# Uncertainty in Physical Measurements

- Page ID
- 5178

This page provides access to a series of documents on uncertainty in physical measurements intended for science undergraduates. They are intended to be worked on by a Team of 2 - 4 students, and most involve using physical apparati. Each Module assumes some knowledge of the content of previous ones. Modules 1 - 6 include a summary at the end. All documents are in pdf format and will appear in a separate tab/window.

For some Modules, there are two versions. The main version uses *Python* for data analysis, and the second version uses *Excel*. The content of the versions are the same except for the computing environment.

Name & Link | Description |
---|---|

0 - Introduction | An introduction intended for students, and another section introducing the Modules to Instructors. |

Statistics of rolling a pair of dice, including an Activity of repeating the roll of dice 36 times. The module introduces: - Histograms
- Probability distributions
- Triangular probability distributions
- The mean or average
- The deviation
- The variance
The modules also include links to videos using the computing environment ( | |

2 - Digital Instruments | Uncertainty associated with digital instruments, including an Activity of measuring the diameter of a coin with a digital caliper. The module introduces: - Rectangular or uniform probability distributions
- The standard deviation
- The uncertainty associated with a measurement
- Accuracy
- Quadrature
It also discusses significant figures in an experimental context. |

3 - Analog Instruments | Uncertainty associated with analog instruments, including an Activity of measuring the diameter of a coin with a ruler. The module also discusses systematic errors. |

Uncertainty associated with repeated measurements that do not give the same values. It includes an Activity of measuring the time it takes a piece of paper to fall to the floor, and another Activity of measuring the ratio of the circumference to the radius of a number of metal hoops. The module introduces: - Scattered or dispersed measurements
- Bell-shaped curves, also called Gaussian distributions or normal distributions
- Gaussian probability distributions.
- The uncertainty in the estimated standard deviation
- Propagation of uncertainties
- The uncertainty in the mean
A 10 minute PowerPoint review of bell-shaped curves that is useful after the students have read the Module but before they begin work on it. Primarily intended for instructors. | |

This Module is mostly concerned with fitting data to models, both by hand and by using least-squares techniques. It emphasises the visual display of data. The module introduces: - Independent and dependent variables of a dataset
- Residuals of a fit
- Sum of the squares of the residuals
- Method of least squares
- Degrees of freedom of a fit
- Chi-squared
- Effective variance method
| |

Miscellaneous topics that don't fit nicely into the previous Modules. The Sections are: - Confidence Intervals
- Uncertainty in a Count
- Systematic Effects and Calibration
- Outliers and Robust Estimators
- Accepted Values
The Module includes an Activity of calibrating a force sensor, and another Activity using robust measures of data to determine the period of a pendulum. | |

Summary | A summary of the Modules. |

Equipment and Software | A list of equipment and software needed for each of the modules. |