Statistical Analysis of Experimental Data

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

Mean (best estimate):

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

Standard deviation (spread of individual measurements):

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

Standard error of the mean (uncertainty in $\bar{x}$):

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

Key relationships

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

Outlier detection — A point more than $3s$ from the mean is suspicious. Use Chauvenet's criterion or the Grubbs test to decide whether to reject it.

Tip: Increasing $N$ from 4 to 16 halves the standard error. But going from 100 to 400 only halves it again — diminishing returns set in.

Common pitfall: The standard deviation describes the scatter of single readings; the standard error ($\sigma/\sqrt{N}$) describes the reliability of the mean. Quoting the wrong one misstates your precision by a factor of $\sqrt{N}$.