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In the academic journal article "The Role of Deliberate Practice in Chess Expertise" from APPLIED COGNITIVE PSYCHOLOGY Appl. Cognit. Psychol. 19: 151– 165 (2005)

At the top of page 162, the authors state that

"...each log unit of serious study alone yields about 200 rating points compared to 33 rating points for log tournament play. Hence, players ought to devote the majority of their time to that activity."

I have gone over this academic paper several times, but I cannot decipher what "each log unit" represents. What time span is a "log unit" in this context where studying chess alone gave the chess players in this academic paper 200 ratings points vs 33 ratings points from playing in tournaments? Is it 6 months, 10 months, a year?

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The first paragraph of the RESULTS section on p. 155 says

We applied log-10 transformations to three variables (cumulative study alone, cumulative tournament play, and chess book library size) prior to correlation and regression analyses.

The log-10 (logarithm base 10) of a number is what power you have to raise 10 to in order to produce that number; e.g. log 100 = 2, log 1000 = 3, etc.

So the sentence

each log unit of serious study alone yields about 200 rating points

means that, all other things being equal, if player A studies ten times as long as player B, his rating will be about 200 points higher. (Another student who studies a hundred times as long as player B will have a rating of about 200 points higher than player A.) So there is no constant time span per 200-point gain; the required time to gain a particular amount of rating improvement gets larger and larger as you improve. (Another way to put this is that chess study produces diminishing returns.)

(In a sense, the Elo rating system is logarithmic as well, so you could claim that the relationship between study time and playing skill is actually linear. But since everybody measures playing skill by Elo rating, that's not so productive.)

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  • 1
    Please clarify the "If player A studies 10 times as along as player B..." analogy. If it takes 1,000 hours of solitary study to get a 1600-1799 rating, then it would take 10,000,000 hours (1142 years) of study to get to 2400+! But in the paper, they state that a grandmaster accumulated 5000 hours on serious study alone "nearly five times the average amount reported by intermediate-level players" (and not 10,000 times 1,000 hours, using log analogy, see Figure 1, page 161). – MorphyWormhole Sep 12 '13 at 3:48
  • I don't have time to delve into the paper deeply, but 1) these are just correlations, so it's not surprising that the best players outperform them; 2) there were 9 other input variables other than "log hours of study" which contribute as well; 3) the section you quote is specifically trying to understand why grandmasters' rating gain over time is faster than the average slope, hypothesizing that earlier accumulation of knowledge accelerates the process - note the "by the tenth year" that you omitted in your quote. – dfan Sep 12 '13 at 12:03
  • So let's forget about the quote and the Figure 1 that I added in the comment. Back to your answer and log explanation. If it takes 1,000 hours (which seems reasonable) of solitary study (which is what is being singled out) to get to 1600, would it then take on average 10,000,000 hours (following your log explanation) to get to 2400 using solitary study alone, which is the most efficient way to get better (which is the whole point of the paper, see DISCUSSION section, pg 161)? – MorphyWormhole Sep 12 '13 at 12:32
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    Yes, but that is extrapolating their curve to an absurd degree. Nobody is going to get to 2400 using solitary study alone. Also, because of the log factor, a single hour of solitary study is going to have less and less cumulative effect over time (as you note). The conclusion to draw from the paper is that doubling your study time will have more effect than doubling your tournament time, not that adding an hour of study time will have more effect than adding an hour of tournament time. – dfan Sep 12 '13 at 12:46
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    If multiplying your study hours by 10 increases your rating by 195 on average, and multiplying your tournament hours by 10 increases your rating by 33 on average, and you have already studied for 1000 hours and played in tournaments for 100 hours, then you could study another 9000 hours (going from 1000 to 10000) and increase your rating by 195 (0.02 points/hour) or play in tournaments for another 900 hours (going from 100 to 1000) and increase your rating by 33 (0.04 points/hour). I've spent as much time as I'm going to on this question; maybe other people can pick it up from here. – dfan Sep 12 '13 at 13:34
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The "log unit" term refers to a unit increment change in that variable, converted into the logarithmic equivalent.

The Table 3 on p. 158 shows how the variables that the researchers investigated are related to the output variable, which is ELO rating.

The variables included:

v1: Hours of serious study

v2: Hours of tournament play

v3: Years of private instruction

v4: Years of group instruction

v5: Current hours/week serious study

v6: Current hours/week tournament play

Each of these is converted into a logarithm before the equation's factors are calculated. (The factor of a variable is the multiplier of that variable that produces the closest prediction to actual results, when all factors and variables are included.)

Their regression model shows that (if you use only these variables and the logarithmic approach) the peak ELO of a player will be best predicted by the equation:

PeakELO = 1145 +

195 x log(v1) + 32.7 x log(v2) +
9.4 x log(v3) +  4.3 x log(v4) +
6.3 x log(v5) + 20.3 x log(v6)

where 1145 is just a constant that helps improve the prediction.

They point out that the variables with the strongest effect on the prediction were those with the highest factors; v1 (total hours of serious study), and v2 (total hours of tournament play), but the first was much stronger than the second (195 vs 32.7, which is nearly 6 times as strong a factor), and dominated the other factors in their effect on the prediction.

The short answer to your question is that a log unit for the variable of total hours of serious study is an increment of 1, which transforms into different numbers of actual hours, depending on what the increment is being added to.


To illustrate what this means, if I've studied 1,000 total hours, then the log of 1000 = 3. My predicted PeakELO (for that variable alone) becomes

PeakELO = 1145 + 195 x log(1000) = 1145 + 195 x 3 = 1145 + 585 = 1730 ELO

(I'm ignoring the other variables for simplicity, but the results show that they're not substantial anyway.)

If I work 1 more hour, then

PeakELO = 1145 + 195 x log(1001) = 1145 + 195 x 3.0004 = 1145 + 585.086

which is an indistinguishably small change.

If I work 100 more hours, though:

PeakELO = 1145 + 195 x log(1100) = 1145 + 195 x 3.041 = 1145 + 593 = 1738 ELO

So, while I'm at 1730, 100 hrs more of study will boost my ELO by 8 points.


But look at what happens if I've only studied 500 hours in total so far:

PeakELO = 1145 + 195 x log(500) = 1145 + 195 x 2.70 = 1145 + 526 = 1671.

Now, if I study an additional 100 hrs,

PeakELO = 1145 + 195 x log(600) = 1145 + 195 x 2.78 = 1145 + 542 = 1687.

The same 100 hours boosts my ELO by 16 points.


What the formula is clearly doing is reflecting the diminishing returns of the same number of hours of study time as your ELO increases. It accomplishes this accurate depiction of what actually happens by using the logarithm of the underlying data.

However, I would question the simple interpretation that the investigators offer, which is that a log unit increment of 1 results in 200 ELO increase. Well, log of 1,000 is 3. Log of 10,000 hrs is 4.

So, they claim that I would get 200 ELO improvement if I had already studied 1,000 hours, and now committed to studying another 9,000 hours (yes, that's about 4.5 years of full-time study). In their model, I would advance from 1730 ELO to 1930 ELO in no less than that time.

Clearly, this cannot be true for every player. I would suspect the following:

a) some confounding variables are having an effect, such as effectiveness of study (as determined by periodic assessment), ability to implement knowledge in tournament games (as determined by post-game analysis vs established knowledge base), age of player which the investigators acknowledge the model does not take into account, and so on...

b) the model's coefficient of correlation estimate (R^2) is only 0.34. Perfect correlation has 1.0; chaos (no correlation) is 0. So, 0.34 is promising, but not wholly effective by a long shot. Also, the Standard Estimate of the Error is 222, meaning the predicted ELO could be off by over 200 ELO for a particular player, and it wouldn't be remarkable. That's a pretty wide margin. So, in short, the model needs substantial improvement to be useful as a predictor of ELO.

What the model does accomplish at the moment is to suggest that study is 6 times more effective than tournament play, which is in turn more effective than any other training method, including coaching, for the average player.

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  • It's interesting that, beside these two variables, v6 (current hours/week tournament play) is also very important, near to the total hours spent in tournament play. I'd think there's some correlation between v6 and v1, so the difference between study and tournament play is not so large. On the other hand v5 is not so great, so one could conclude that the diference between study and play is not so large, but positive effects of study are actually more permanent while playing must be practised or otherwise its effect is partially lost. – sharcashmo Sep 28 '16 at 8:00
  • @sharcashmo The factors for v1, v2 and v6 are 195, 32.5 and 20, respectively. So, v6's factor is about 60% as influential on the Predicted ELO as v2's. Adding v2 and v6 together results in 52.5. This combined factor is only 27% as influential as total hours of serious study, so it takes 4 hours of tournament play (total and weekly combined) to accomplish what you can achieve with 1 hour of total serious study time. Interestingly, the number of hours per week of study is insignificant; it's the total hours that matter. Don't rush. – jaxter Sep 28 '16 at 17:54
  • I would agree that the relative sizes of v5 (6.3) and v6 (20.3) suggest that it's 3 times more valuable to spend an hour playing in a tournament game per week than it is to study for that hour. It's intuitive to suspect, though, that this variable has its own point of diminishing returns; in other words, if you play for 40 hours a week, do you get twice the benefit of 20 hours / week? The model says you should not. What if I switch the extra 20 hours to study? Is that more beneficial? If not, what are the right crossover numbers? More research is needed; it seems the factors must vary... – jaxter Sep 28 '16 at 18:04

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