Error Analysis

Class Information

The first few lectures of the Intro to Modern Physics Lab section were dedicated to error analysis
Material covered is based on John Taylor’s “An Introduction to Error Analysis”

Measured value of $x$ = $x_{best} \pm \delta x$

Types of Errors
- Random: You can use statistical analysis (described in this section) on random uncertainties
  - Random uncertainty is decreased by taking repeated measurements
- Systematic: Can’t treat these statistically
  - Systematic uncertainty is not decreased by taking repeated measurements

Mean ($\overline{x}$): Gives average value of $x$
- Gives the best estimate for the true value of $x$
  $$ \overline{x} = \frac{1}{N} \sum_{i=1}^{N} x_i = \text{ mean }$$

Standard Deviation of a Single Measurement ($\sigma_x$): Measure of variance or dispersion - how spread out the data is from the mean
- Gives average uncertainty of measurements $x_1, x_2, …, x_N$
- We often estimate $\sigma_x$ since the true value is usually unknown because we often only get a small sample of the larger population
- If we only have a sample dataset, we introduce more uncertainty, so we estimate standard deviation with the sample Standard Deviation (SD) by introducing $N-1$ in the denominator
  $$ \sigma_x = \sqrt{\frac{1}{N-1}\sum (x_i - \overline{x})^2}$$
- If our dataset is the full population, our sample=population, so there is less uncertainty, and we can use $N$ in the denominator
  $$ \sigma_x = \sqrt{\frac{1}{N}\sum (x_i - \overline{x})^2}$$
- If we make one measurement of $x$, then
  $$\delta x = \sigma_x$$
- We can be $68%$ sure the single measurement of $x$ lies between $x - \sigma_x$ and $x + \sigma_x$
Variance $\sigma_x^2$

Standard Deviation of the Mean ($\sigma_{\overline{x}}$):
- Also known as Standard Error
The quantity $x_{best} = \overline{x}$ represents a combination of the $N$ measurements, so $\overline{x}$ should be more reliable than any single measurement taken alone. Thus, the uncertainty for the mean, $\overline{x}$, is smaller than the average uncertainty for a single measurement, $\sigma_x$.
The uncertainty of the mean is denoted by $\sigma_\overline{x}$
$$ \sigma_\overline{x} = \sigma_x/\sqrt{N} $$
$$ \text{true value of } x = x_{best} \pm \delta x $$
$x_{best} = \overline{x}$
$\delta x = \sigma_\overline{x}$
Since $\delta x = \sigma_x / \sqrt{N}$, notice how repeated measurements will lead to a decrease in the random uncertainty
Note that $\sigma_{\overline{x}}$ is usually only 1 sig fig. Use normal sig fig rules when calculating the mean and $\sigma_x$. Note that the right most significant digit’s place in the mean should match the right most significant digit’s place in the standard error.