How do I calculate a combined data error?

Question

I have a time series of data points read from an instrument, I also get an estimate of the error ($\sigma$) for each of these readings. e.g.

$10.12 \pm 0.05$

$10.09 \pm 0.02$

$10.20 \pm 0.12$ etc.

To reduce the noise inherent in the data I'm binning (averaging) them with respect to time intervals, all fine and simple even with my basic maths knowledge. But what is the correct way of combining the errors readings?

Answer 1

I found the answer myself in the end on Wikipedia under weighted arithmetic mean I've also used a weighted mean instead of a simple mean using the reciprocal of sigma squared as the weighting factor for each value.

Answer 2

I will show two calcultions.

Example: w = (4.52 ± 0.02) cm, x = ( 2.0 ± 0.2) cm, y = (3.0 ± 0.6) cm. Find z = x + y - w and its uncertainty.

z = x + y - w = 2.0 + 3.0 - 4.5 = 0.5 cm

METHOD 1

Delta z = Delta x + Delta y + Delta w = 0.2 + 0.6 + 0.02 = 0.82 rounding to 0.8 cm

So z = (0.5 ± 0.8) cm

METHOD 2

Solution with standard deviations

Delta z = SQRT((Delta x)^2 + (Delta y)^2 + (Delta w)^2)

Delta z = 0.633 cm

z = (0.5 ± 0.6) cm

Notice that we round the uncertainty to one significant figure and round the answer to match.

Now you can apply this to your measurements.