Calculating marginal improvement in the 1% Rule
I got curious as to how the 37% improvement in James Clear’s book “Atomic Habits” was calculated. As such, I went about figuring out how and tried to generalize it to different time periods (rather than just a year) and with variable improvement and regression rates for each day.
Improvement at a fixed percentage
This calculation is made on the assumption that there is a compounding involved, that the improvement is consistent every day without gaps, and no regression happens during that time. With $\Delta p$ as the improvement in percent and $P(t)$ as the cumulative improvement over time, we arrive at the following:
$$\begin{aligned} \left( 1 + \Delta p \right)^t &= P(t) \ \left( 1 + 0.01 \right)^{365} &= 37.7834 \end{aligned}$$
Here is where we get the 37% cumulative improvement in the 1% Rule.
Improvement and Regression at a fixed percentage
Now, what if we have regression on some days? As such a much more general equation where regression can be expected is as follows:
$$P(t) = \left( 1 + \Delta p + \Delta r \right)^{t}$$
where
 $P(t)$ is still the cumulative improvement over time
 $\Delta p$ is the improvement increment for a day in percent: $\Delta p \in (0,1)$
 $\Delta r$ is the regression decrement for a day in percent: $\Delta r \in (1,0)$
 $t$ as the total period (in days) of which the cumulative improvement is being tracked
Improvement and Regression at a variable percentage
This equation, however, only applies to improvement and regression increments that are constant. Now what if I do not consistently improve at 1% but have days where I improve 2%, 3%, or 0.25%? Or days where things did not go well and I regress by 0.5%, 1%, or 5%?
To convert it to discrete variable improvement and regression percentages per day we have the following (which we can further take its limits to arrive at a product integral):
$$P(t) = \prod_{i=1}^{t} \left( 1 + \Delta p_i + \Delta r_i \right)$$
Calculating $P(t)$ with Python
Given some data (maybe a CSV file with date, $\Delta p$, and $\Delta r$ fields for each date in a given time period), we can code up a quick calculation for $P(t)$. For example in python
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