Statistical Analysis of Experimental Data

Mean and Spread

When you repeat a measurement $N$ times, statistics summarises the results.

The mean ($\bar{x}$) is your best estimate:

$$\bar{x} = \frac{1}{N}\sum_{i=1}^N x_i$$

The standard deviation ($s$) measures the spread of individual measurements:

$$s = \sqrt{\frac{1}{N-1}\sum_{i=1}^N (x_i - \bar{x})^2}$$

The standard error of the mean ($\sigma_{\bar{x}}$) measures the uncertainty in $\bar{x}$:

$$\sigma_{\bar{x}} = \frac{s}{\sqrt{N}}$$

Distributions and Pitfalls

For random errors, a Gaussian distribution forms a bell curve where $\approx 68\%$ of values lie within $\bar{x} \pm s$ and $\approx 95\%$ within $\bar{x} \pm 2s$.

An outlier more than $3s$ from the mean is suspicious. Use Chauvenet's criterion or the Grubbs test to decide whether to reject it.

Diminishing returns: Increasing $N$ from 4 to 16 halves the standard error. But going from 100 to 400 only halves it again.

Common pitfall: The standard deviation describes single-reading scatter, while the standard error describes mean reliability. Quoting the wrong one misstates precision by $\sqrt{N}$.