# Stabilization curve

**Stabilization curve** is the single most important curve in learning.

The **forgetting curve** has been known for a century even though it is not clear who was first to plot its graphical representation. **Forgetting curve** tells you how much you remember after learning. **Stabilization curve** tells you how much you gain in memory stability when you actively retrieve memories (e.g. in spaced repetition).

The formula for the **stabilization curve**:

**SInc**=*SInc**e_{Max}^{-Gain*R}

where:

*SInc*: stabilization (aka*stability increase*) is the ratio of memory stability after and before the repetition*SInc*: maximum possible increase in memory stability._{Max}*SInc*depends on current stability and memory complexity (i.e. item difficulty). Maximum stabilization can be achieved at the verge of forgetting_{Max}*Gain*: a constant that expresses the gain in stabilization with passing time (*gain*is a measure of the spacing effect)- R: retrievability, i.e. the expected probability of recall at a given point in time. It is determined by the forgetting curve

**Forgetting curve** describes passive memory. It speaks of the properties of memory when it is left unexercised to decay over time. However, the forgetting curve is not a call to action (unless you are a user of SuperMemo, who uses the forgetting index to optimize learning).

**Stabilization curve** describes the effect of review on memory. It tells you how much you can gain in learning, and how much you can potentially remember. It tells you how to optimally use the weapon of repetition.

Originally, SuperMemo was only interested in *optimum review* and stabilization at retrievability of around 90% (R=0.9). After 2002, SuperMemo used heuristic formulas to approximate the **stabilization curve** for advanced or delayed repetitions. Only recently, SuperMemo 17 made it possible to finally establish the exact formula. The newest spaced repetition algorithm is pretty accurate at estimating changes to stability over time. Those changes make it possible to plot the **stabilization curve**.

Notice that stabilization greater than 1 for retrievability of 1 is, in theory, implausible. The *gain* should exceed -ln(1/SIncMax). However, in SuperMemo, the value of *SIncMin* derived from the formula for the **stabilization curve** is often greater than 1.00. This can be a result of the imprecision of the approximation, boundary conditions imposed on SInc, noise in data, imprecision in dividing R in quantiles, etc. This can also be a result of the fact that no memory is perfectly retrievable, and the real curve kinks away from the approximated ideal at the very high, and at the very low levels of retrievability (as explained in the picture below).

See also:

This glossary entry is used to explain SuperMemo, a pioneer of spaced repetition software since 1987

Figure:Stabilization curvecomputed with SuperMemo. The horizonal line shows time expressed as memory retrievability. The vertical line shows stabilization, i.e. the increase in the durability of memory expressed as an increase in memory stability. Blue circles show how much stability increases with review that takes place at a given level of retrievability. The size of the blue circles depends on the number of data points collected. The graph has been produced using 31,721 repetitions in SuperMemo in data quantiles of difficulty=0.53 and stability=26 [days]. Stabilization increases from 1.36 at retrievability=100% (SIncMin=1.36) to 26.31 at R=0% (SIncMax=26.31). Optimum review at R=90% in SuperMemo produces stabilization of 1.86 (Stab90 is an equivalent of O-Factor in older SuperMemos). The Gain constant that expresses the spacing effect equals 2.96, i.e. relatively high as befits a low stability quantile. Stabilization in this dataset can be computed accurately with the formula (yellow line): 26*e^{-2.96*R}, which produces a tiny deviation of 0.5069. The consolidation curve is shown in purple, and indicates that the forgetting index is a plausible learning criterion for the presented dataset. For R approaching 100%, the actual stabilization for the presented graph is 0.879 as obtained with 2074 measurements. This is not clearly visible in the picture, but can be investigated with stabilization matrix exports in SuperMemo. This means that the stabilization curve formula may be inaccurate at the extremes of retrievability approaching 100% and 0%. This can be explained by the fact that no memory is perfectly retrievable or verifiably obliterated (see: We never forget)

Figure:SuperMemo is the first place in the world where you can see the spacing effect measured precisely in an actual learning process.Spacing effect gainis derived from the stabilization curve. SuperMemo can compute thespacing effect gainfor 400 stabilization curves, however, the graph shows only the data points with no less than 35 repetitions. The entire graph was based on 225,678 repetition cases. The horizontal X axis on the right represents item difficulty. The horizonal Y axis represents memory stability. Thespacing effect gainis displayed on the vertical axis with its maximum value of 3.8 (this value corresponds with a slope of exponential regression). This graph show thatgainincreases slightly with item difficulty and drops significantly with memory stability. The lower value at low stabilities may be an algorithmic artifact caused by difficulty with estimating startup stability for new items in the learning process