I'm feeling than that all these calculators do have a little flaw: they do not take into account the tolerance.

I mean, as stated before, E12 value 12 can be achieved as a series of E3 values 10 + 2.2. But E12 values have a tolerance of 10 %, so we want to be inside 12 ± 10 % ⇒ between 10.8 and 13.2. In order for a series of two resistors of 10 and 2.2 to be inside such margin, they must have a tolerance of 5 %, otherwise there is a possibility of their series being outside such interval.

In doing these calculations I'm considering/assuming that:

- (assumption) The resistance of resistors is distributed normally (Gaussian) with mean equal to the nominal value of the resistor and standard deviation proportional to the product of nominal value times tolerance
- Thus, conductance of a given resistor, provided that the tolerance is small, is also distributed normally with mean the inverse of the resistor nominal value and equal tolerance than the resistance
- (assumption) The true resistance values of two resistors are uncorrelated
- Thus, the resistance of a series of two resistors distributes normally with mean the sum of both resistors nominal values and \(\sigma=\sqrt{\sigma_1^2 + \sigma_2^2}\)
- Thus, the series of two resistors of equal tolerance \(T_0\) has tolerance \(T=T_0{\sqrt{R_1^2+R_2^2}\over R_1+R_2}\)
- The «tolerance» in resistance of a parallel combination of two resistors of tolerance \(T_0\) has exactly the same expression as the series case

It is interesting to note that the tolerance of a series/parallel combination of two resistors has a

**tighter **tolerance than the original resistors. In spite of this, there are cases (the 12 ≈ 10 + 2.2 above) when we need resistors of tighter tolerance than that of the target E series. In this example case, if using 10 % resistors, the nominal value of the series is 12.2, and its tolerance 8.4 %, so can expect values between 11.26~13.22, outside by a little of the desired 10.8~13.2. If the resistors are 5 %, the final range would be 11.71~12.71 ⇒ OK.

Maybe I'm a bit picky here, but had a fun time this morning doing these calculations