A question that arises from time to time is:

Can we improve the tolerance of a load resistance by paralleling several identical resistors?

It seems plausible, lets following a thread of thinking.

## A statistical approach

Let’s assume that we manufacture a batch of 1 million resistors, and measuring them we find the measured values are normally distributed with mean \(\mu=1000\) and the standard deviation \(\sigma=10\) (1% of the mean).

If we draw a sample of n resistors randomly from the lot (the population), then the expected mean of the sample is normally distributed with mean=μ and \(s=\frac{\sigma}{\sqrt n}=\frac{10}{\sqrt 20}=2.24\).

So if we parallel 20 of these resistors, we have created a resistance of mean value 50Ω with a standard deviation of 0.111Ω (0.224% of the mean).

So, by paralleling resistors of those specifications, we did obtain a new value with reduced standard deviation (or standard error).

## But I have not mentioned tolerance

Resistors are normally supplied to a tolerance specification (eg ±1%) meaning that ALL resistors will fall within the tolerance limits.

The tolerance specification does not imply a population mean nor standard deviation.

In fact it is likely that the standard deviation is way less than the tolerance limits, and the mean might be anywhere such that no values fall outside the tolerance limits. It is likely that supplied product does not have a strictly normal distribution, though for very large n, the error in assuming normal distribution should be small.

Assumption that the population mean is equal to the nominal value of the resistors will usually be unjustified, and so the outcome of reducing tolerance by paralleling resistors is not necessarily achieved.