Ebbinghaus Forgetting Curve
See step-by-step calculation
In 1885, Hermann Ebbinghaus became the first scientist to measure memory decay experimentally. His finding is uncomfortable: without reinforcement, the brain loses roughly 50% of new information within 1 hour, 70% within 24 hours, and more than 90% within a week. The forgetting curve is exponential and predictable. The central formula is R = e^(−t/S), where R is retention (0 to 1), t is days elapsed since the last study session, and S is memory stability in days. The ratio t/S is the forgetting exponent: the larger it is, the lower the retention. At t/S = 0 retention is 100%; at t/S = 0.22 retention has fallen to 80% (the standard review trigger); at t/S = 1 retention is only 37%. This calculator computes that exponent directly. Enter how many days have passed since you last studied (t) and the current memory stability of that item (S). The result tells you how "loaded" the forgetting curve is and whether it's time to review. To get the exact retention percentage, apply R = e^(−result). Useful for students, language learners, certification candidates (AWS, CFA, PMP), and anyone who wants to know whether a memory is still intact or already in the danger zone.
In 1885, Hermann Ebbinghaus became the first scientist to measure memory decay experimentally. His finding is uncomfortable: without reinforcement, the brain loses roughly 50% of new information within 1 hour, 70% within 24 hours, and more than 90% within a week. The forgetting curve is exponential and predictable.
When to use this calculator
- A medical student reviewed a pharmacology fact three times, reaching memory stability S = 15 days. She wants to know if she can wait another 10 days. She enters t = 10, S = 15; the result is 0.67, implying R = e^(−0.67) ≈ 51%. She should review sooner than 10 days.
- An AWS Solutions Architect candidate studied a new service concept today for the first time (active flashcard review), establishing S = 2 days. She enters t = 1, S = 2; the result is 0.50, implying R ≈ 61%. Tomorrow morning is the ideal first review, not in 3 days.
- A language learner wants to know when vocabulary from today's lesson (S = 1, passive reading) will drop below 80% retention. She calculates: t = S × 0.22 = 1 × 0.22 ≈ 5 hours. She schedules a quick review before bedtime.
- A high school teacher checks how much students retain from a lesson taught 14 days ago without any review (S = 1). She enters t = 14, S = 1; the result is 14.0, implying R ≈ 0%. She plans to restart that unit from scratch.
Example: English vocabulary after 2 reviews
- You learned 30 new words 5 days ago. After two successful reviews, your memory stability S has reached 10 days. What is the current forgetting exponent?
- Exponent t/S = 5 ÷ 10 = 0.50. Estimated retention: R = e^(−0.50) ≈ 61%. You are already below the 80% threshold (reached at t/S = 0.22). Time to review.
How it works
2 min readHow It's Calculated
The Ebbinghaus Forgetting Curve models memory retention as an exponential decay:
R(t) = e^(−t / S)Where:
This calculator computes the exponent t/S. Once you have the exponent, apply R = e^(−exponent) to find the current retention.
What Is Memory Stability S?
S is the key parameter that grows with each successful review. Reference values:
| Situation | Estimated S |
|---|---|
| First passive exposure (reading) | 0.5 – 1 day |
| First active recall (flashcards, practice test) | 1.5 – 2 days |
| After 1st successful review | 3 – 5 days |
| After 2nd successful review | 7 – 12 days |
| After 3rd successful review | 15 – 25 days |
| After 4th successful review | 35 – 60 days |
| After 5th successful review | 90 – 150 days |
Each successful review multiplies S by approximately 2.5× (SuperMemo SM-2 algorithm). Anki uses the FSRS algorithm (2022) which adjusts that multiplier per item based on your actual recall responses.
Reference Table: Exponent → Retention
| Exponent t/S | Retention R | Status |
|---|---|---|
| 0.00 | 100% | Just studied |
| 0.11 | 90% | Excellent |
| 0.22 | 80% | Recommended review threshold |
| 0.36 | 70% | Review soon |
| 0.51 | 60% | Urgent review |
| 1.00 | 37% | Critical zone |
| 2.00 | 14% | Nearly forgotten |
| 3.00 | 5% | Almost the same as relearning from scratch |
How to Use the Result to Plan Reviews
To schedule a review at exactly X% retention, solve for t:
t_review = S × (−ln(X))
Example for 80% retention:
t_review = S × 0.2231
Example for 90% retention:
t_review = S × 0.1054This gives you the exact number of days from the last review until the next one.
Common Mistakes
1. Using S = 1 always: S grows with each review. After 3 reviews S can be 15–25 days; scheduling every 24 hours is unnecessary and counterproductive.
2. Passive review (re-reading) instead of active recall: Active recall (flashcards, questions, practice problems) produces a higher S than passive rereading.
3. Skipping a review without resetting S: If you go past double the scheduled interval without reviewing, the effective S has dropped. Treat that material nearly as new.
4. Cramming large batches in a single session: Working memory is limited (~7 items without chunking). Reviewing 200 items in one session is less effective than 4 sessions of 50 items each.
Frequently asked questions
What exactly is the Ebbinghaus Forgetting Curve?
It is a mathematical function describing how memory retention decays over time without reinforcement. Ebbinghaus formulated it as R = e^(−t/S), where R is retention (0 to 1), t is time elapsed, and S is the 'stability' of the memory. The key finding: without review, retention falls to 37% after just 1 day (when S = 1), and to under 5% after 3 days. Ebbinghaus documented this using nonsense syllables; meaningful content decays more slowly (higher initial S), but the exponential shape still holds.
What does this calculator actually compute?
It computes the forgetting exponent t ÷ S — the number that goes into the Ebbinghaus formula R = e^(−t/S). Once you have the exponent, the retention is R = e^(−exponent). If the result exceeds 0.22 (= −ln 0.80), retention has already dropped below 80% and a review is overdue. If it exceeds 1.0, retention is below 37% — the material is nearly forgotten.
How do I estimate my memory stability S?
For a first passive exposure (reading once), S ≈ 0.5–1 day. For a first active recall exposure (flashcard, practice test), S ≈ 1.5–2 days. After each successful review, multiply S by approximately 2.5: after the 1st review S ≈ 3–5 days; after the 2nd ≈ 7–12 days; after the 3rd ≈ 15–25 days. If you use Anki, you can see each card's current stability in the card info panel.
When should I schedule my first review after learning something?
For a first learning session with S = 1, the review at 80% retention should happen at t = S × 0.22 = 0.22 days ≈ 5–6 hours after initial study. At 24 hours, retention has already fallen to ~37%. In practice, the first review is best done the same evening or the next morning — never skipped for 3 or more days if the material is new.
What happens if I miss a scheduled review?
It depends on how much time passed. If you miss the Day 3 review and do it on Day 5, retention has fallen further but the review is still beneficial — just proceed from that point. If you went past double the scheduled interval (e.g., should have reviewed on Day 7 with S = 7, but reviewed on Day 20), the exponent t/S ≈ 2.86 implies under 6% retention. Treat the material as essentially new and reset S.
Is spaced repetition backed by science?
Yes, by a century of research. Ebbinghaus's original 1885 experiments have been replicated many times. Cepeda et al. (2006, Psychological Bulletin) meta-analyzed 254 studies confirming that spaced practice consistently outperforms massed practice. Piotr Wozniak's 2008 analysis of millions of SuperMemo reviews confirmed the exponential decay model in real-world conditions. Robert Bjork at UCLA documented the 'desirable difficulty' mechanism explaining why spaced retrieval outperforms rereading.
What retention threshold should I target for scheduling reviews?
Most spaced repetition systems (Anki, SuperMemo) default to an 80–90% retention target, meaning you review just before the memory decays to 80–90%. Targeting higher (95%) means more reviews but less forgetting per session; targeting lower (70%) means fewer reviews but stronger retrieval effort, which can actually reinforce learning — the 'desirable difficulty' effect (Bjork, UCLA). For high-stakes material (exams, safety procedures) stick to 85–90%.
How many reviews does it take to remember something for a year?
Using S × 2.5 per review, starting at S = 1: Review 1 → S = 2.5, Review 2 → S = 6.25, Review 3 → S = 15.6, Review 4 → S = 39, Review 5 → S = 97.5, Review 6 → S = 244, Review 7 → S = 610 days. About 7 well-spaced reviews are enough to maintain 80% retention for a full year. The total time investment is far less than re-reading or cramming.
Does sleep affect the forgetting curve?
Yes, significantly. During slow-wave and REM sleep the hippocampus transfers memories to the cortex, effectively increasing S. Walker et al. (UC Berkeley) found sleeping within 12 hours of learning can boost next-day retention by 20–40%. Practically: review briefly before sleep, then test yourself after waking. Skimping on sleep the night after a study session noticeably increases the forgetting rate — the equivalent of a lower S for the material learned that day.
Can I use the forgetting curve formula for physical skills, not just facts?
Yes, but procedural (motor) memories decay much more slowly than declarative (factual) ones. A well-practiced physical skill can have S in the hundreds or thousands of days. However, precision skills (surgical procedures, emergency protocols, instrument flying) do show measurable decay and benefit from periodic spaced practice. The FAA mandates instrument proficiency checks every 6 months for pilots (14 CFR § 61.57) — a real-world acknowledgment of procedural memory decay.
What apps implement spaced repetition automatically?
Anki is the most recommended free option (desktop/Android free; iOS paid). It uses the FSRS algorithm (2022), the most evidence-based implementation available. SuperMemo (Piotr Wozniak, 1987) is the original system. Quizlet offers spaced repetition with some limitations on the free tier. RemNote and Mochi Cards are solid alternatives. The advantage of this calculator is that it works without loading individual cards — useful for planning whole study blocks or material not easily converted to flashcard format.