The Forgetting Curve

Abstract

The forgetting curve describes how retention of learned material declines with the time elapsed since learning. Hermann Ebbinghaus first charted it in 1885 by memorizing lists of nonsense syllables and measuring, through his savings method, how much of the original effort survived at delays from twenty minutes to a month. The curve falls steeply at first and then flattens, so that most loss occurs soon after learning and the remainder is comparatively durable. A century of work has refined the picture: forgetting is better described by a power or logarithmic function than by a simple exponential, averaging across people distorts the true form, and the rate of loss depends on how the material was learned and what happened afterward. Three interactive demonstrations model the savings method, the mathematical form of the curve, and how spaced review slows forgetting.

Keywords: forgetting curve, retention, savings method, spacing effect, memory decay

The forgetting curve is the empirical function relating the retention of learned material to the time that has passed since it was learned, first charted by Hermann Ebbinghaus in the 1885 monograph that founded the experimental study of memory (Ebbinghaus, 1913). Its defining shape is a steep early decline that decelerates: a large fraction of what is learned is lost within hours, after which the survivors are shed far more slowly. This regularity is among the most reproducible in psychology, recovered in a careful modern replication of Ebbinghaus's own procedure more than a century later (Murre & Dros, 2015). What the curve does not settle is its own mechanism or its exact mathematical form, and both questions have driven a long and productive research literature.

Ebbinghaus and the First Forgetting Curve

Hermann Ebbinghaus set out to bring memory into the laboratory at a time when it was thought to lie beyond experiment. To strip away the confounding effect of prior knowledge, he invented the *nonsense syllable*, a consonant-vowel-consonant trigram such as *ZOK* or *BIV* with no established meaning, and used lists of these syllables as material that carried no associations he could not control (Ebbinghaus, 1913). Serving as his own and only subject, he learned list after list to a fixed criterion, waited a measured interval, and then determined how much remained. Repeating this thousands of times under tightly held conditions, he produced the first quantitative description of forgetting over time. The resulting curve showed loss to be rapid and then slow: in his own data, savings fell from roughly 58 percent about twenty minutes after learning to about 44 percent after an hour and about 34 percent after a day, then declined only gradually to about 21 percent after a month, so much of a freshly learned list was gone within the first hour while what survived that initial collapse decayed at a diminishing rate over the following weeks. A single-subject replication using Ebbinghaus's methods reproduced his retention values closely, confirming that the curve was a real property of memory rather than an artifact of one man's habits (Murre & Dros, 2015).

The Savings Method

Ebbinghaus's central methodological insight was that memory can be measured even when nothing can be consciously recalled. Rather than ask how many syllables a subject could reproduce, he measured *savings*: the reduction in effort required to relearn a list to the original criterion (Ebbinghaus, 1913). If a list first took thirty trials to master and, after a delay, required only eighteen to bring back to criterion, then twelve trials, or forty percent, had been saved. Savings is a more sensitive index than recall or even recognition, because it detects traces too weak to support deliberate retrieval yet strong enough to speed the second learning. This sensitivity matters for the shape of the curve: measured by recall alone, forgetting looks nearly complete within a day, but measured by savings, a substantial residue persists, which is why the choice of retention measure partly determines the curve one observes (Loftus, 1985). The demonstration below computes savings from an original and a relearning score.

The Mathematical Form of Forgetting

Once forgetting could be measured, the question became what mathematical function it obeys, and the answer proved surprisingly hard to pin down. Ebbinghaus himself fit a logarithmic expression to his data, and later investigators proposed exponential, power, hyperbolic, and other functions (Rubin & Wenzel, 1996). The distinction matters theoretically. A simple *exponential* function, in which retention loses a constant proportion per unit time, would imply that a memory is equally likely to fail at any moment regardless of its age. The data instead show that older memories are more resistant, a deceleration captured better by a *power function* or a logarithmic one, in which the rate of forgetting itself declines over time. In a broad quantitative survey of more than two hundred retention data sets, no single two-parameter function fit every case, but power and logarithmic forms consistently outperformed the exponential (Rubin & Wenzel, 1996). Early single-trace models had already argued on independent grounds that forgetting follows a power-like rather than exponential course (Wickelgren, 1974). The generalization that older memories decay more slowly than young ones is old enough to have a name, Jost's law, and any adequate function must reproduce it.

Modeling Retention Over Time

The contrast between exponential and power forgetting is easiest to grasp by watching the two functions diverge. Both can be made to fall steeply at first, but they part company in the tail: the exponential continues to shed a fixed fraction and drives retention toward zero, while the power function bends and flattens, preserving a long-lived remnant. Because the two agree closely over short intervals and separate only at long delays, distinguishing them empirically demands data collected across a wide range of retention intervals rather than a few closely spaced ones (Wixted & Ebbesen, 1991). The demonstration below plots both functions on the same axes and lets their parameters vary, so the reader can see where they coincide and where the choice of model changes the predicted retention substantially.

Why Averaged Curves Mislead

A subtle statistical trap complicated the search for the true form of forgetting. Retention curves are usually built by averaging over many subjects or many items, and averaging can create a shape that no individual actually produced. If each person forgets exponentially but with a different rate, the average of their curves is not exponential but takes on the flattened, power-like appearance of the group (Anderson & Schooler, 1991). This means the widely observed power form might be an averaging artifact rather than a property of any single memory. Analyses of individual-subject forgetting functions, collected densely enough to fit each person separately, were therefore decisive: examined one learner at a time, forgetting still departs from the exponential and follows a power-like course, so the deceleration is genuine and not merely a consequence of pooling (Wixted & Ebbesen, 1997). Later hierarchical modeling that estimates the form while properly separating individual variation from the group concluded that the fate of an individual memory is best described by an exponential-power function, again confirming that the flattening is real (Averell & Heathcote, 2011). The episode is a standard caution in cognitive science: a curve fit to averaged data can answer a question about no one.

Decay Versus Interference

Describing the curve is not the same as explaining it, and two broad accounts compete over why forgetting happens at all. *Decay theory* holds that memory traces fade with the passage of time itself, an autonomous weakening that the curve then charts directly. *Interference theory* holds that time is largely a proxy: memories are lost not because the clock runs but because other learning, acquired before or after the target material, competes with it at retrieval (Underwood, 1957). Underwood showed that much of what looked like spontaneous decay in list-learning experiments was in fact proactive interference from the many earlier lists the same subjects had learned, so that a naive subject forgets far less over a day than the curve from a practiced one implies. The modern review of the psychology and neuroscience of forgetting treats the two mechanisms as complementary rather than exclusive, with genuine trace change and retrieval competition both contributing, and notes that the shape of the curve alone cannot arbitrate between them (Wixted, 2004). A prominent theoretical synthesis recasts the problem in terms of retrieval access, distinguishing the storage strength of a memory from its momentary retrieval strength and attributing much of forgetting to loss of the latter (Bjork & Bjork, 1992).

Slowing the Curve: Spacing and Testing

The forgetting curve is not fixed; several manipulations reliably flatten it, and they are the practical payoff of a century of study. The most robust is *distributed practice*: spreading repetitions across time rather than massing them produces far more durable retention, and a quantitative synthesis of the literature shows the benefit grows as the spacing between study sessions increases, up to an optimal gap that scales with the retention interval required (Cepeda, Pashler, Vul, Wixted, & Rohrer, 2006). Each review restores retention and, crucially, slows the subsequent decline, so a small number of well-timed repetitions can hold material near ceiling for far longer than massed study of the same total duration. A second lever is *retrieval practice*: testing memory, rather than restudying it, produces better long-term retention, an effect strong enough that a single test can outperform repeated reading over a week's delay (Roediger & Karpicke, 2006). A third is the *degree of learning*: material learned more thoroughly at the outset is forgotten more slowly, though the curves for weakly and strongly learned lists are more nearly parallel than one might expect once degree of learning is properly equated (Slamecka & McElree, 1983). The demonstration below shows how spaced reviews reset and flatten the curve across a fixed horizon.

The Permastore: What Survives

If the curve flattens, does anything last essentially forever? Bahrick's studies of very-long-term retention suggest that it can. Testing adults on Spanish they had learned in school decades earlier, he found that after an initial period of loss lasting a few years, retention stabilized into what he termed the *permastore*, a durable residue that changed little over the following decades regardless of whether the knowledge had been used (Bahrick, 1984). The permastore is the natural endpoint of a decelerating forgetting curve: material that survives the early, steep phase enters a regime in which further loss is negligible. What determines entry into this durable store is chiefly the original degree of learning and the distribution of practice, which is why the levers that flatten the curve in the short term also govern what remains after a lifetime. The finding connects the laboratory curve of Ebbinghaus to the retention of knowledge across the whole span of ordinary life.

Note. No single two-parameter function fits every data set, but functions whose rate of loss decelerates outperform the simple exponential (Rubin & Wenzel, 1996; Averell & Heathcote, 2011).

Worked Example

Consider the savings method applied to a single list. A learner masters a list of nonsense syllables, reaching the criterion of one perfect recitation after 15 study trials. Twenty-four hours later the learner attempts to recall the list and can produce none of it, so measured by recall, forgetting appears complete. The savings method asks a different question: how much effort does relearning now require? The learner relearns the list to the same criterion in 9 trials. The saving is the reduction in trials expressed as a proportion of the original: the difference between 15 and 9 is 6 trials, and 6 divided by 15 is 0.40, so 40 percent of the original learning was saved. Despite recall of zero, forty percent of the memory, by this measure, survived the delay (Ebbinghaus, 1913).

The example exposes why the choice of measure shapes the curve. A recall test would place this point at zero and imply total forgetting within a day; the savings measure places it at forty percent and reveals a substantial durable trace. The two measures disagree because recall requires a trace strong enough to support deliberate retrieval, whereas savings detects traces too weak for that but still able to speed relearning, which is the more sensitive index (Loftus, 1985). Had the same learner instead studied the list in three sessions spread across three days rather than in one massed session, the relearning score would be lower still and the savings higher, because distributed practice slows the subsequent decline (Cepeda, Pashler, Vul, Wixted, & Rohrer, 2006). The SavingsMethodDemo above computes this proportion for original and relearning scores the reader sets, and the arithmetic it reports is exactly the calculation carried out here.

Discussion

The forgetting curve endures as a landmark because it converted an intuition, that we lose what we learn, into a measurable function with a characteristic shape, and because that shape has survived more than a century of scrutiny nearly intact (Murre & Dros, 2015). Its practical importance is large. The finding that forgetting is rapid at first and then slow, and that spaced and tested learning flattens the decline, underwrites the design of study schedules, spaced-repetition software, and curricula that revisit material at expanding intervals (Cepeda, Pashler, Vul, Wixted, & Rohrer, 2006; Roediger & Karpicke, 2006). What the curve alone cannot do is reveal its own cause: the same descending shape is compatible with traces that fade and with traces that are crowded out by other learning, and disentangling decay from interference has required experimental designs that hold time constant while varying what else is learned (Underwood, 1957; Wixted, 2004).

The curve is also a methodological parable. Its exact form was obscured for decades by the practice of averaging over subjects, which can turn a population of exponential forgetters into an apparently power-law group and answer a question about no individual (Anderson & Schooler, 1991; Wixted & Ebbesen, 1997). The resolution, fitting the curve one learner at a time and modeling individual variation explicitly, is now a template for how to study any process whose form might differ across people (Averell & Heathcote, 2011). Between Ebbinghaus's nonsense syllables and Bahrick's fifty-year Spanish lies a single continuous function, steep then flat then nearly level, that connects a laboratory afternoon to the memory of a lifetime (Bahrick, 1984).

Glossary

Consolidation.

The process by which a newly formed memory becomes progressively more stable over time, one basis for the resistance of older memories to loss.

Decay theory.

The account that memory traces weaken and fade with the passage of time itself, independent of subsequent learning.

Degree of learning.

The thoroughness with which material was originally learned; more thoroughly learned material is generally forgotten more slowly.

Distributed practice.

Spreading repetitions across time rather than massing them, which produces more durable retention and flattens the forgetting curve.

Exponential function.

A decline that loses a constant proportion of what remains per unit time, implying an age-independent probability of loss; it fits short intervals but decays too fast in the tail.

Forgetting curve.

The empirical function relating retention of learned material to the time elapsed since learning, steep at first and then flattening.

Interference theory.

The account that forgetting results largely from competition among memories rather than from the passage of time.

Jost's law.

The generalization that of two memories of equal current strength, the older is forgotten more slowly.

Nonsense syllable.

A consonant-vowel-consonant trigram without established meaning, devised by Ebbinghaus to control for prior associations.

Permastore.

The durable residue of knowledge that survives the early phase of forgetting and persists with little further loss for decades.

Power function.

A decline whose rate of loss itself decreases over time, flattening in the tail and fitting individual forgetting data better than the exponential.

Proactive interference.

Disruption of new learning by material learned earlier, a major source of what can look like spontaneous decay.

Retrieval practice.

Strengthening memory by testing it rather than restudying it, which improves long-term retention; also called the testing effect.

Savings.

The reduction in trials or time needed to relearn material to its original criterion, a sensitive measure of retention that detects memory too weak for recall.

Spacing effect.

The finding that repetitions separated in time produce more durable memory than the same repetitions massed together.

Key Researchers

Hermann Ebbinghaus (1850-1909). Worked at the University of Berlin and later at Breslau and Halle; using nonsense syllables on himself, he charted the first forgetting curve and introduced the savings method and the spacing effect, founding the experimental study of memory. Wikipedia

John T. Wixted. Distinguished Professor of Psychology at the University of California, San Diego; showed that individual forgetting functions follow a power rather than exponential course and reviewed the psychology and neuroscience of forgetting. Faculty Page - Google Scholar - ORCID

Jaap M. J. Murre. Professor of Brain and Cognition at the University of Amsterdam; conducted the definitive modern replication of Ebbinghaus's original experiment, confirming the shape of the classic curve with contemporary analysis. Faculty Page - Google Scholar - ORCID

David C. Rubin. Professor of Psychology and Neuroscience at Duke University; with Wenzel synthesized a century of retention data across hundreds of data sets to characterize the mathematical form of forgetting. Faculty Page - Google Scholar - Wikipedia)

Harry P. Bahrick. Helen Whitelaw Jackson Professor of Psychology at Ohio Wesleyan University; documented the permastore, showing that knowledge such as school-learned Spanish survives largely intact for decades after the early phase of loss. Faculty Page

Harold Pashler. Distinguished Professor of Psychology at the University of California, San Diego; with Cepeda and colleagues produced the definitive meta-analysis of distributed practice, quantifying how spacing flattens the forgetting curve. Faculty Page - Google Scholar - ORCID - Wikipedia

Andrew Heathcote. Professor of Psychology at the University of Tasmania; with Averell used hierarchical modeling to separate the true form of the forgetting curve from the averaging artifact, arguing for an exponential-power fate of individual memories. Faculty Page - Google Scholar - ORCID

Frequently Asked Questions

What is the forgetting curve?
The forgetting curve is the empirical function relating how much of learned material is retained to the time that has passed since learning. It falls steeply soon after learning and then flattens, so most loss happens early and the remainder is comparatively durable (Ebbinghaus, 1913).

Who discovered the forgetting curve?
Hermann Ebbinghaus charted it in his 1885 monograph on memory. Serving as his own subject, he learned lists of nonsense syllables, waited measured intervals, and recorded how much survived, producing the first quantitative description of forgetting over time (Ebbinghaus, 1913).

What is the savings method?
The savings method measures retention by the reduction in effort needed to relearn material to its original criterion. It is more sensitive than recall because it detects traces too weak for deliberate retrieval yet strong enough to speed relearning (Ebbinghaus, 1913).

Is the forgetting curve exponential?
No. Although an exponential fits short intervals, forgetting decelerates more than an exponential predicts, so older memories are lost more slowly. Power and logarithmic functions describe the curve better across a wide range of delays (Rubin & Wenzel, 1996).

Why can averaging distort the forgetting curve?
If different people each forget exponentially but at different rates, the average of their curves takes on a flattened, power-like shape that no individual actually showed. Fitting the curve one learner at a time avoids this artifact (Wixted & Ebbesen, 1997).

Does forgetting happen because of decay or interference?
Both contribute. Decay theory attributes loss to traces fading over time, while interference theory attributes it to competition from other learning; the shape of the curve alone cannot decide between them, and current accounts treat them as complementary (Wixted, 2004).

How can forgetting be slowed?
Spacing repetitions across time, testing memory rather than restudying it, and learning material more thoroughly all flatten the curve. Distributed practice is the most robust of these, with benefits that grow as the gap between sessions increases (Cepeda, Pashler, Vul, Wixted, & Rohrer, 2006).

Does anything ever stop being forgotten?
Some knowledge does. After an early period of loss, well-learned material can enter a durable permastore that persists with little further decline for decades, as shown by retention of school-learned Spanish across a lifetime (Bahrick, 1984).

References