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”

How to Report and Use Uncertainties

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

  • $x_{best} = $ best estimate of $x$
  • $\delta x$ = uncertainty or error in the measurement
  • Fraction uncertainty = $\frac{\delta x}{\left|x_{best}\right|}$

Statistical Analysis of Random Uncertainties

  • 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

  • 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

  • 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.

Propagation of Uncertainties

Least-Squares Fitting