The Forgetting Curve, Explained
After reading this you will be able to predict how fast a memory fades, calculate the retention gain from a well-timed review, and choose review dates that keep more material in your head for fewer repetitions.
What the forgetting curve is
Learn something today, test yourself in a week, and you remember only part of it. The forgetting curve describes that decline. Hermann Ebbinghaus measured it on himself in 1885 by memorizing lists of nonsense syllables and re-testing at fixed delays. His data showed a fast early drop followed by a long slow tail, the shape of an exponential decay.
Here is the hook. Suppose you study a fact and it starts at 100% retrievable. With a memory stability of about 2 days, one day later you can recall it with probability 61%, and after 4 days only 13%. Review it once at the right moment and that same memory might sit above 80% a week out. The visualizer draws this: each review snaps the curve back to 100% and stretches the next decay so it falls more slowly.
When to use this model, and when not
The single-curve exponential model is the right mental tool when you are studying discrete, testable items: vocabulary, dates, formulas, definitions, anatomy labels. For those, retrievability behaves like a decaying probability, and spacing reviews changes that probability in a measurable way.
It is the wrong tool for skills that improve through doing rather than recall: riding a bicycle, sight-reading music, conversational fluency. Motor and procedural skills decay far more slowly and follow different curves. It also breaks down for a single item over months, because real memory stability grows in jumps at each review rather than staying fixed.
Ebbinghaus used nonsense syllables precisely to strip out meaning. Material you understand decays more slowly than his syllables did, so treat the raw numbers here as a clean baseline, not a promise about your history exam.
The formula and why it decays
The core equation for retrievability R after a delay t is a single exponential.
Here R(t) is the probability you can recall the item, a number between 0 and 1. The delay t is time since the last successful review, measured in days. The stability S is the time constant: it is exactly the delay at which retrievability falls to e^{-1} \approx 0.368, that is 36.8%.
The intuition is that memory decays at a rate proportional to how much is left. When a lot remains, you lose a lot per day; when little remains, you lose little. That proportional loss is what produces an exponential rather than a straight line. Double the stability S and you double the time needed to reach any given retention level.
The spacing effect enters through S. Each successful review multiplies stability. A simple model uses a fixed factor: S_{new} = m \cdot S_{old} with m around 2. Review when retrievability is still high and you gain little, because the memory was not close to failing. Review when it has dropped near 30% to 40% and the same repetition buys far more added stability per unit effort. That is why expanding intervals win.
A worked example reproducing the demo
The demo starts a single item with stability S = 2 days and applies an expanding schedule with reviews on days 1, 3, 7 and 14. Each review resets R to 1.0 and multiplies S by 2. Work it out step by step.
Expanding reviews on days 1, 3, 7, 14
- Start: S = 2. On day 1, elapsed time is 1 day, so R = e^{-1/2} = 0.607. Review lands here.
- Review 1 resets R to 1.0 and sets S = 4. Next review is day 3, so elapsed is 2 days: R = e^{-2/4} = 0.607.
- Review 2 resets R and sets S = 8. Next review is day 7, elapsed 4 days: R = e^{-4/8} = 0.607.
- Review 3 resets R and sets S = 16. Next review is day 14, elapsed 7 days: R = e^{-7/16} = 0.646.
- Review 4 resets R and sets S = 32. On day 20, six days later: R = e^{-6/32} = 0.829.
Notice the pattern. Because the intervals roughly track the growing stability, each review catches the curve near 60%, the sweet spot, and the final decay is gentle. By day 20 you still hold 83% with only four reviews.
Compare that with cramming: four reviews all on day 1 push stability to 2 \times 2^4 = 32 immediately, but then nothing until day 20. On day 20 elapsed time is 19 days, so R = e^{-19/32} = 0.552. Same four reviews, but retention at the horizon is 55% instead of 83%.
Try moving the review time
The one thing a static chart cannot show is how the payoff of a review depends on when you place it. Move a single review earlier or later and watch the stability gain and the final retention change.
Reading and interpreting the results
The visualizer reports one summary number: average retention over the horizon, the area under the sawtooth curve divided by the horizon length. This matters more than the value at a single instant, because it captures how much you knew across the whole period, not just on test day.
For the expanding schedule above, averaging R across the 20 days gives roughly 0.80. For the crammed schedule, the curve sits near 1.0 for a day then decays for 19 days, and the average comes out near 0.60. The gap of 0.20 is the spacing effect stated as a single figure.
| Schedule | Review days | Final S | R at day 20 | Average R |
|---|---|---|---|---|
| None | - | 2 | 0.0000454 | 0.100 |
| Cramming | 1,1,1,1 | 32 | 0.552 | 0.600 |
| Expanding | 1,3,7,14 | 32 | 0.829 | 0.800 |
All three of the last two rows use the same four repetitions and reach the same final stability of 32. The difference in outcome comes entirely from timing.
Common mistakes
The most frequent error is reviewing too early. If you re-study while R is still 0.95, the memory was barely fading, so the added stability is small and you spent a repetition for almost nothing. Anki users who compulsively re-check "easy" cards fall into this trap.
The opposite error is reviewing too late. Once R drops below about 0.2, the item is nearly a fresh relearn, and you lose the efficiency of the spacing effect. A three-week vacation that skips reviews can push a whole deck into this zone at once.
Do not read the exact percentages as your personal recall. The stability S and the multiplier m vary by item, by person, and by how well you understood the material. Use the model to compare schedules, not to guarantee a score.
A third mistake is confusing the storage strength of a memory with its retrievability. A memory can be highly stable (hard to forget) yet momentarily hard to retrieve, or easy to retrieve now but fragile. The exponential model tracks retrievability; the multiplier on S is a coarse stand-in for storage strength.
Related tools
Once you accept that timing sets stability, the natural next question is what algorithm should pick the dates. The Spaced Repetition Simulator runs SM-2, FSRS and half-life regression side by side over a year so you can compare their curves and workload. If you already have a deck and want to know the daily cost, the Flashcard Workload Planner estimates reviews per day and how large the backlog grows after time off. For a different model of learning, where a tutor infers hidden mastery from right and wrong answers rather than from a decay curve, see the Knowledge Tracing Simulator.
Frequently asked questions
Is the forgetting curve really exponential?
For rote items it is close. Some studies fit a power law almost as well, and the two are hard to separate from noisy data. The exponential form is used here because it has a clean interpretation: constant stability S and proportional loss per day.
What stability value should I assume?
For fresh nonsense-like material, a first stability of 1 to 2 days is typical. Meaningful, well-understood material can start at a week or more. The visualizer defaults to S = 2 days so the decay is visible on a short horizon.
Why do the same four reviews give different retention?
Because stability only grows at a review, and the decay between reviews depends on how long that gap is. Bunching all reviews on day 1 wastes 19 idle days. Spreading them keeps R high across the whole window, raising the average from 0.60 to 0.80.
Does forgetting ever stop?
In this model R approaches 0 but never reaches it, since e^{-t/S} is always positive. In practice a memory reviewed enough times reaches a stability of years, at which point daily loss is negligible.
How is average retention computed?
It is the area under the curve over the horizon divided by the horizon length. For a plain exponential from day 0 to day T, that area equals S(1 - e^{-T/S}), which you can check by hand against the reported figure.
Related tools
Spaced Repetition Simulator
SM-2, FSRS, HLR, Leitner and Pimsleur side by side — workload, retention and knowledge per study hour.
Flashcard Workload Planner
How many daily reviews will your Anki deck really cost — vacation strategies, easy days, exam pacing, SM-2 vs FSRS.
Knowledge Tracing Simulator
Bayesian Knowledge Tracing live — each right / wrong answer shows the Bayes math in plain numbers.
Final Grade Calculator
What do you need on the final exam to reach your target grade? Instant answer for any weight.
GPA Calculator
Compute your GPA from course grades and credits — or ask what term GPA you need to hit a cumulative goal.
Math Worksheet Generator
Printable worksheets: arithmetic, fractions, algebra, order of ops, decimals, mixed numbers and percents.