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# Uncertainty in Physical Measurements

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.

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

1 - Backgammon 101

1 - Excel Version

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 (Python or Excel) for analysis of the data.

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

It also discusses significant figures in an experimental context.

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

4 - Repeated Measurements

4 - Excel Version

Review of Bell-shaped Curves

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.

5 - Data With Two Variables

5 - Excel Version

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

6 - Miscellanous Topics

6 - Excel Version

Miscellaneous topics that don't fit nicely into the previous Modules. The Sections are:

1. Confidence Intervals
2. Uncertainty in a Count
3. Systematic Effects and Calibration
4. Outliers and Robust Estimators
5. 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.