# Productivity and ability

What makes some people more productive then others?  Is it innate ability, better training, hard work?  Although the meaning of productivity is subjective,  there are quantifiable differences between researchers in measures of productivity such as the  h-index.    Here I will argue that a small difference in ability or efficiency can lead to great differences in output.

Let’s consider a simple and admittedly flawed model of productivity.  Suppose we consider productivity to be the number of tasks you can complete and let $P$ represent the probability that you can accomplish a  task (i.e. efficiency).  A task could be anything from completing an integral, to writing a program, to sticking an electrode into a cell, or to finishing a paper.  The probability of completing $N$ independent tasks is $T=P^N$.  Conversely, the number of steps that can be completed with probability $T$ is $N = \log T/\log P$.  Now let $P = 1-\epsilon$, where $\epsilon$ is the failure probability.  Hence, for high efficiency (i.e. low failure rate),  we can expand the logarithm for small $\epsilon$ and obtain $N \propto \epsilon^{-1}$.  The number of tasks you can complete for a given probability  is inversely proportional to your failure rate.

The rate of change in productivity with respect to efficiency increases even faster with

$\frac{dN}{d P}\propto \epsilon^{-2}$

Hence, small differences in efficiency can lead to large differences in the number of tasks that can be completed and the gain is more dramatic if you have higher efficiency.  For example, if you go from being $90\%$ efficient (i.e. $\epsilon = .1$) to $95\%$ efficient (i.e. $\epsilon = .05$) then you will double the number of tasks you can complete. Going from $98\%$ to $99\%$ is also a doubling in productivity.  The model clearly disregards the fact that tasks are often correlated and have different probabilities for success.  I know  some people who have great trouble in revising and resubmitting papers to get published and thus they end up having low measured productivity even though they have accomplished a lot.   However, it seems to indicate that it is always worth improving your efficiency even by a small amount.