bouts they fight in the current tournament.

So you can’t predict exactly how many bouts a rikishi will actually win. This is because rikishi always changing… they get a little bit better, they get a little bit worse.

So if your rikishi wins 60% of the 75 bouts used to calculate DoS for the current basho and then goes on to win 75% of the bouts in the current basho, you shouldn’t feel that DoS has let you down. DoS isn’t meant to predict what rikishi will actually do, it is meant to predict what they would do if they were exactly the same as they were in the past.

The main thing that DoS is good for is figuring out if a rikishi is overranked or ready for promotion. It is good for comparing past performance to current performance. And, with a little bit of ingenuity, it is also good for calculating a Ganbatta Factor for each tournament.

**Setting Expectations for rikishi using their bout history against opponents (in other words, using their DoS score)**

In order to calculate how much above or below expectations a rikishi performs in a given basho, we need some way of converting a DoS score into a number of expected wins.

You remember from above that the DoS is the number of bouts a rikishi has lost out of 75 bouts fought against all of the opponents he will face in a given tournament. Above, I calculated the percentage of these total 75 possible losses in increments of 5%

In order to match a DoS score to a number of expected wins, we need to make an assumption. That assumption is that the percentage of losses a rikishi has gotten on average against all the opponents he will face in the current basho (in other words this is the cumulative DoS divided by 75) will reflect to some extent the percentage of losses the rikishi will experience int he current basho.

So, instead of increments of 5%, we need to figure out what percentage each loss equals out of a total of 15 bouts. I have calculated this here:

- 0/15= 0%
- 1/15 = 6.7%
- 2/15 = 13.3%
- 3/15 = 20%
- 4/15 = 26.7%
- 5/15 = 33.3%
- 6/15 = 40%
- 7/15 = 46.7%
- 8/15 = 53.3%
- 9/15 = 60%
- 10/15 = 66.7%
- 11/15 = 73.3%
- 12/15 = 80%
- 13/15 = 86.7%
- 14/15= 93.3%
- 15/15= 100%

Now, to convert the DoS scores into numbers of expected wins, we simply have to multiply by the percentages listed above. If a rikishi loses 0% of the bouts in the past, their record for the tournament should be 0 losses or 15 wins. So for example, 0% of 75 is 0, so a DoS of 0 should tell us to expect 15 wins (assuming nothing has changed about the rikishi over time and that luck is not a factor). 100% of 75 is 75, so a DoS of 75 should tell us to expect 0 wins (again nothing has changed about the rikishi over time and that luck is not a factor). Because each win is worth 6.7% so we need to cacluate DoS in increments of 6.7%. As it happens 6.7% of 75 is 5. So for every 5 points increas in Difficulty of Schedule, we can expect one less win.

- 75*0 (DoS=0) = 15 extpected wins
- 75*.067 (DoS=5) = 14 expected wins
- 75 *.133 (DoS=10) = 13 expected wins
- 75*.2 (DoS=15) = 12 expected wins
- 75*.267 (DoS=20) = 11 expected wins
- 75*.33.2 (DoS=25) = 10 expected wins
- 75*.4 (DoS=30) = 9 expected wins
- 75*.467 (DoS=35) = 8 expected wins
- 75*.533 (DoS=40) = 7 expected wins
- 75*.6 (DoS=45) = 6 expected wins
- 75*.667 (DoS=50) = 5 expected wins
- 75*.733 (DoS=55) = 4 expected wins
- 75*.8 (DoS=60) = 3 expected wins
- 75*.867 (DoS=65) = 2 expected wins
- 75*.933 (DoS=70) = 1 expected wins
- 75* 1 (DoS=75) = 0 expected wins

1 win out of 15 is 6.7%

5 DoS points out of 75 is also 6.7%

So if you divide DoS by 5, you will get the expected number of losses for your rikishi. Subtract that from 15 and you have your expected number of wins.

**The Ganbatta Factor**

Now, you have to remember that this is the expected number of wins assuming nothing changed over time. In reality, it is almost certain that your rikishi got better or worse over time. So to figure out if your rikishi has had a good tournament or a bad tournament, you need to divide the number of actual wins by the number of expected wins.

If your rikishi gets less actual wins than expected wins, then he underperforms, if your rikishi gets more actual wins than expected wins, then he overperforms.

By dividing the actual wins by the expected wins, you can figure percentage of over/underperformance:

- 10 actual wins out of 10 expected wins means the rikishi performed exactly as expected (or at 10/10 – 100% of expectations)
- 7 actual wins out of 10 expected wins means the rikishi performed worse than expected (or at 7/10 – 70% of expectations)
- 14 actual wins out of 10 expected wins means the rikishi performed far better than expected (or at 14/10 – 140% of expectations)

And that is how you can compare rikishi’s performance in a basho compared to expectations. A rikishi who was expected to win 3 bouts and wins 5 does better compared to expectations than a rikishi who is expected to win 10 bouts and only wins 8.

Nigel – I think the idea of dividing actual wins by expected wins measures rikishi performance in the way you were hoping for back in our discussion in the comments section of the Yoshikaze post. What do you think?

Hello Haj,

Actually you had won me over to your point of view, vis-a-vis using only DoW. (I have a sheet printed out here in front of me for Takayasu). At the bottom of it you have a box for `Cumulative DoW` and `Cumulative DoS`. I was going to work out the % after the basho.

I am NOT much of a statistician but to my mind the figure in the `DoS cumulative total` box is the absolute highest that that rikishi can score in the basho. Therefore the rikishi who gets closest to his individual possible maximum (done the best job of overcoming the difficulty of his schedule) should be the shukun-sho (“Fighting Spirit”) winner.

By expressing that in terms of a percentage it avoids difficulties where one rikishi has a much higher `DoS cumulative total` than another. Getting 24 DoW points out of a maximum possible of 25 is much more impressive an effort than getting 35 out of 50. In this example I would say that 24 did better than 35 and should be the winner. (96% vs 70%)

But, like I say, I`m not a statistician and maybe I`m barking up the wrong tree.

At the end of this basho let us rank the make-koshi (from Hatso) who kachi-koshi (in Haru) both by DoW and by %. Then – with the evidence in front of us – we can discuss the merits and de-merits ready for the next basho.

Really looking forward to Sunday.

And so we begin…..

Day 1: Takayasu [M7e] vs Toyohibiki [M7w] (LLWWL) = 2/3*5

= DoS of 3.33.

Day 2: Takayasu [M7e] vs Homasho [M5w] (LW) = 1/2*5

= DoS of 2.50.

Hey Nigel,

The reservation I have is that it is much harder to get 50 out of 50 DoS points than it is to get 25 out of 25 DoS points.

If you take it to near the extreme, its easy to see, if a rikishi has lost 70 times out of the last 75 bouts he has fought against his opponents in the basho (5 bouts with each opponent), then that means they will have a DoS of 70. Such a DoS score means that out of 15 opponents, the rikishi has managed to beat one of those opponents only once in recent history. So if that rikishi turns around their sumo and gets 70 out of 70 DoS points by beating all of their opponents, that is huge… its unbelievable, incredible.

On the other hand, if a rikishi has lost 1 time out of the last 75 bouts he has fought against his opponents in the basho (5 bouts with each opponent), then that means they will have a DoS of 1. That means the rikishi has a perfect record (5-0) against 14 of his opponents and a 4-1 record against one opponent.

So if this rikishi then goes Zensho-yusho and gets all of their points, then their records against their opponents barely changes at all. The record definitely doesn’t change at all for 14 opponents, and for the 15th opponent, the record will either change from 4-1 to 5-0 or it will just stay the same.

A rikishi who gets 1 out of 1 possible DoS points displays no real improvement. By contrast, a rikishi who gets 70 out of 70 points shows a huge change.

Thats why I think it is important to divide actual wins by expected wins rather than DoW by DoS.

In such an example, if we divide DoW by DoS, we end up with 100% for both rikishi. But if we refer to the chart at the beginning of this post, and divide actual wins by expected wins, we get 100% for our rikishi who got 1 out of 1 DoS points (because based on his DoS of one, we expect him to win 15 bouts and he does, so 15/15 = 100%). By contrast, based on the other rikishi’s DoS (70), we expect him to win 1 fight. In fact he wins 15, which means that he performed at 1500% of what we expected – its a ridiculous number, but it reflects how unlikely it would be for a rikishi to go Zensho-yusho with a DoS of 70.