# How to calculate the average value of indirect measurements

Using a pendulum, I’ve calculated the value of the gravitational acceleration. In my experiments, I was using pendulums with three different lengths, therefore, I’ve calculated three different values which unsurprisingly are quite similar. Each result has a slightly different absolute error though, let’s say $$9.79\pm 0.3,\,\, 9.81\pm 0.5,\,\, 9.82\pm 0.2\frac{\mathrm{m}}{{\mathrm{s}^{2}}}$$ All the results belong to the same confidence interval, let’s say: $$90$$%. I’m just wondering how to figure out the final result? Should I just calculate $$\frac{9.79+9.81+9.82}{3}\pm \frac{0.3+0.5+0.2}{3}$$ or is there something more?

Typically you would use the equation for propagation of uncertainties (shown here), if your uncertainties are uncorrelated. In your case you calculate the average value $$\bar{g} = \frac{g_1 + g_2 + g_3}{3}$$ with the corresponding uncertainties $$s_1$$, $$s_2$$, and $$s_3$$. So the uncertainty in $$\bar{g}$$, is $$s = \sqrt{\left(\frac{\partial \bar{g}}{\partial g_1}s_1\right)^2 + \left(\frac{\partial \bar{g}}{\partial g_2}s_2\right)^2 + \left(\frac{\partial \bar{g}}{\partial g_3}s_3\right)^2 } = \sqrt{\left( \frac{1}{3}s_1 \right)^2 + \left( \frac{1}{3}s_2 \right)^2 + \left( \frac{1}{3}s_3 \right)^2}$$ or simply $$s = \frac{1}{3}\sqrt{s_1^2 + s_2^2 + s_3^2}$$