Learning Retention Rate Calculator

Estimate memory retention over time using the Ebbinghaus Forgetting Curve formula. Understand how much information is retained after initial learning and how spaced repetition improves long-term memory.

How many days since the material was first learned
Stability constant S — average is ~1.84 days for new material; increases with repetitions
Each repetition multiplies stability by ~2.5 (SuperMemo model)
Score immediately after learning session (baseline retention)

Formulas Used

Ebbinghaus Forgetting Curve:

R(t) = e−t / S_eff

Where:

  • R(t) = proportion of material retained at time t (0–1)
  • t = time elapsed since learning (days)
  • S_eff = effective memory stability (days)

Effective Stability with Spaced Repetition (SuperMemo SM-2 model):

S_eff = S₀ × 2.5n

  • S₀ = initial stability (~1.84 days for new material)
  • n = number of successful repetitions
  • 2.5 = ease factor (default in SM-2 algorithm)

Actual Retention Rate:

Retention (%) = R₀ × e−t / S_eff

Memory Half-Life:

t½ = S_eff × ln(2)

Next Optimal Review (at 90% retention threshold):

treview = S_eff × ln(1 / 0.9)

Assumptions & References

  • Based on Hermann Ebbinghaus's Über das Gedächtnis (1885) — the foundational forgetting curve research.
  • The ease factor of 2.5 per repetition follows the SuperMemo SM-2 algorithm (Wozniak, 1987), widely used in spaced repetition software like Anki.
  • Default initial stability of 1.84 days is derived from empirical spaced repetition datasets (Settles & Meeder, 2016 — Duolingo half-life regression study).
  • The model assumes successful recall at each repetition. Failed recalls reset or reduce stability.
  • Retention is bounded by the initial learning score — if you scored 80% immediately after learning, maximum possible retention is 80%.
  • The 90% threshold for next review timing is a common target in spaced repetition systems to balance efficiency and retention.
  • Individual differences in memory (sleep, stress, prior knowledge, material complexity) are not modeled but significantly affect real-world retention.
  • Reference: Ebbinghaus, H. (1885). Memory: A Contribution to Experimental Psychology. Columbia University.
  • Reference: Wozniak, P.A. (1990). Optimization of Learning. SuperMemo World.
  • Reference: Settles, B. & Meeder, B. (2016). A Trainable Spaced Repetition Model for Language Learning. ACL.

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