Ebbinghaus forgetting curve: what the 1885 data really show

The forgetting curve is the graph showing how what we learn fades over time: a lot in the first hours and less and less afterwards, until what remains barely changes. The German psychologist Hermann Ebbinghaus drew it in 1885 from an experiment run on Ebbinghaus's own memory, and in 2015 it was replicated with very similar results.¹³ What is rarely explained is what it measured: not the percentage of what is remembered, but the time saved when learning it again. The two previous articles were about the art of memory, from the Roman memory palace to Giordano Bruno; this one is about the first attempt to measure forgetting, which Ebbinghaus deliberately made without that art. I read the book in the original German.

What is the forgetting curve?

It is seven numbers. Ebbinghaus learned lists of syllables by heart, left them alone for a while and then learned them again. What was recorded was how much work the second learning saved, as a percentage of what the first one had cost:¹

• after 20 minutes, a saving of 58.2%;
• after 1 hour, 44.2%;
• after 9 hours, 35.8%;
• after 1 day, 33.7%;
• after 2 days, 27.8%;
• after 6 days, 25.4%;
• after a month (31 days), 21.1%.

Ebbinghaus's own summary, translated from the German: "One hour after the end of learning, forgetting had already gone so far that more than half of the work originally spent had to be renewed before the series could be reproduced again; after 8 hours, what had to be made up was almost 2/3 of the first effort. Gradually, however, the process slowed down. After 24 hours about a third still remained; after 6 days a quarter, and after a whole month still a good fifth."¹ At that pace, the book adds, the effect of the first learning would have vanished completely only "after, so to speak, an infinitely long time".

How it was measured

Ebbinghaus wanted material without meaning, so that interest, the beauty of a line of verse or a personal memory would not get in the way. All possible syllables made of a consonant, a vowel and another consonant were formed, "about 2300" of them, mixed and drawn at random to make lists.¹ For the forgetting curve, between 1879 and 1880, lists of 13 syllables were learned until they could be recited twice in a row without error, and relearned after seven different intervals: 163 double tests in all.¹² Ebbinghaus was 29, according to the authors of the 2015 replication,³ and was both the researcher and the only subject.

And here is the bridge to the memory palace. Among the conditions of the experiment, Ebbinghaus writes: "No attempt was ever made to connect the nonsense syllables through imagined relations, for example according to the rules of the mnemonists; learning happened purely through the effect of mere repetition on natural memory. Since I do not have the slightest practical knowledge of the mnemonic arts, meeting this condition gave me no difficulty."¹ The forgetting curve is, by design, what happens without the art of memory: the "natural" memory of the Rhetorica ad Herennium, not the "artificial" one.

Ebbinghaus also saw a danger that has a name today: experimenter bias. Someone who studies their own memory ends up expecting certain results, and that expectation can lead to trying a little harder here and a little less there without meaning to. The remedy was to keep the results hidden from oneself "as long as possible", while admitting that the real solution was two people working together.²

What the forgetting curve does not say

It does not say what percentage you remember. The 33.7% of the first day does not mean that a third of the syllables were remembered the next day: it means that relearning them took two thirds of the original time. It is a measure of savings, not of recall. If a graph puts "% remembered" on its vertical axis and cites Ebbinghaus, it is mixing two different things. The authors of the 2015 replication measured the percentage of syllables recalled correctly separately, and treat it as a different measure.³

It is not a general law. Ebbinghaus is blunt: given the "special, individual and moreover uncertain" character of the numbers, no one should want to know right away what "law" they reveal. They are one person's data, with nonsense syllables, in lists of 13.¹

The formula that circulates is not the original. Ebbinghaus did propose a formula, but only as a shorthand for the seven data points. The Spanish Wikipedia gives an exponential, R = e^(−t/S), as the formula of the curve; it does not appear in the 1885 book. The Portuguese Wikipedia copies Ebbinghaus's formula, but with a mistake (the exponent c multiplies the logarithm instead).⁴ The real one is in the next section.

The Ebbinghaus formula

With b as the saving in percent and t as the minutes since the end of learning, Ebbinghaus wrote:

b = 100 k / ((log t)^c + k), with k = 1.84 and c = 1.25 (common logarithms).

The constants, the book says, were set "by approximate estimate and without exact calculation by least squares", and the formula is only a short way of stating results "found only once and under the circumstances described".¹ It fits five of the seven points well. It misses two: the one at one hour and, above all, the one at one day, where the real value (33.7) is 3.3 points above the calculated one (30.4).

That one-day point was a nuisance. Between 9 and 24 hours the saving had dropped by only 2.1%, and between 24 and 48 hours by 6.1%: forgetting went faster during the second day than during the night before, the opposite of the rest of the curve. Ebbinghaus suspected the 33.7 was one or two points too high, but other tests from a different period (33.4% after a day) supported it, and the conclusion was: "I cannot decide."¹ A detail for readers of the 1913 English translation: it says "2 1/2 per cent" where the German original says 2.1%.²

Has the forgetting curve been replicated?

Yes. In 2015 Jaap Murre and Joeri Dros, of the University of Amsterdam, published a replication in which Dros spent about 70 hours learning and relearning lists of syllables, between December 2011 and February 2012, in a Dutch-language setting.³ Their conclusion: Ebbinghaus's curve has been replicated, and it also resembles a German replication from 1991 with two subjects. The biggest difference is at one month: Dros saved only 4%, far below the others, perhaps because many more lists were studied between learning and relearning, causing interference.³

And the awkward point turned out not to be an error. According to Murre and Dros's analysis, the curve is not entirely smooth: it most probably jumps upwards from the 24-hour point. Jenkins and Dallenbach had already attributed it to sleep in 1924, and that was why they ran their experiment on forgetting during sleep and waking. The authors are cautious: the jump exists, but whether sleep itself causes it needs more research.³ The data point Ebbinghaus did not dare to correct is, almost a century and a half later, the start of a whole line of study.

How often to review

In another chapter, Ebbinghaus compared two ways of repeating. A 12-syllable series repeated 68 times in a row on one day could be recited the next day after 7 repetitions. With the repetitions spread over three days, 38 were enough for the same effect.¹ The conclusion, with every caution: with many repetitions, "a suitable distribution of them over a certain time is decidedly more advantageous than massing them at one time". And it matches what people already did: a schoolboy does not learn vocabulary all at once at night, knowing it has to be gone over again in the morning.¹

That is the origin of what is now called spaced repetition. What I did not find, in what I read of the book, is a review schedule: the "review after a day, a week and a month" plans that circulate under Ebbinghaus's name do not come from these pages.

Look at the data

This chart shows Ebbinghaus's seven data points, the formula and, if you like, the data from the 2015 replication. Time is on a logarithmic scale, as in Murre and Dros's paper, so that twenty minutes and a month fit in the same drawing.

Look at the one-day point. Between 9 and 24 hours, Ebbinghaus's saving hardly drops (from 35.8 to 33.7) and Dros's rises (from 27.6 to 31.7): neither series follows the fall seen in the rest of the curve.

What I could not read

I read in German, in the 1885 edition, the preface, the sections on method (§§ 11-14), the whole of chapter VII (§§ 26-30) and the passage on distributing repetitions (§ 34); I checked the rest against the English translation by Ruger and Bussenius (1913), of which I read chapters III, VII and VIII.¹² I read Murre and Dros's article in full.³ I did not read the German replication by Heller, Mack and Seitz (1991), Jenkins and Dallenbach's experiment (1924), or the 1880 manuscript Ebbinghaus submitted for the habilitation; I know them only through Murre and Dros. The biographical details (born in Barmen in 1850, died in 1909) come from Wikipedia.⁴ Translations from the German are mine.

What we do with this

On the title page of the book, Ebbinghaus put a sentence in Latin: "De subiecto vetustissimo novissimam promovemus scientiam", "on a most ancient subject, we advance a most recent science".¹ That is what this part of the Diario tries to do: memory, with two thousand years of art behind it, and the way to measure it. Anima Mundis measures too, and the Observatory shows how much happens in each hour. Ebbinghaus's lesson holds for us: say how many we were, how it was measured and which number looks odd, instead of turning a data point into a law.