<p>Really great question.</p>
<p><strong>Source:</strong></p>
<p>Papers I have found featuring such graphs tend to credit <a href="https://en.wikipedia.org/wiki/SuperMemo" rel="nofollow noreferrer">SuperMemo</a>, an early computer program implementing spaced repetition for general memorization. Piotr Woźniak, lead developer of SuperMemo, <a href="https://www.supermemo.com/en/blog/did-ebbinghaus-invent-spaced-repetition" rel="nofollow noreferrer">explains</a> that this graph was based on intuition rather than evidence, and attributes its Internet popularity to a <a href="http://web.archive.org/web/20220506023029/https://www.wired.com/2008/04/ff-wozniak/" rel="nofollow noreferrer">2008 Wired Magazine</a> article:</p>
<p><a href="https://i.sstatic.net/266HTKyM.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/266HTKyM.jpg" alt="Original version of graph" /></a></p>
<p>Ironically, the article emphasizes how the public tends to ignore scientific evidence, relying on intuition instead. For what it's worth, the image was <a href="https://www.wired.com/2008/04/ff-wozniak/" rel="nofollow noreferrer">removed</a> from the article sometime in 2022.</p>
<p>Presumably, the graph became popular because it is intuitive and easy to understand. In fact, Woźniak was admittedly not the first to draw this graph, as <a href="https://doi.org/10.1111/j.1540-4781.1967.tb06700.x" rel="nofollow noreferrer">Pimsleur (1967)</a> already included this:</p>
<p><a href="https://i.sstatic.net/Wx5nRnqw.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/Wx5nRnqw.png" alt="Earlier version of graph" /></a></p>
<p>But Woźniak had not done any literature review and was not aware of this work at the time. I suspect that other authors featuring this type of graph have also not done any review. All other academic papers using it that I found so far are outside the field of psychology (mostly in computer science), and none suggest that the graph is evidence-based. That said, it doesn't necessarily mean that the graph is wrong.</p>
<p><strong>Evidence:</strong></p>
<p>The study of how rehearsal affects memory retention is called <a href="https://en.wikipedia.org/wiki/Distributed_practice" rel="nofollow noreferrer">distributed practice</a>, and this line of research has yielded at least one very robust finding: Spaced repetition substantially outperforms short-interval (massed) practice for retention, at the expense of acquisition. As reviewed by <a href="https://doi.org/10.1038%2Fnrn.2015.18" rel="nofollow noreferrer">Smolen, Zhang, &amp; Byrne (2016)</a>, this is due to a net gain in memory reinforcement, which they illustrate with these familiar-looking graphs, based on early research in the field:</p>
<p><a href="https://i.sstatic.net/3KQk90wl.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/3KQk90wl.png" alt="Evidence-based graph" /></a></p>
<p>The above graphs assume that the forgetting curve is unaffected by rehearsal. Whether or not this is actually the case however, seems to depend on the type of rehearsal (eg, <a href="https://doi.org/10.1146/annurev-psych-010419-051019" rel="nofollow noreferrer">McDermott, 2021</a>; <a href="https://doi.org/10.3758/BF03194050" rel="nofollow noreferrer">Pashler et al, 2007</a>; <a href="https://doi.org/10.1037/xlm0001110" rel="nofollow noreferrer">Radvansky et al, 2022</a>). For example, simple repetition or reminders, as in the Internet meme, do not affect the rate of forgetting, while <a href="https://en.wikipedia.org/wiki/Testing_effect" rel="nofollow noreferrer">retrieval and testing</a> do. <a href="https://doi.org/10.1073/pnas.2114377119" rel="nofollow noreferrer">Bäuml &amp; Trißl (2022)</a> used the following figure to summarize the results of their experiment, showing that in some cases the forgetting curve is flattened, while in others it is unaffected:</p>
<p><a href="https://i.sstatic.net/M2NESxpB.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/M2NESxpB.png" alt="Experimental evidence" /></a></p>
<p>Another issue is whether an expanding interval schedule, as depicted in the popular graph, outperforms a uniform interval schedule: This question remains heavily debated (eg, <a href="https://psycnet.apa.org/record/2007-02165-006" rel="nofollow noreferrer">Balota, Duchek, &amp; Logan, 2007</a>; <a href="https://doi.org/10.1007/s10648-020-09572-8" rel="nofollow noreferrer">Latimier, Peyre, &amp; Ramus, 2021</a>; <a href="https://psycnet.apa.org/record/2010-23868-002" rel="nofollow noreferrer">Roediger &amp; Karpicke, 2011</a>), and the answer may again be that ... it depends. Based on results of their experiment, <a href="https://doi.org/10.3758/s13423-014-0636-z" rel="nofollow noreferrer">Kang et al (2014)</a> argue that although these different schedules produce the same outcomes, expanding interval schedules are superior during the learning process, as depicted by this figure:</p>
<p><a href="https://i.sstatic.net/YjIehKQx.webp" rel="nofollow noreferrer"><img src="https://i.sstatic.net/YjIehKQx.webp" alt="Experimental evidence" /></a></p>
<p>There are a number of additional caveats to the superiority of spaced repetition that have to be outside of the scope of this already long answer.</p>
<p><strong>Formula:</strong></p>
<p>The <a href="https://en.wikipedia.org/wiki/Forgetting_curve" rel="nofollow noreferrer">forgetting curve</a> is affected by the schedule, material, student, sleep, interference, feedback, and other factors. Accordingly, the research literature features several <a href="https://en.wikipedia.org/wiki/Spacing_effect#Causes" rel="nofollow noreferrer">competing theories</a> regarding how the forgetting curve ought to be affected by these various factors. As <a href="https://doi.org/10.1037/xlm0001110" rel="nofollow noreferrer">Radvansky et al (2022)</a> put it:</p>
<blockquote>
<p>A major implication of Ebbinghaus’s (1885) original work is that
memory accessibility changes over time in a systematic way. However, a
great embarrassment of memory research is that we cannot
quantitatively predict these changes.</p>
</blockquote>
<p>Thus, I would not anticipate a simple model. For examples of complex models, see <a href="https://doi.org/10.1111/cogs.12602" rel="nofollow noreferrer">PPE</a>, <a href="https://www.researchgate.net/publication/221619316_Predicting_the_Optimal_Spacing_of_Study_A_Multiscale_Context_Model_of_Memory" rel="nofollow noreferrer">MCM</a>, <a href="https://doi.org/10.1073/pnas.1815156116" rel="nofollow noreferrer">MEMORIZE</a>, and <a href="https://en.wikipedia.org/wiki/Spaced_repetition#Algorithms" rel="nofollow noreferrer">Wikipedia</a>. My bet is that a machine learning algorithm would ultimately be needed to make accurate predictions on a per-student per-subject basis.</p>

CC BY-SA 4.0
2024-09-10T09:33:14.967Z
Funny that you would bring up this exact Pimsleur paper, I was about to search for it to answer. I always found it wild how Wozniak so brazenly claimed having invented this plot. Do you happen to know if Pimsleur was the first one to do the "shark fin"-style graph?
Kolja Sam
CC BY-SA 4.0
2024-09-10T13:51:19.903Z
Woźniak himself [posted a challenge](https://www.supermemo.com/en/blog/did-ebbinghaus-invent-spaced-repetition) to find earlier versions of the graph, and Pimsleur is the earliest found so far.
Arnon Weinberg
CC BY-SA 4.0
2024-09-13T16:56:22.890Z
I think it's also worth considering that commonly these sorts of "forgetting curves" are studied with quite artificial stimuli for experimental reasons. [Extinction](https://en.wikipedia.org/wiki/Extinction_(psychology)) as a behavioral paradigm often involves more ecologically relevant stimuli (risks of harm, rewards like food or drugs with abuse potential) and very different time courses for memory.
Bryan Krause
CC BY-SA 4.0
2024-09-10T03:43:28.960Z
2024-09-13T16:28:57.300Z
2024-09-13T16:28:57.300Z
Arnon Weinberg
<p>Forgetting curves are a well understood and well studied subject. The original research results from Ebbinghaus (the black line) have been replicated by <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4492928/" rel="nofollow noreferrer">third parities</a>.</p>
<p>However in many illustrations, people add curves for repetition count such as <a href="https://www.researchgate.net/publication/261952026_WHAT_TO_LEARN_NEXT_CONTENT_SELECTION_SUPPORT_IN_MOBILE_GAME-BASED_LEARNING" rel="nofollow noreferrer">this paper</a>. These lesser slopes of the red curves have been shared widely, and are cited as motivation for card scheduling algorithms.</p>
<p><img src="https://www.researchgate.net/profile/Florian-Schimanke/publication/261952026/figure/fig3/AS:667596047269891@1536178709388/Alteration-of-the-forgetting-curve-through-repetition-according-to-Ebbinghaus-1885-and.png" alt="Alteration of the forgetting curve through repetition according to Ebbinghaus (1885) and estimations from Paul (2007) taken from WHAT TO LEARN NEXT? CONTENT SELECTION SUPPORT IN MOBILE GAME-BASED LEARNING " /></p>
<p>I have failed to find any paper that has such a plot with actual experimental data. I also have not found a formula predicting the slope of the curve, depending on repetition spacing.</p>
<p>Can someone find a paper that contains data to support the shape of this plot? (Or alternatively, a paper that shows different curves)</p>
<p>Can someone find a paper/resource proposing an experimentally informed model for forgetting given a list of events and the timing of spaced repetitions?</p>
<p>I imagine an ODE model with jumps could be an appropriate formalism.</p>
<p>I am interested in models that allow computation and prediction. Not &quot;models&quot; that roughly explain the shape of the curve.</p>

CC BY-SA 4.0
2024-09-01T16:50:02.377Z
This question is similar but destinct from https://psychology.stackexchange.com/questions/15504/how-accurate-is-this-graph-about-learning-curve-and-spaced-repetition-system
worldsmithhelper
CC BY-SA 4.0
2024-09-01T17:17:31.043Z
Nice question, but before I look into it, can you check if [this](https://psychology.stackexchange.com/a/21309/7001) answers it?
Arnon Weinberg
CC BY-SA 4.0
2024-09-02T15:32:34.883Z
This answer makes the same claim "After every review, s increases" without backing this up with data or giving formula how "s" increases. I am also deeply dubious of these charts, as from my experience i sometimes can't recall a card, even if it is the same card i just checked during the first few reviews. Thus is suppose the jump to 100% right after review would not bear out in data (assuming that point was sampled at all).
worldsmithhelper
<learning><memory><reference-request><cognitive-modeling><long-term-memory>
Evidence of flattening forgetting curves with spaced repetition and formulas to predict them
worldsmithhelper
# Evidence of flattening forgetting curves with spaced repetition and formulas to predict them

(Asked by: username_0 on 2024-09-01)

Forgetting curves are a well understood and well studied subject. The original research results from Ebbinghaus (the black line) have been replicated by [third parities](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4492928/).

However in many illustrations, people add curves for repetition count such as [this paper](https://www.researchgate.net/publication/261952026_WHAT_TO_LEARN_NEXT_CONTENT_SELECTION_SUPPORT_IN_MOBILE_GAME-BASED_LEARNING). These lesser slopes of the red curves have been shared widely, and are cited as motivation for card scheduling algorithms.

![Alteration of the forgetting curve through repetition according to Ebbinghaus (1885) and estimations from Paul (2007) taken from WHAT TO LEARN NEXT? CONTENT SELECTION SUPPORT IN MOBILE GAME-BASED LEARNING ](https://www.researchgate.net/profile/Florian-Schimanke/publication/261952026/figure/fig3/AS:667596047269891@1536178709388/Alteration-of-the-forgetting-curve-through-repetition-according-to-Ebbinghaus-1885-and.png)

I have failed to find any paper that has such a plot with actual experimental data. I also have not found a formula predicting the slope of the curve, depending on repetition spacing.

Can someone find a paper that contains data to support the shape of this plot? (Or alternatively, a paper that shows different curves)

Can someone find a paper/resource proposing an experimentally informed model for forgetting given a list of events and the timing of spaced repetitions?

I imagine an ODE model with jumps could be an appropriate formalism.

I am interested in models that allow computation and prediction. Not "models" that roughly explain the shape of the curve.

**Tags:** learning, memory, reference-request, cognitive-modeling, long-term-memory

> **Comments:**
> **username_0**: This question is similar but destinct from https://psychology.stackexchange.com/questions/15504/how-accurate-is-this-graph-about-learning-curve-and-spaced-repetition-system  
> **username_1**: Nice question, but before I look into it, can you check if [this](https://psychology.stackexchange.com/a/21309/7001) answers it?  
> **username_0**: This answer makes the same claim "After every review, s increases" without backing this up with data or giving formula how "s" increases. I am also deeply dubious of these charts, as from my experience i sometimes can't recall a card, even if it is the same card i just checked during the first few reviews. Thus is suppose the jump to 100% right after review would not bear out in data (assuming that point was sampled at all).  

---

## 1 answers

### Answer by username_1 on 2024-09-10. Score: 4 | Accepted answer

Really great question.

**Source:**

Papers I have found featuring such graphs tend to credit [SuperMemo](https://en.wikipedia.org/wiki/SuperMemo), an early computer program implementing spaced repetition for general memorization. Piotr Woźniak, lead developer of SuperMemo, [explains](https://www.supermemo.com/en/blog/did-ebbinghaus-invent-spaced-repetition) that this graph was based on intuition rather than evidence, and attributes its Internet popularity to a [2008 Wired Magazine](http://web.archive.org/web/20220506023029/https://www.wired.com/2008/04/ff-wozniak/) article:

[![Original version of graph](https://i.sstatic.net/266HTKyM.jpg)](https://i.sstatic.net/266HTKyM.jpg)

Ironically, the article emphasizes how the public tends to ignore scientific evidence, relying on intuition instead. For what it's worth, the image was [removed](https://www.wired.com/2008/04/ff-wozniak/) from the article sometime in 2022.

Presumably, the graph became popular because it is intuitive and easy to understand. In fact, Woźniak was admittedly not the first to draw this graph, as [Pimsleur (1967)](https://doi.org/10.1111/j.1540-4781.1967.tb06700.x) already included this:

[![Earlier version of graph](https://i.sstatic.net/Wx5nRnqw.png)](https://i.sstatic.net/Wx5nRnqw.png)

But Woźniak had not done any literature review and was not aware of this work at the time. I suspect that other authors featuring this type of graph have also not done any review. All other academic papers using it that I found so far are outside the field of psychology (mostly in computer science), and none suggest that the graph is evidence-based. That said, it doesn't necessarily mean that the graph is wrong.

**Evidence:**

The study of how rehearsal affects memory retention is called [distributed practice](https://en.wikipedia.org/wiki/Distributed_practice), and this line of research has yielded at least one very robust finding: Spaced repetition substantially outperforms short-interval (massed) practice for retention, at the expense of acquisition. As reviewed by [Smolen, Zhang, & Byrne (2016)](https://doi.org/10.1038%2Fnrn.2015.18), this is due to a net gain in memory reinforcement, which they illustrate with these familiar-looking graphs, based on early research in the field:

[![Evidence-based graph](https://i.sstatic.net/3KQk90wl.png)](https://i.sstatic.net/3KQk90wl.png)

The above graphs assume that the forgetting curve is unaffected by rehearsal. Whether or not this is actually the case however, seems to depend on the type of rehearsal (eg, [McDermott, 2021](https://doi.org/10.1146/annurev-psych-010419-051019); [Pashler et al, 2007](https://doi.org/10.3758/BF03194050); [Radvansky et al, 2022](https://doi.org/10.1037/xlm0001110)). For example, simple repetition or reminders, as in the Internet meme, do not affect the rate of forgetting, while [retrieval and testing](https://en.wikipedia.org/wiki/Testing_effect) do. [Bäuml & Trißl (2022)](https://doi.org/10.1073/pnas.2114377119) used the following figure to summarize the results of their experiment, showing that in some cases the forgetting curve is flattened, while in others it is unaffected:

[![Experimental evidence](https://i.sstatic.net/M2NESxpB.png)](https://i.sstatic.net/M2NESxpB.png)

Another issue is whether an expanding interval schedule, as depicted in the popular graph, outperforms a uniform interval schedule: This question remains heavily debated (eg, [Balota, Duchek, & Logan, 2007](https://psycnet.apa.org/record/2007-02165-006); [Latimier, Peyre, & Ramus, 2021](https://doi.org/10.1007/s10648-020-09572-8); [Roediger & Karpicke, 2011](https://psycnet.apa.org/record/2010-23868-002)), and the answer may again be that ... it depends. Based on results of their experiment, [Kang et al (2014)](https://doi.org/10.3758/s13423-014-0636-z) argue that although these different schedules produce the same outcomes, expanding interval schedules are superior during the learning process, as depicted by this figure:

[![Experimental evidence](https://i.sstatic.net/YjIehKQx.webp)](https://i.sstatic.net/YjIehKQx.webp)

There are a number of additional caveats to the superiority of spaced repetition that have to be outside of the scope of this already long answer.

**Formula:**

The [forgetting curve](https://en.wikipedia.org/wiki/Forgetting_curve) is affected by the schedule, material, student, sleep, interference, feedback, and other factors. Accordingly, the research literature features several [competing theories](https://en.wikipedia.org/wiki/Spacing_effect#Causes) regarding how the forgetting curve ought to be affected by these various factors. As [Radvansky et al (2022)](https://doi.org/10.1037/xlm0001110) put it:

> A major implication of Ebbinghaus’s (1885) original work is that
> memory accessibility changes over time in a systematic way. However, a
> great embarrassment of memory research is that we cannot
> quantitatively predict these changes.

Thus, I would not anticipate a simple model. For examples of complex models, see [PPE](https://doi.org/10.1111/cogs.12602), [MCM](https://www.researchgate.net/publication/221619316_Predicting_the_Optimal_Spacing_of_Study_A_Multiscale_Context_Model_of_Memory), [MEMORIZE](https://doi.org/10.1073/pnas.1815156116), and [Wikipedia](https://en.wikipedia.org/wiki/Spaced_repetition#Algorithms). My bet is that a machine learning algorithm would ultimately be needed to make accurate predictions on a per-student per-subject basis.

> **Comments:**
> **username_2**: Funny that you would bring up this exact Pimsleur paper, I was about to search for it to answer. I always found it wild how Wozniak so brazenly claimed having invented this plot. Do you happen to know if Pimsleur was the first one to do the "shark fin"-style graph?  
> **username_1**: Woźniak himself [posted a challenge](https://www.supermemo.com/en/blog/did-ebbinghaus-invent-spaced-repetition) to find earlier versions of the graph, and Pimsleur is the earliest found so far.  
> **username_3**: I think it's also worth considering that commonly these sorts of "forgetting curves" are studied with quite artificial stimuli for experimental reasons. [Extinction](https://en.wikipedia.org/wiki/Extinction_(psychology)) as a behavioral paradigm often involves more ecologically relevant stimuli (risks of harm, rewards like food or drugs with abuse potential) and very different time courses for memory.  

---

