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Replication and Analysis of Ebbinghaus' Forgetting Curve

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We present a successful replication of Ebbinghaus' classic forgetting curve from 1880 based on the method of savings. One subject spent 70 hours learning lists and relearning them after 20 min, 1 hour, 9 hours, 1 day, 2 days, or 31 days. The results are similar to Ebbinghaus' original data. We analyze the effects of serial position on forgetting and investigate what mathematical equations present a good fit to the Ebbinghaus forgetting curve and its replications. We conclude that the Ebbinghaus forgetting curve has indeed been replicated and that it is not completely smooth but most probably shows a jump upwards starting at the 24 hour data point.
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RESEARCH ARTICLE
Replication and Analysis of Ebbinghaus
Forgetting Curve
Jaap M. J. Murre*, Joeri Dros
University of Amsterdam, Amsterdam, The Netherlands
* jaap@murre.com
Abstract
We present a successful replication of Ebbinghaus classic forgetting curve from 1880
based on the method of savings. One subject spent 70 hours learning lists and relearning
them after 20 min, 1 hour, 9 hours, 1 day, 2 days, or 31 days. The results are similar to
Ebbinghaus' original data. We analyze the effects of serial position on forgetting and investi-
gate what mathematical equations present a good fit to the Ebbinghaus forgetting curve
and its replications. We conclude that the Ebbinghaus forgetting curve has indeed been rep-
licated and that it is not completely smooth but most probably shows a jump upwards start-
ing at the 24 hour data point.
Introduction
This paper describes a replication of one of the most important early experiments in psychol-
ogy, namely Ebbinghaus' classic experiment on forgetting from 1880 and 1885. We replicated
the experiment that yielded the famous forgetting curve describing forgetting over intervals
ranging from 20 minutes to 31 days. Ebbinghaus' goal was to find the lawful relation between
retention and time-since-learning. This is why he fitted the data to two different functions (a
power function, 1880, and a logarithmic function, 1885), as have many theorists since (e.g.,
[1,24]). This papers also includes an analysisincluding one with a new modelof th e shape
of the Ebbinghaus' forgetting curve and its replications. Do the replicated forgetting curves
have the same shape, or must we conclude that Ebbinghaus' forgetting curve was idiosyncratic
and that quite different shapes may occur?
There is currently an increasing interest in replication studies in psychology, motivated by a
growing uneasiness in the community about unreliable findings in psychology. It seems partic-
ularly important to try to replicate classic studies that are included in every textbook on cogni-
tive psychology and may also be known by the general public. A good example of this is the
classic study by Bartlett [5], which until 1999 had only had unsuccessful replication attempts,
until finally Bergman and Roediger [6] succeeded in replicating the basic findings. One of the
reasons earlier replications may have failed is because not all details were well-documented in
the original study from 1932. The exact instructions, for example, were not included. This may
explain why Wynn and Logie [7] had found the forgetting gradient in their experiment to be
quite different from the one in Bartlett 's experiment. Bergman and Roediger [6 ] also argue that
PLOS ONE | DOI:10.1371/journal.pone.0120644 July 6, 2015 1/23
OPEN ACCESS
Citation: Murre JMJ, Dros J (2015) Replication and
Analysis of Ebbinghaus Forgetting Curve. PLoS
ONE 10(7): e0120644. doi:10.1371/journal.
pone.0120644
Editor: Dante R. Chialvo, National Scientific and
Technical Research Council (CONICET).,
ARGENTINA
Received: November 22, 2014
Accepted: January 25, 2015
Published: July 6, 2015
Copyright: © 2015 Murre, Dros. This is an open
access article distributed under the terms of the
Creative Commons Attribution License, which permits
unrestricted use, distribution, and reproduction in any
medium, provided the original author and source are
credited.
Data Availability Statement: Data are available from
the Open Science Framework at DOI: osf.io/6kfrp.
Funding: This experiment was conducted as an
employee (JM) and student (JD) of the University of
Amsterdam. The authors received no specific funding
for this work.
Competing Interests: The authors have declared
that no competing interests exist.
this may have been caused by certain differences in the study design. Replication of classic
experiments, thus, serves the dual purpose of verifying the reliability of the original results and
uncovering more precisely how the original experiment was conducted.
It is hard to overestimate the importance of Hermann Ebbinghaus' contribution to experi-
mental psychology. Influenced by the work of the German philosopher Herbart, he was the
first to carry out a series of rigorous experiments on the shape of forgetting, which he com-
pleted in 1880. The experiment itself was preceded by a period in which he tried out a variety
of materials and methods. After having tested himself with tones, numbers, and poem stanzas,
he decided that none of these served his purposes. Tones were too cumbersome to handle and
too difficult to reproduce for him, he did not find digits zero to nine suitable as basic units for
the long-running experiments he envisioned, and the poem fragments he tried to learn (from
Byrons Don Juan) were deemed too variable in the meanings they evoked and therefore likely
to cause measurement error [8] (p. 1417). He, therefore, introduced nonsense syllables, which
had more uniform characteristics than existing words or other verbal material. In his later
experiments on learning, however, he did verify his results with the Don Juan verses, confirm-
ing both his main results on learning and his intuition that the latter stimuli did indeed yield
much more variance in the data [9]. Since his introduction of nonsense syllables, a large num-
ber of experiments in experimental psychology has been based on highly controlled, artificial
stimuli.
In all experiments reported by Ebbinghaus [9], he used only himself as a subject. Single-
subject designs are not unusual in memory psychology. Especially in the study of autobio-
graphical memory we find several diary studies based on one persons personal memories (e.g.,
[10,11,12]). They have the advantage that there is no inter-subject variability, although they
still require hundreds of trials to reduce the variance due to differences in stimuli and other fac-
tors. This places a great burden on the subject. Indeed, Ebbinghaus forgetting curve is based
on seven months of experimenting, often up to three sessions per day. Wagenaar [13] meticu-
lously recorded one daily memory during six years and spent several months recalling these.
A disadvantage of a single-subject design is that it remains unclear what the shape of for-
getting would be with other subjects. Are the results universal or did the subject happen to
have a memory that was exceptional in some way [14,15]. The generality of the results can be
assessed with a faithful replication. There have been a number ofmostly earlyreplications
of Ebbinghauss forgetting curve, notably by Radossawljewitsch [16] and Finkenbinder [17],
but these authors used a much slower presentation rate of the stimuli of 2 s per stimulus, where
Ebbinghaus learned at 0.4 s per stimulus. This was partially the result of the development of
devices for mechanical presentation by Müller and colleagues [18,19], who presented materials
at a rate one stimulus per second. Slowing down the presentation this much alters the nature of
the processing with more time to generate meaningful associations to otherwise meaningless
syllables. Though the resulting forgetting curves are clearly of interest to the field, we feel th at
the slow method of presentation form a large departure from Ebbinghaus original study. Also,
Finkenbinders[17] longest retention interval is 3 days, instead of 31 days and though in the
experiment by Radossawljewitsch [16] the retention interval range extends up to 120 days, his
design suffers from an uneven distribution of intervals throughout time and time-of-day. Sti-
muli were learned in order: in the first few days of the study all 5 min intervals were learned,
then the 20 min intervals, and so on. Because he did not use a pre-experimental practice phase,
the early interval s took longer to learn while the subjects were still getting used to the materials
and the procedure; it is likely that this has affected the shape of the forgetting curve reported by
him. There are other differences between these two studies and Ebbinghaus, for example, the
degree to which was learned and whether the subjects were allowed to pause between lists.
Replication and Analysis of Ebbinghaus' Forgetting Curve
PLOS ONE | DOI:10.1371/journal.pone.0120644 July 6, 2015 2/23
There are several unanswered questions about Ebbinghaus results that formed part of the
motivation for us to undertake this replication. His basic stimulus was a row of thirteen non-
sense syllables, which he studied until he could correctly recall it in the correct order twice in
succession. A question that seems pertinent is how stimuli at different serial positions were
learned and how these were forgotten over time. Another question is how his measure of
choice, namely savings (see below) is related to the nowadays more common measure of per-
centage correct. Finally, we were interested in the role of interference or fatigue in the course of
the experiment.
To help answer these questions, we consulted not only the widely published text of 1885 [9],
which was transla ted into English in 1913 [20], but also an earlier report of 1880 [8]. This is a
handwritten manuscript that he submitted for his Habilitation, which in Germany is a require-
ment to be considered for a full professorship. This text (the so called Urmanuscript or original
manuscript) has been typeset and republished in German in 1983. Even with this additional
source, however, we still could not answer the questions above.
For these reasons, we decided to replicate Ebbinghaus forgetting experiment. If our replica-
tion yielded similar results, this would support the generality of Ebbinghaus curve and through
a more detailed analysis of our data, we would be able to address the issues above. In the course
of preparing for our study, we found that there has been at least one other replication study,
namely by Heller, Mack, and Seitz [21]. This study has been published only in German, without
an English abstract, and is not easily accessible; at the time of writing, it is not available in elec-
tronic format (i.e., it is not available online) and it has never been cited in international journals
in English. It is, however, a thorough study and an excellent replication attempt. Where the
Ebbinghaus [8,9] texts are unclear about certain details, we have mostly followed Heller et al.
[21] as a guideline so that we can also compare our results with theirs. Because we feel th is is an
important study that has not received the readership it deserves, we will mention more of its
details here than we would have had it been more accessible at this point in time.
In 1885, Ebbinghaus introduces the savings measure of learning and memory (it does not
appear in this form in his earlier text from 1880). Savings is defined as the relative amount of
time saved on the second learning trial as a result of having had the first. Suppose, one has to
repeat a list for 25 times in order to reach twice perfect recollection and that after one day, one
needs 20 repetitions to relearn it. This is 5 less than the original 25; we can say that on relearn-
ing we saved 20% with respect to the original 25 rehearsals (5/25 = 0.2 or 20%). If it takes just
as long to relearn the list as it took to learn it originally, then savings is 0. If the list is still
completely known at the second trial (i.e., no forgetting at all), then savings is 1 or 100%.
Ebbinghaus prefers to express savings in terms of time spent learning and relearning but the
principle remains the same. After Ebbinghaus publication in 1885, the savings measure
remained popular for several decades [1619,2224]. Eventually, researchers found the savings
method too unreliable compared with other methods of measuring memory [24] and in the fol-
lowing decades it was used much less with some exceptions (e.g., [25]). Later, an important
improvement was suggested [26,27 ], where learning is not to the 100% criterion but to a much
lower one, such as 50% correct. These improved versions of the method are used nowadays, for
example, when studying forgetting of foreign languages [2830].
In the following, we will first report our replication experiment. Then, in the Discussion sec-
tion we will revisit the shape of forgetting, analyze the effects of serial position on forgetting,
and investigate what mathematical equations present a good fit to the Ebbinghaus forgetting
curve and its replications. Finally, we will study whether there is evidence for a jump at 24
hours in these curves, which some authors have attributed to th e effect of sleep.
Replication and Analysis of Ebbinghaus' Forgetting Curve
PLOS ONE | DOI:10.1371/journal.pone.0120644 July 6, 2015 3/23
The Replication Experiment
The current study was set up to replicate the findings by Ebbinghaus [8,9]. Despite a quite
detailed account of his experiment, we found some information to be lacking and we had to
estimate or guess these details, as outlined below. Also, we did not have the seven months avail-
able that Ebbinghaus invested in the experiment, but we had to accommodate our design to a
75 day period. We nonetheless believe that our experiment is close enough to his to be still
called a replication.
There were a few differences between Ebbinghaus study, Heller et al.s[21] replication, and
ours. (i) Because we were limited in time, like Heller et al. [21], we ran only 10 replications per
time interval, instead of the 12 to 45 by Ebbinghaus. This means the variance in our data is
larger than in Ebbinghaus especially at th e longest time intervals; apart from that no systematic
differences were introduced. (ii) We were not able to experiment at a fixed time of day. Ebbin-
ghaus (1880), who started experimenting in the morning at (A) 10:00, and then sometimes also
at (B) 12:00 and usually at (C) 19:00 to 20:00, noticed that there was a difference between these
times of the day in his ability to acquire a list. He subtracted 5% for B and 13% for C from the
learning times at these hours to normalize the data with respect to time A. Heller et al. [21]
were also able to conduct experimental sessions at specific times throughout the day, but they
did not find such a time-of-day effect and hence did not implement a correction. (iii) Our stim-
ulus material conformed to the phonotactics of the Dutch language and thus differs from both
Ebbinghaus and Heller et al. Also, in contrast to both Ebbinghaus and Heller et al. we removed
syllables that had too much meaning in order to further balance the level of difficulty of the
stimuli. (iv) Our subject, J. Dros, was younger than H. Ebbinghaus, who was 29 during his
experiments in 18791880. The ages of the two subjects in Heller et al. [21] are not given. (v)
Ebbinghaus [8] gives exact testing dates for each of the short time-intervals but not for the lon-
ger ones (24 hours and up). Hence, we do not know exactly when he learned and relearned the
lists for the longer intervals. This makes it impossible to calculate the number of interfering
lists between learning and relearning. It also makes it nearly certain that our schedule differed
from his (and from that of Heller et al. [21] who also do not supply such a schedule).
Subject
The second author, J. Dros, (22 years, male) was the only subject in the experiment. This exper-
iment was reviewed and approved by th e Review Board of the Psychology Department of the
University of Amsterdam (see www.lab.uva.nl). The project is filed with case number
2014-BC-3879 (contact is Dr. R.H. Phaf). Consent was implicit as the second author of the
paper was also the only subject on which we report. This was also app roved by the Review
Board. The subject's native language is Dutch, making this the first non-German replication of
Ebbinghaus forgetting experiment.
Materials
The learning material consisted of 70 lists. Each list consisted of 104 nonsense syllables, which
in turn consisted of 8 rows of 13 syllables.
Nonsense syllables. Each syllable consisted of 3 or 4 lower-case letters. The structure of a
syllable was a lower-case consonant-vowel-consonant (CVC) structure. The consonant of the
syllable was always one of b, d, f, g, h, j, k, l, m, n, p, r, s, t, or w. The vowel could be one of e, i,
o, u, aa, uu, ee, ei, eu, oe, ie, oo or ui. The double-letters stand for standard Dutch vowels. The
last consonant of the syllable was one of f, g, k, l, m, n, p, r, s, or t.
The number of different possible consonant-vowel-consonant combinations on the basis of
these letter combinations is 2100 (15 × 14 × 10). Not every possible consonant-vowel-consonant
Replication and Analysis of Ebbinghaus' Forgetting Curve
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combination was included in the learning material; we removed words that had too much mean-
ing in Dutch, in order to further balance the difficulty of the stimuli. Syllables with meanings in
other languages spoken by the subject, such as English and German, were not excluded.
Row and list construction. Using the pseudo-random generator of Excel 2010, rows of 13
syllables were constructed. Within a row we did not allow two syllables with the same vowel in
direct succession. We also did not allow two identical syllables within one row, but we did
allow them in different rows within a single list. When syllables needed to be adjusted we first
tried changing only the first or second letter of a syllable until the criteria were met. If this did
not suffice, additional letters were changed. The adjustment process was not purely random
but was carried out by hand during stimulus preparation by the authors.
Procedure
The only independent variable in this experiment was the time-interval, which started at the
end of learning a list for the first time. The time-interval ended at the beginning of learning a
list for the second time. The time-intervals between learning and relearning were the same as
Ebbinghaus [8]: 20 minutes, 1 hour, 9 hours, 1 day, 2 days, 6 days and 31 days. For each time
interval, 10 lists were learned and relearned (for the 9 hour interval only 9 lists were learned
due to unforeseen circumstances).
We need to elaborate on the choice of these time intervals as there is some confusion about
the exact length of the shorter retention intervals used by Ebbing haus. He mentions both 15
min (in 1880 [8]) and 19 min (in 1885 [9]) for the shortest interval, and 63 min and 8.75 hours
(525 min) for the longer intervals. He also states that relearning took place after about one
third of an hour, after 1 hour, after 9 hours, one day, two days, six days, or 31 days. ([20],
p. 66). Heller et al. [21] followed the latter intervals, using 20 min, 60 min and 9 hours, etc. We
have also used these, given that this seems to have been the intended lengths of Ebbinghaus
retention intervals. The deviations these values by Ebbinghaus are based on corrections after
the experiment.
Ebbinghaus shortest interval ( 20 minutes) is based on almost immediately relearning a list
of eight rows and hence the interval depends on how long it took to learn the eight lists. When
relearning the lists so soon it takes much less time to relearning them, than the original learn-
ing, so that the intervals between learning and relearning of lists is not constant, with List 8
being relearned earliest (e.g., 20 min) and List 1 latest (e.g., 10 min). Ebbinghaus [8] (p. 50)
states that the average time is about 15 min and argues that that whereas he does not know
exactly how to correct for these variable learning times, the error will be small. We recalculated
the average of the times stated on page 51 of the 1880 text and find it to be 1010 s or 16.8 min.
Ebbinghaus keeps using the value of 15 min throughout his text from 1880, including for fitting
his power function equation (see below). The learning and relearning times given in his 1885/
1913 volume [20] are the same as in 1880 [8], but to each interval he has now added 88 s for
reasons that are not made clear. The average of these learning intervals then becomes 18.3 min.
Given that he also remarks that for the shortest interval relearning of the first series of a test
followed almost immediately or after an interval of one or two minutes upon the learning of
the last series of the same test ([20], p. 66) may explain rounding 18.3 min to 19 min, the
value used throughout the text from 1885. In general, it seems he made more or less intuitive
corrections for the variable learning times and changed his mind from 1880 to 1885 about the
most appropriate method to approach this. We have used 19 min, 63 min, and 8.75 hours in
the graphs and tables (and fits), for Ebbinghaus data. For the other data sets, we use 20 min,
1 hour, and 9 hours.
Replication and Analysis of Ebbinghaus' Forgetting Curve
PLOS ONE | DOI:10.1371/journal.pone.0120644 July 6, 2015 5/23
Measurement of repetitions and time. The main measurement was the number of repeti-
tions needed to correctly reproduce the syllables in a row in the correct order.
For the forgetting curve experiment, Ebbinghaus [8] learned until twice correct, but in later
experiments switched to once correct [9], because he found there to be no essential difference
in the outcomes. We chose to also learn to once correct. Heller et al. [21] (p. 8) seem to be
using the once-correct criterion as well but this is not made entirely clear.
Ebbinghaus [8] uses elapsed time to calculate the number of repetitions, because he finds
keeping count too distracting. Heller et al. [21] use a chain with wooden, colored beads, much
like a rosary to keep count. We found that word processing software (Microsoft Word) was
handy to keep track of the number of repetitions. During learning, each single repetition of a
row was counted by pressing the button 1 on the keyboard at the beginning of every single
repetition of a row. At the end of learning a list, the total number of 1s for each row was
counted and entered into the database.
We also measured the time in seconds needed to learn a list. A clock was shown on the com-
puter screen during the task and we recorded the begin times and end times of learning a list.
Subtracted from the time were the pauses of 15 s between two rows (cf. [8], p. 19). When
relearning a list, the extra time (15 s) introduced by the voice recording of the first retrieval
attempt was subtracted from the total relearning time.
The practice phase and experimental phase. Following Ebbinghaus, we preceded the
experimental phase of the experiment with a practice phase to prevent as much as possible gen-
eral learning effects due to growing experience with the task and materials. The practice phase
took place between 08-11-2011 and 29-11-2011. A total of 14 lists was learned and relearned
after 20 minutes (Heller et al. [21] relearned lists after one hour). After these, a further 19 lists
were learned only (i.e., not relearned later) for additional familiarization with the task.
In the experimental phase, which took place between 01-12-2011 and 13-02-2012, a total of
69 lists was learned and relearned (9 for the 9 hour interval). The total time spent on data col-
lection in the experimental phase amounted to about 70 hours. We distributed the ten lists for
each time interval as much as possible over the whole experimental period. Due to the limited
time available to run the whole experiment, we were not able to achieve this for the 31 days
time interval condition, so that we decided to learn these lists near the beginning of this experi-
mental period.
List learning phase. All lists were printed on paper (black ink, font Calibri, 11 points)
(eight rows per page; a row was actually printed in a column format for easier studying). The
non-studied rows were covered by sheets of paper. The subject was seated behind a desk in a
quiet room. The main goal was to learn a list as quickly as possible, to learn each row until it
could be reproduced correctly once.
Following Ebbinghaus [8] (p.18) , the syllables were softly spoken from the first syllable to
the 13
th
syllable at a constant speed of 150 beats per minute. The repetition of a row took 5.2 s
on average. Our subject preferred to speak the syllables in a jambus-like manner, where sylla-
bles were paired so that the emphasis always was on the second syllable (i.e. wes-hóm, niem-
hág, etc...). The last syllable (13
th
syllable) was not paired to another syllable and was not
spoken with an emphasis. Here, we use an approach similar to Heller et al. [21] and not to
Ebbinghaus, who prefers a ¾ rhythm, stressing the first syllable in each group of three (this is
in fact the reason he gives for his preference for rows of leng th 10, 13, or 16 syllables, see Ebbin-
ghaus [8], p. 19).
During the learning phase, the subject had a continuous choice to either read or reproduce
the syllables. Towards the end of the learning process, occasional attempts were made to pro-
duce an entire row by heart. When there was a moment of hesitation during such blind repro-
duction, the rest of the list was read (i.e., not blind) to the end. Blind reproduction always
Replication and Analysis of Ebbinghaus' Forgetting Curve
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started with the first syllable of a row. Rows we re not learned in parts. Each time the 13
th
sylla-
ble had been reached and the row still contained errors, it was read again from the beginning.
After having learned one row and before starting the next one, there was an interval of 15 sec.
The interval serv ed as a moment of rest and pause. All of this was aimed at following Ebbin-
ghaus as closely as possible. Each row was thus learned to a 100% correct criterion before mov-
ing to learning the next row.
During the pre-pretraining phase, a metronome was used at first to achieve a recitation rate
of 150 beats, but this was found to be too intrusive and distracting. Eventually, the rhythm was
internalized and the metronome was only used for occasional rate checks. During the experi-
mental phase, it was not used. After each rehearsal of a row, there was a little transition-pause
of about 3 beats to take a breath before the next repetition of the row.
Learning of a list was considered complete, if all rows had thus been learned in order. The
retention interval was started at the time a list had been learned. On most days two or three
lists were learned or relearned with a maximum of four. The full learning schedule is given in
Fig 1.
Relearning phase. We added one additional measurement to Ebbinghaus procedure: the
number of correct syllables at first reproduction of a row at relearnin g, not necessarily at the
right location. We recorded the recall of the first time a row was relearned with an Olympus
WS-450s voice-recorder. After the last relearning session of the experimental phase, the sound
files were transcribed and scored.
Rows were relearned in the same order as during original learning. During relearning, the
subject was seated at a desk with a computer. A word processing program was opened on the
computer and a clock was visible on the screen. A sheet of paper with a list printed on it was
laid in front of the subject with only the syllables from the row to be learned visible. The other
rows were covered by a piece of paper. During learning, the subject used the 1 button on the
keyboard to count the number of repetitions. Following Ebbinghaus [8] (p. 19), after successful
relearning there was a 15 second pause. Then, the row learned last was covered, a next row
uncovered and the procedure was repeated.
Relearning of a row started with turning on the voice recorder. Then the row was read once
as described above. Twenty seconds after turning the voice recorder on, the subject stopped
recall attempts and turned the voice recorder off. After that, relearning continued in the usual
fashion. This procedure was repeated for every row in the list. After a list had been relearned,
the audio file of the recording was saved on a computer.
Results
The forgetting curve
The main objective was to replicate Ebbinghaus famous forgetting curve. The average number
of repetitions is given in Table 1 and the number of seconds spent on learning and relearning
each list, with the calculated savings scores, is given in Table 2. The raw data of this experiment
are freely available online at the website of the Open Science Foundation (URL: https://osf.io/
6kfrp/). To see whether there were differences between the time-intervals in the average num-
ber of repetitions at first learning, a one-way independent ANOVA with the average number of
repetitions per list as the dependent variable and the time-interval as the independent variable.
There was no significant effect for the time-interval, F(6, 69) = 0.691, p = 0.658. This means
that the average number of repetitions per list did not differ significantly per time interval, indi-
cating there were no randomization confounds.
Savings scores (based on time in s) are compared with those of Ebbinghaus [8], and Mack
and Seitz [21]inTable 3 and plotted with error bars in Fig 2 using loglog coordinates and in
Replication and Analysis of Ebbinghaus' Forgetting Curve
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Fig 3 using log coordinates only for the time axis. Despite the fact that the original experiment
dates from 1880 and replications were done over a century later, and despite the fact that our
replication was carried out in a Dutch language context, the four forgetting curves share many
characteristics. To facilitate a direct comparison we have overlaid the four curves in Fig 4
where we have normalized the savings scores such that the first data point (at 20 min) was
always equal to 1.0. Given expected individual differences, we find the resemblance of the four
graphs remarkable. The greatest deviation is by Dros at 31 days; his savings score is much
Fig 1. Learning schedule during 20112012 for all lists, where labels in bold indicate when each of the lists 1 to 10 was first learned for each
retention interval. Relearning times are not shown but can be derived by adding the retention interval (e.g., 6 days).
doi:10.1371/journal.pone.0120644.g001
Table 1. Average number of repetitions until once correct.
n Learning SD n Relearning SD
20 min 10 30.77 2.90 10 16.26 2.27
1 hour 10 30.64 2.28 10 19.22 1.56
9 hours 9 31.07 2.08 9 22.48 2.67
1 day 10 31.22 2.34 10 21.33 2.11
2 days 10 31.42 2.49 10 24.19 2.30
6 days 10 31.21 3.19 10 25.97 3.37
31 days 10 29.44 2.26 10 28.23 3.48
doi:10.1371/journal.pone.0120644.t001
Replication and Analysis of Ebbinghaus' Forgetting Curve
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Table 2. Time spent learning (session S1) and relearning (session S2) for each list with savings (Q) by Dros.
20 min 1 hour 9 hours 1 day 2 days 6 days 31 days
List S1 S2 Q S1 S2 Q S1 S2 Q S1 S2 Q S1 S2 Q S1 S2 Q S1 S2 Q
1 1405 670 0.523 1690 1280 0.243 1815 1240 0.317 1670 1105 0.338 1710 1195 0.301 1780 1370 0.230 1480 1380 0.068
2 1840 1210 0.342 1790 1330 0.257 1780 1350 0.242 1840 1325 0.280 1635 1340 0.180 1605 1560 0.028 1680 1450 0.137
3 1830 1100 0.399 2070 1235 0.403 1935 1350 0.302 1930 1205 0.376 1950 1580 0.190 1870 1545 0.174 1770 1530 0.136
4 2180 960 0.560 1875 1130 0.397 1525 975 0.361 1740 1365 0.216 1935 1440 0.256 2020 1805 0.106 1440 1510 -0.049
5 1800 840 0.533 1775 1245 0.299 1770 1275 0.280 1875 1410 0.248 1830 1500 0.180 2090 1785 0.146 1650 1760 -0.067
6 1815 1345 0.259 1765 1170 0.337 1815 1335 0.264 1710 1215 0.289 2130 1485 0.303 1740 1585 0.089 1890 1785 0.056
7 2040 1110 0.456 1680 1125 0.330 1635 1220 0.254 1905 1325 0.304 1890 1440 0.238 1710 1350 0.211 1815 1745 0.039
8 1725 865 0.499 1905 1250 0.344 1845 1380 0.252 2095 1235 0.411 2085 1460 0.300 2025 1665 0.178 1910 1505 0.212
9 1935 1320 0.318 1805 1155 0.360 1950 1585 0.187 1860 1290 0.306 1740 1335 0.233 2100 1415 0.326 1490 1260 0.154
10 1830 1235 0.325 2065 1325 0.358 1980 1275 0.356 1695 1375 0.189 1890 1275 0.325 1710 1395 0.184
Average 1840 1066 0.421 1842 1225 0.335 1786 1301 0.271 1861 1275 0.315 1860 1415 0.239 1883 1536 0 1684 1532 0.090
doi:10.1371/journal.pone.0120644.t002
Replication and Analysis of Ebbinghaus' Forgetting Curve
PLOS ONE | DOI:10.1371/journal.pone.0120644 July 6, 2015 9/23
lower that any of the other three. We can only speculate at the reason for this. It may be that
this subject simply has more long-term forgetting. Another explanation bears on the fact that
the lists for the 31 day interval were all learned early in the experiment (see Fig 1) and learning
times were shorter at the beginning, due to greater initial enthusiasm, less pro-active interfer-
ence, or yet another reason. It is also possible that the low savings score on the 31 day point is
an effect of the relatively short time period in which initial learning took place for the 31-day
data point (i.e., massed learning); learning for other intervals was more widely spaced. We
were forced to place the learning sessions at the beginning because of the limited time available
by the subject for data collection. Ebbinghaus could spread his sessions of a seven-month inter-
val, though we do not know the exact schedule for the intervals past 9 hours, nor do we know
Table 3. Savings for Ebbinghaus ([8], p. 56, see Note 1 for the comments on the intervals 20 min, 1
hour and 9 hours), Mack and Seitz [21], and Dros (this paper).
Ebbinghaus Mack Seitz Dros
20 min 0.582 0.544 0.442 0.472
1 hour 0.442 0.432 0.325 0.373
9 hours 0.358 0.285 0.270 0.276
1 day 0.337 0.316 0.270 0.317
2 days 0.278 0.365 0.286 0.230
6 days 0.254 0.309 0.205 0.168
31 days 0.211 0.258 0.201 0.041
doi:10.1371/journal.pone.0120644.t003
Fig 2. Four loglog graphs with savings as a function of retention interval with fitted power function curves and curves with best fitting power
functions with boost at 1 day (see text for an explanation).
doi:10.1371/journal.pone.0120644.g002
Replication and Analysis of Ebbinghaus' Forgetting Curve
PLOS ONE | DOI:10.1371/journal.pone.0120644 July 6, 2015 10 / 23
anything about the schedule followed by Mack and Seitz. Finally, we considered where the
number of interve ning lists had influenced retention at the 31-day-point. For this time point,
the number of intervening lists varied from 23 to 33. Our analysis, however, revealed virtually
no relationship with savings score as a function of intervening lists (R
2
was about 8.4%). Our
data did not allow us to further disentangle the effect of number of intervening lists versus time
because list number and time were too strongly confounded in our design.
We found a gradual increase in learning time throughout the course of the experiment as
can be seen in Fig 5, where averaged learning time in s has been plotted for consecutive ten-day
periods (bins). In the course of the 75 days of the experimental phase there was an average
increase in learning time of 2.67 s per day for a list (this linear regression explained 56.18% of
the variance). If we correct for this steady increase, which mostly affects the 31 day interval, the
corrected savings measure would be 0.137 for the 31 day interval instead of 0.0410. This, how-
ever, is still well below the values for the three others, which are in the 0.20 range. This steady
increase in learning time may be due to pro-active interference or fatigue. Ebbinghaus [8] and
Heller et al. [21] do not report or analyze this.
We also analyzed the false alarms and correct answers measured on first relearning. We did
a one-way independent ANOVA with the number of false alarms per row as the dependent
variable and the retention interval as th e independent variable. There was no significant effect
for the time interval, F(6, 512) = 0.753, p > 0.608, indicating that the number of false alarms
was not significantly different for the time-intervals. A one-way independent ANOVA with the
number of hits per row as the dependent variable and retention interval as the independent
Fig 3. Four log graphs with savings as a function of retention interval with best-fitting Memory Chain Model retention functions (see text for an
explanation).
doi:10.1371/journal.pone.0120644.g003
Replication and Analysis of Ebbinghaus' Forgetting Curve
PLOS ONE | DOI:10.1371/journal.pone.0120644 July 6, 2015 11 / 23
variable, however, yielded a significant effect for the time-interval, F(6, 512) = 3.85, p < 0.01,
which we will analyze further in the next section.
Serial position effects
In Fig 6, we have plotted the average serial position curves for each retention interval and for
the grand avera ge. Even if a correct syllable was not mentioned at the correct position, it was
still scored as correct for its intended position (this was rare and had only a small effect on the
data). Fig 6 shows clear serial position curves for all retention intervals.
In Fig 6, it also seems that there is more forgetting with time in the middle positions. In
Fig 7A, overall forgetting is shown, where the average of all positions is shown for each reten-
tion interval. Though there is forgetting, these curves are much shallower than the savings
curves, which are shown as well for comparison. In Fig 7B, forgetting is shown for four groups
of serial positions, indicating that indeed there is virtually no forgetting in the final positions
11 to 13. A regression line is nearly horizontal for positions 1113 (slope -0.000183) and 12
(slope is -0.000112) with the largest decrease over time found in positions 38 (slope is
-0.0114) and 910 (slope is -0.0119). The averaged curve has a slope of -0.00930, with all slopes
calculated over the untransformed scores (i.e., not on a logarithmic scale).
Fig 4. Normalized savings scores as a function of retention interval on a logarithmic scale, rescaled so the first data point is 1.0 for all curves.
doi:10.1371/journal.pone.0120644.g004
Replication and Analysis of Ebbinghaus' Forgetting Curve
PLOS ONE | DOI:10.1371/journal.pone.0120644 July 6, 2015 12 / 23
Fig 5. Learning time per list as a function of day of experiment with a fitted straight line.
doi:10.1371/journal.pone.0120644.g005
Fig 6. Serial position for correct relearning scores for each retention interval and for the average of all retention intervals (see text for an
explanation).
doi:10.1371/journal.pone.0120644.g006
Replication and Analysis of Ebbinghaus' Forgetting Curve
PLOS ONE | DOI:10.1371/journal.pone.0120644 July 6, 2015 13 / 23
Discussion
We believe that we may conclude that our attempt to replicate Ebbinghaus classic forgetting
was successful. We were able to follow his method quite closely and the resulting curve is very
similar to both that of Ebbinghaus and that of the two subjects in an earlier German replication
[21], with the exception of the savings value at 31 days, which in our case is much lower than
the others. The latter difference remains even if we correct for increased learning time over the
course of the experiment. It is possible that with Ebbinghaus and Heller et al. there were far
fewer intermediary lists learned between learning and relearning and hence much less interfer-
ence. Unfortunately, this information is not available, so this must remain speculation.
Effects of serial position on forgetting
Ebbinghaus does not say anything about serial position curves or indeed about the order in
which he acquired the syllables. Our data allow us to say a little more about this. When inter-
preting the serial position scores in Fig 6, one has to bear in mind the nature of the savings
methods with lists of nonsense syllables. With Ebbinghaus-type relearning, a row is always first
studied before it is relearned. Savings experiments are very different from normal memory
retention experiments where the subject learns something at after some time interval is tested
for retention. With savings, the retention measurement itself consists of relearning the original
material in repeated recall trials each of which is preceded by prior exposure to the stimulus
materials. In theory it is, therefore, possible in extreme cases that there are subjects who can
Fig 7. Proportion correct as a function of retention interval on a logarithmic scale. (a) Proportion correct, averaged over all serial positions, shown with
Dros savings scores for comparison. (b) Proportion correct curves for different groups of serial positions and for the average over all 13 positions.
doi:10.1371/journal.pone.0120644.g007
Replication and Analysis of Ebbinghaus' Forgetting Curve
PLOS ONE | DOI:10.1371/journal.pone.0120644 July 6, 2015 14 / 23
learn a row of 13 syllables in a single learn-recall trial but who subsequently always forget the
learned materials in say 10 min. By conventional (non-savings) retention tests, these subjects
would be reporte d as have 0% retention past 10 min, but these same subjects would show 100%
savings for all retention intervals past 10 min because when testing th eir savings performance
they would go through the exposure-learn sequence, always (re)learning the materials immedi-
ately: at t = 0, they take 1 trial to learn the material and at t = 1 day they also take 1 trial, show-
ing perfect savings (i.e., seemingly no forgetting). Thus, when stimuli are very easy to learn true
forgetting becomes impossible to measure with the savings method, because original learning
will be at ceiling (one-trial learning) as will be relearning after a time-interval (again: one-trial
learning), no matter how long the interval.
Here, it seems that the first two and the last three syllables were very easily learned (and
relearned), probably because of primacy and recency effects. This effect was also noticed by
other early researchers who adapted Ebbinghaus method (e.g., [17]). Relearning the harder
parts of a list, in particular the middle syllables, benefits most from recently having learned
these sometime before (i.e., before the relearning phase of the savings experiment), as is evident
from the savings scores. The discrepancy between savings and recall or recognition has also
been found by other authors (e.g.,[24,31,32,33]) and appears here as a function of serial
position.
The fairly evident primacy and recency effects also suggest that the despite Ebbinghaus
efforts to construct equivalent stimuli, there is a great variation in how well they eventually
were learned: During the first phase (t = 0), stimuli in the first and last positions were very eas-
ily and hence very well learned, while those in the middle were learned much less. The effect
seems quite constant, however, so that it need not affect the validity of the shape of the for-
getting curve, but we should be aware that its shape is based on a combination of very well
learned items and just barely learned items. In that sense, the forgetting curve by Ebbinghaus is
an average over different forgetting curves of items in various serial positions, which have been
learned to varying degrees. This in itself may explain part of the characteristic shape of the
curve, which we will explore in the next section.
Curve fitting
Hermann Ebbinghaus [8] was the first to try to find a mathematical equation that describes the
shape of forgetting. Many researchers who used his method have followed suit, also trying to
summarize the forgetting curve in a concise equation. In his first manuscript, from 1880,
Ebbinghaus proposes the equation x =[1 (2/t)
0.099
]
0.51
, where x equals 1 minus savings at
time t (in min). It is of some interest that Ebbinghaus [8] (p. 5763) puts the entire section in
which he fits this equation to the data between square brackets, making it an aside: something
that is also interesting but not belonging to the main text. Nonetheless, he gives a well-moti-
vated derivation based on a differential equation of a gradually slowing forgetting process.
Interesting is that in the write-up of the experiment in 1885 [9], the equation has been changed
to the very different one, which has been become generally known as the Ebbinghaus For-
getting Equation, rather than the first one, namely Q(t) = 1.84 / ((log t)
1.25
+ 1.84), where the
log is taken with base 10. This equation is lacking a derivation and Ebbinghaus remarks on it:
Of course this statement and the formula upon which it rests have here no other value than
that of a shorthand statement of the above results which have been found but once and under
the circumstances described. Whether they possess a more general significance so that, under
other circumstances or with other individuals, they might find expression in other constants I
cannot at the present time say. [20]
Replication and Analysis of Ebbinghaus' Forgetting Curve
PLOS ONE | DOI:10.1371/journal.pone.0120644 July 6, 2015 15 / 23
We are now in a better position to verify Ebbinghaus question about the general signifi-
cance of his equation by fitting his equations to the other data. The results are given in Table 4
and were obtaine d with a nonlinear fitting procedure in Mathematica 9 (the Mathematica code
is available from the author). We calculated several goodness-of-fit measures including the
Akaike Information Criterion or AIC [49,58], which contrary to, for example, varian ce
explained (R
2
) or sum of squared differences ( SSD), takes into account (and penalizes for) the
number of free parameters. It also allows a comparison of the goodness-of-fit of different mod-
els, even if they have different numbers of parameters. Lower values indicate better fit, where a
difference of more than 2 is seen as a meaningful difference in goodness-of-fit [49].
In the table, we see that in the case of his power function from 1880 [8], Ebbinghaus' calcu-
lations, carried out by hand, were quite close the computer-optimized parameter values: he
found values 0.51 and 0.099 for the parameters, whereas we found 0.523 and 0.101, respec-
tively. For the logarithmic function from 1885 [9], we also found similar parameters to those
parameters Ebbinghaus reported: 1.8 and 1.21 for his values of 1.85 and 1.25 respectively.
The goodness-of-fit of his functions is quite good, in both cases explaining 98.8% of the var-
iance (R
2
) for his own data. The equation from 1885 has a slightly smaller SSD value (i.e., fits
better), which in fact is the lowest value for an individually fitted curve we obtained (also see
below and Table 5). Though the equations found by Ebbinghaus fit his own data very well, they
do not always fit the other curves well, with especially Mack and Dros showing a relatively bad
fit on these classic equations. This is perhaps not concluded from the variance explained (R
2
),
which is very high for all studies, but if we base our judgment on the AIC we observe large dif-
ferences where the AIC for the Ebbinghaus data is almost twice as low as on the Dros data.
This suggests that the general applicability of Ebbinghaus equations may be lacking. We further
investigate this by comparing Ebbinghaus functions with some other functions that have been
proposed in the literature.
Ebbinghaus function from 1880 is a type of double power function. A normal power func-
tion is described by the equation QðtÞ¼ð1 þ m
1
tÞ
a
1
, where Q(t) is savings at time t and μ
1
and a
1
are parameters. The latter equation has been proposed by several authors to describe the
time-course of forgetting (e.g., [1,3,4,34]). The forgetting mechanism typically associated with
a power function is a constant slowing down of the forgetting rate with time (cf. Ebbinghaus'
Table 4. Fits of two equations proposed by Ebbinghaus in 1880 and 1885 to data from his own study
and from three replication studies. See text for the meaning of the parameters. SSD is the sum of squared
differences between data and fitted curve, R
2
is proportion variance explained, and AIC is the Akaike Informa-
tion Criterion. To stay close to Ebbinghaus own estimates, the parameters are fitted for time expressed in
minutes.
Ebbinghaus Mack Seitz Dros Average
Ebbinghaus 1880 Power Function
μ
1
0.523 0.325 0.248 0.516
a
1
0.101 0.0518 0.0525 0.14
SSD 0.00224 0.0107 0.0043 0.0177 0.00871
R
2
0.998 0.989 0.993 0.972 0.988
AIC -30.5 -19.5 -26 -16 -23
Ebbinghaus 1885 Logarithmic Function
μ
1
1.8 1.34 0.9 1.36
a
1
1.21 0.873 0.82 1.34
SSD 0.00218 0.00976 0.00403 0.0212 0.00928
R
2
0.998 0.99 0.993 0.966 0.987
AIC -30.6 -20.2 -26.4 -14.7 -23.
doi:10.1371/journal.pone.0120644.t004
Replication and Analysis of Ebbinghaus' Forgetting Curve
PLOS ONE | DOI:10.1371/journal.pone.0120644 July 6, 2015 16 / 23
account from 1880 mentioned above). Whereas this is certainly a viable mechanism of for-
getting, it can be proven mathematically that (spurious) power functions may emerge from
averaging over different subjects or items [35,36]. This has also been shown in simulations con-
sidering a wide range of circumstances [37,38]. As argued in the previous section, the forgetting
curve (also) averages over items that have been learned to various degrees, due to their serial
position. There are, therefore, several reasons to expect and consider the power function.
The goodness-of-fit of the simple power function to our data is given in Table 5. As can be
seen, the fit to Ebbinghaus data is still impressive, though somewhat less good than either of
his own equations. The goodness-of-fit of the power function, as expressed by the SSD, aver-
aged over all four subjects is comparable to the Ebbinghaus 1885 logarithmic function and it
is somewhat worse than his 1880 power function. The AIC is slightly worse, but probably not
meaningfully so as the difference in AIC measures is only 1.
Heller et al. [21] also fitted the Ebbinghaus 1885 logarithmic equation to the Mack and
Seitz data and noticed that it did not fit the Mack and Seitz data well. They therefore proposed
Table 5. Fits of a number of equations to data from Ebbinghaus and replication studies. See text for
the meaning of the parameters. SSD is the sum of squared differences between data and fitted curve, R
2
is
proportion variance explained, and AIC is the Akaike Information Criterion. The parameters are fitted for time
expressed in seconds.
Ebbinghaus Mack Seitz Dros Average
Power Function
μ
1
1.4 0.965 0.822 1.56
a
1
0.13 0.0926 0.099 0.167
SSD 0.00285 0.0129 0.00523 0.0163 0.00932
R
2
0.997 0.987 0.991 0.974 0.987
AIC -28.8 -18.2 -24.5 -16.6 -22
Power Function with Boost
μ
1
1.65 2.13 1.27 1.67
a
1
0.152 0.194 0.155 0.176
Boost 0.0303 0.131 0.0631 0.011
SSD 0.00232 0.00355 0.0031 0.0162 0.00628
R
2
0.998 0.996 0.995 0.974 0.991
AIC -28.2 -25.2 -26.2 -14.6 -23.6
Summed Exponential Function
μ
1
0.383 0.315 0.304 0.262
a
1
0.000319 0.000296 0.000457 0.000353
μ
2
0.321 0.323 0.266 0.3
a
2
1.79E-07 7.99E-08 1.22E-07 1.00E-06
SSD 0.00469 0.00356 0.00276 0.00295 0.00349
R
2
0.995 0.996 0.995 0.995 0.996
AIC -21.3 -23.2 -25 -24.5 -23.5
MCM Exponential Function
μ
1
0.704 0.639 0.57 0.563
a
1
0.000319 0.000296 0.000457 0.000353
μ
2
0.000145 0.00015 0.000213 0.000188
a
2
1.79E-07 7.99E-08 1.22E-07 1.00E-06
SSD 0.00469 0.00356 0.00276 0.00295 0.00349
R
2
0.995 0.996 0.995 0.995 0.996
AIC -21.3 -23.2 -25 -24.5 -23.5
doi:10.1371/journal.pone.0120644.t005
Replication and Analysis of Ebbinghaus' Forgetting Curve
PLOS ONE | DOI:10.1371/journal.pone.0120644 July 6, 2015 17 / 23
a different equation, the sum of two exponen tials: QðtÞ¼m
1
e
a
1
þ m
2
e
a
2
A similar function
has independently been proposed by Rubin, Hinton and Wenzel [39] to succe ssfully t a for-
getting curve with very large numbers of observations per data point, which could not be tted
satisfactorily with any of the more than hundred functions studied in Rubin and Wenzel [2].
None of these authors gave a forgetting mechanism associated with these functions.
Though providing us with a superior fit, a disadvantage of the summed exponential is that
there are no memory models that explain why forgetting might have this shape. As remarked
by several authors investigating the shape of learning and forgetting [35,38,40], simply fitting
sums of exponentials is expected to yield progressively better fits for the simple reasons that
any function may be approximated by such a sum, which is related to the Laplace
transformation.
A model of forgetting and amnesia developed by our group also yields a summed exponen-
tial function, but with a different parameterization [41]. This so called Memory Chain Model
assumes that a memory passes through several neural processes or stores, from short-term to
very long-term memory. While a memory is (exponentially) declining in intensity in Store 1
(e.g., the hippocampus), its contents is steadily transferred to a Store 2 (e.g., the neocortex)
from which it will decline at a lower rate. We still have two exponentially declining stores, as in
the summed exponential function above, but they are linked by a memory consolidation pro-
cess. The decay rates in Store 1 and Store 2 are given by a
1
and a
2
, respectively. The initial
strength of the memory traces in Store 1 are given by μ
1
and the rate of consolidating the con-
tents of Store 1 to Store 2 is given by μ
2
. In experiments with dementia patients and experimen-
tal animals, Store 1 may typically be identified with the hippocampus and Store 2 with the
neocortex. Lesioning Store 1, will produce a retrograde amnesia gradient that can be modeled
by the Memory Chain Model simultaneously with the forgetting gradient of healthy controls
[42].
The Memory Chain Model (MCM) equation for type of savings studied here is given by
QðtÞ¼m
1
e
a
1
t
þ
m
1
m
2
ðe
a
2
t
e
a
1
t
Þ
a
1
a
2
The MCM function has the same number of parameters but they are arranged differently. The
proof that this equation is a mathematical formalization of the memory consolidation process
can be found elsewhere [42]. As can be seen in Table 5, the summed exponential and the MCM
function give exactly the same ts, though the parameters differ. The gain of using the MCM
function lies primarily in the fact that its parameters can be interpreted more clearly, that it is
associated with a type of consolidation mechanism, and that also explains other types of data
than the savings function [4143]. The MCM function assumes a neural system consolidation
mechanism [44,45] that has been dubbed the 'Standard Consolidation Theory' [46,47], where
the latter authors propose a different theory, the so called Multiple Trace Theory of consolida-
tion. It is here not our goal to evaluate the merits of these theories; we have reviewed these and
other theories of consolidation elsewhere [48]. We merely want to apply the MCM equation to
these four savings curves and evaluate the goodness-of-t, viewing it as a conceptual improve-
ment of the summed exponential.
If we compare the MCM equation or summed exponential function to the other functions,
this only makes sense if we rely on the AIC, which takes into account the varying number of
parameters. The MCM function (or double exponential function) fits two of the four curves
(Mack and Dros) better than Ebbinghaus own equations from 1880 and 1885, it gives about
the same fit on the Seitz data, and it does much worse on Ebbinghaus' own data. The average
AIC is 0.5 less than the average AIC for the classic Ebbinghaus functions, whichthough it
Replication and Analysis of Ebbinghaus' Forgetting Curve
PLOS ONE | DOI:10.1371/journal.pone.0120644 July 6, 2015 18 / 23
indicates a better fitmay not be considered a meaningful difference; a difference of 2 is con-
sidered 'meaningful' [49]. We also fitted a single exponential with only two parameters but this
fared far much worse on all data sets, including Ebbinghaus data (also see [3]).
Summarizing, Ebbinghaus' data fit his own equations and the power function best. The AIC
indicates that on average the MCM equation (or summed exponential function) is on average
better than all equations considered thus far, where the difference with the power function is
1.5. The difference with Ebbinghaus' own equations is only 0.5 but this is partially because his
own data have an exceptionally good fit on his own equations, with a very low AIC of about
-30.5 (and an extremely high 99.8% variance explained). It is likely, however, that Ebbinghaus
actively searched for an equation that achieves such an exceptional fit, which in his eyes was no
more than a 'summary' of the forgetting curve (see citation above). This also explains why he
has no problems substituting a 'logarithmic' equations for the earlier 'power' equation: it shows
a slightly better fit.
Fitting data is always done with a purpose. Ebbinghaus achieved a concise summary of his
forgetting data, the power function is a parsimonious description of the forgetting function
that shows a good or at least adequate fit in many types of forgetting experiments, and the
MCM equation attempts to capture the shape of a hypothetical consolidation process in the
brain albeit at the expense of additional parameters. Taking into account these extra parame-
ters, however, does not give a worse fit on the AIC and approaches a meaningful improvement
over the power function.
The 24-hour point in Ebbinghaus' forgetting curve
When looking at the shapes of the four curves in Fig 2, savings after 1 day (or 2 days) seems
higher than expected. Ebbinghaus [8] notices this as well but merely writes it off as a discrep-
ancy from his fitted curve (see above) that still falls within the error bars ([8], p. 62). He clearly
did not trust this data point because in his text from 1885 [9] he reports that he later had repli-
cated this 24 hour data point. The replicated data for this point gave a very similar score, so we
must consider it a valid measurement. Jenkins and Dallenbach [50], however, interpreted the
discrepancy as an effect of sleep, which motivated them to investigate this closer in an experi-
ment on the effect of sleep on forgetting. They also refer to the forgetting curve by Radossawlje-
witsch [16], who also found higher savings after both 1 and 2 days (0.689 and 0.609, resp.)
compared with after 8 hours (0.474). To them, this is suggestive of a very strong effect of sleep,
but Finkenbinder [17] points out that Radossawljewitsch's 8-hour data point may not be reli-
able, because these lists were all relearned during the afternoon, when there was less rapid
learning resulting in fewer savings. He, therefore, suggests using a corrected savings score at 8
hours of 0.66, which is not unreasonable given that Ebbinghaus also corrected his savings
scores for time-of-day effects, in some cases up to 13%. Even if savings would be 0.66 at 8
hours, however, the 1 day savings score is still higher than the 8 hour score and the 2 day sav-
ings is still higher than what one would expect.
Using free recall and retention up to 8 hours, the seminal study by Jenkins and Dallenbach
[50] yielded a positive effect of sleep on recall. This effect has since been replicated many times,
for example in recent studies on the effects of different sleep stages on both procedural and
declarative memory (e.g., [51,5256]). Whereas the older studies from the 1970s and before
typically confound the sleep manipulation with time-of-day effects or fatigue, this is no longer
the case in the recent studies, so that there is now very strong evidence that sleep does indeed
have an effect on memory independent of the effects of, say, rest or lack of interference. In
some of the sleep-memory experiments cited above, we even see a temporary increase in the
forgetting curve, where subjects score better than after learning in the days following sleep, but
Replication and Analysis of Ebbinghaus' Forgetting Curve
PLOS ONE | DOI:10.1371/journal.pone.0120644 July 6, 2015 19 / 23
not if they skipped the night of sleep after learning (e.g., [53]). This resultand other studies
suggests that the first night of sleep after learning has a particularly important effect on mem-
ory that may continue to evolve for several days afterwards. Such an effect may also be
observed in savings curve by Mack and to some extent in the Seitz curve, both show a tendency
to increase in savings score for two days following learning.
Given that we can trace the history of research on the effects of sleep on memory to the 24
hour point of Ebbinghaus' forgetting curve, we think it is interesting to evaluate this data point
more formally. If we can establish the jump in the curve more formally, it will make a stronger
case that the 'true shape' of the long-term forgetting curve has a jump in at 24 hours (or per-
haps right after the subject has slept), although we may not conclude from this that the local
increase is due to sleep per se, which would require more research is necessary for that.
If we first informally inspect the data points shown in Fig 2 and compare them with the fit-
ted power function, we see relatively less forgetting at either day 1, day 2, or both. In all four
panels, at least one of these points is above the fitted power function curve at a distance of at
least one standard error. The same is true for the fitted curves of the summed exponential,
Ebbinghaus 1880 power function and his 1885 logarithmic function (not shown here). We
also see this effect for the Memory Chain Model curve in Fig 3, though somewhat less pro-
nounced (in the Seitz panel, the fitted curve crosses the error bars at 1 and 2 days). The reason
for this is that the Memory Chain Model already incorporates the effects of a hypothetical con-
solidation process. In short, we observe that there is seems to be a memory boost in the classic
Ebbinghaus forgetting curve and its replications. The current body of research on sleep and
memory would predict such a boost after one or two nights and attribute it to sleep, though for
this particular type of experiments this has to established more firmly in further experiments.
Whatever its cause, we can better quantify the visually observed boost by including it in the
equations fitted. We, therefore, made a variant of the power function that differs only in the
addition of a constant boost factor to the savings of the retention intervals of 1 day and higher.
This power function with boost is also plotted in Fig 2.
The results are mixed, though on average they suggest a trend towards improvement with a
boost parameter. The original power function had a average AIC of -22, a sum-of-squared-dif-
ferences (SSD) of 0.00932 and explained 98.7% of the variance (see Table 5), whereas adding
the boost reduces the AIC to -23.6, the SSD to 0.00628 and increases the average variance
explained to 99.1%, putting its goodness-of-fit on a par with the MCM (or summed exponen-
tial) function. A difference in average AIC of 1.6 may perhaps be called a trend towards a
meaningful difference, though there were large differences between the individual subjects. The
boost parameter in Table 5 shows the size of the upward jump after 24 hours. We see that for
Ebbinghaus, this jump is small (0.030), whereas for Mack it is quite substantial (0.131). The
Dros data show no evidence for a boost but these fits are probably inf luenced strongly by the
very low 31 day data point.
Concluding Remark
In 1880, Ebbinghaus [8] set new standards for psychology experiments, already incorporating
such modern concepts as controlled stimulus materials, counter-balancing of time-of-day
effects, guarding against optional stopping, statistical data analysis, and modeling to find a con-
cise mathematical description and further verify his results. The result was a high-quality for-
getting curve that has rightfully remained a classic in the field. Replications, including ours,
testify to the soundness of his results.
His method can also be seen as a precursor to implicit memory tests in that certain inacces-
sible representations, seemingly forgotten, can still be relearned faster compared with others
Replication and Analysis of Ebbinghaus' Forgetting Curve
PLOS ONE | DOI:10.1371/journal.pone.0120644 July 6, 2015 20 / 23
that do not show such an advantage. This is evidence of implicit memory because the subjects
may not be consciously aware they still possess traces of the memory representations, which
cannot be recalled or recognized but that do show savings. The savings method is still used
today as a sensitive method to study the decline of foreign languages in order to assess the true
extent of linguist ic knowledge retained over a long time [57].
Ebbinghaus [8] also emphasizes the importance of sleep for memory, but these remarks are
limited to how low-quality or insufficient sleep may have inflated his own learning times at cer-
tain dates ([8], p. 66) and as an explanation for the observed time-of-day effects; he learns faster
in the morning than at other times. In other words, he acknowledges the effects of previous
sleep on current learning, but he does not admit to the role of sleep in slowing down long-te rm
forgetting. The formal analysis above suggests that the classic forgetting curve is not completely
smooth but does show a jump at the 1 day retention interval. Current research on the effects of
sleep on memory would predict such a jump, but for this particular type of experiment this
remains to be established.
Acknowledgments
We would like to thank Annette de Groot and Jeroen Raaijmakers for helpful suggestions
when writing this.
Author Contributions
Conceived and designed the experiments: JMJM JD. Performed the experiments: JD. Analyzed
the data: JMJM JD. Contributed reagents/materials/analysis tools: JMJM JD. Wrote the paper:
JMJM JD.
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