accurate measurement from inaccurate tools
Generally, improvement of measurement process has much better positive impact on results, than sophisticated mathematical processing of low quality data. But, sometimes we are forced to data what we have.

There are 3 quantities:

resolution = the smallest difference of directly measured value, distinguishable by the tool, 

precision = Level of uncertainty of repeated measurement of the same value, due random fluctuances. It is affected by the resolution as well, if comparable with precision.

accuracy = Level of agreement of the estimated and real value, due the tool/method bias.

There can be a very precise but inaccurate tool/method and vice versa,
even if one usually comes with the other.



Precision can be improved by measurement repetition. The arithmetic mean of $n^2$ measurements has $n$ times better precision than a single measurement.

The estimation of the standard deviation of normal Gauss distribution of measurements is 

$$s=\sqrt{\frac{\Sigma ( x - \overset{-}x )^2}{n-1}}$$

The estimation of the standard deviation of normal Gauss distribution of real value estimation by arithmetic mean is:

$$s_{\overset{-}x}=\sqrt{\frac{\Sigma ( x - \overset{-}x )^2}{n(n-1)}}$$

For a method consisting of multiple measurement steps, each with  their own error, the error propagation rule has to be followed.

If 2 quantities, subjected to random errors, are added or subtracted, the squares of their standard deviation estimations are additive

$$(s_{A\pm B})^2=(s_{A})^2+(s_{B})^2$$

If quantities are multiplied or divided, their relative standard deviation estimations are propagated:

$$\left(\frac{s_{A\cdot B}}{A\cdot B}\right)^2 = \left(\frac{s_{A}}{A}\right)^2 + \left(\frac{s_{B}}{B}\right)^2$$
$$\left(\frac{s_{A/B}}{A/ B}\right)^2 = \left(\frac{s_{A}}{A}\right)^2 + \left(\frac{s_{B}}{B}\right)^2$$

If we e.g. determine the difference of the standard solution volumes $A - B$,
where $A=20.00 \pm 0.03$, $B=10.00 \pm 0.04$, 

then $A-B=10.00 \pm 0.05$, 

as $\sqrt{0.03^2 + 0.04^2}=0.05$.



Accuracy is harder to improve by mathematical means. It can be improved by measurement bias analysis, by calibration of tools or methods and by combination of more tools / methods / labs.

There is an additional decrease of the measurement method precision due statistical error of calibration via rules above.

$$(s_\mathrm{rel,method})^2 =  (s_\mathrm{rel, meas})^2+(s_\mathrm{rel, cal})^2$$

More tools/methods/labs lead to the  "mean of the means" and partially decreases the possible bias,
similarly as averaging of measurements improve precision. 

But it is more costly.