# Uncertainty in the average of two measurements (with their respective uncertainty)

I know that generally the uncertainty in the mean of a sample should be equal to:

$\frac{V_{max} - V_{min}}{2}$

where $V_{max}$ is the maximum value and $V_{min}$ the minimum value of the sample of data. However, what if each value has its own uncertainty? For example, I have to values:

$R1 = 12.8 \pm 0.2$ m

$R2 = 13.6 \pm 0.4$ m

The mean would be $13.2$ m, but what about the uncertainty? Will be it be the range $1.4/2$ or will it be the combined uncertainty of each measurment?

If you have two uncorrelated quantities $x$ and $y$ with uncertainties $\delta x$ and $\delta y$, then their sum $z=x+y$ has uncertainty

$$\delta z = \sqrt{(\delta x)^2 + (\delta y)^2}$$

The average would then have uncertainty $$\frac{\delta z}{2} = \frac{\sqrt{(\delta x)^2 + (\delta y)^2}}{2}$$

Intuitively, one might imagine that

$$\delta z = \delta x + \delta y$$

However, this overestimates the uncertainty in $z$. If $x$ and $y$ are uncorrelated, then it is very unlikely that their errors would constructively add in this way. It is of course possible that $x$ and $y$ are correlated, but then more complicated analysis is required.

• Could you provide a reason (or a reference to a reputable source) to why that is the case? – daljit97 Mar 13 '18 at 20:29
• The reason is that measured quantities are typically assumed to correspond to normally distributed random variables, and the uncertainty is the standard deviation. Adding two such random variables results in a random variable with standard deviation given by the above formula. This can be found in essentially any reference on experimental techniques, such as this one. – J. Murray Mar 13 '18 at 21:05