# Measures & Significant Figures

## I. Introduction

The following discussion should be thoroughly understood by the student before going ahead with any experiment in the course. A through knowledge of significant figures and the determination of experimental uncertainty will be of great assistance both in reporting your measurements and in evaluating your results.

The sections below are laid out in the order you will need when first learning about this material. Proceed from one section to the next working the practice exercises along the way. Afterword you can browse the topics as needed to refresh yourself on any topic.

## II. Determining the uncertainty of an experimentally deduced result.

Many experiments will have a numerical result as the outcome. The process of measurement will always involve some uncertainty and therefore the result computed from these measurements will have some uncertainty. Our goal is to be able to estimate the size of this uncertainty and report it along with the experimental result. A variety of concepts and procedures are needed for this task. They are taken up in the sections below.

### A. Significant Figures

Rounding and doing math with significant figures.

### B. Accuracy vs. Precision and Error vs. Uncertainty.

Important definitions with examples.

How do you decide the number of significant figures when you make a measurement?

### D. Mean Values

Increased confidence through averaging.

### E. The Average Deviation of the Mean.

The Average Deviation provides an uncertainty with less guesswork.

### F. Relative Uncertainty

A good way to compare uncertainties.

### G. Precision of Computed Results

What to do when the final result is computed from several measurements, each with some uncertainty.

### H. Precision of Computed Results (Formal method using Calculus)

What to do when the final result is computed from several measurements, each with some uncertainty.

### III. Comments on Sources of Uncertainties

Experimental uncertainty due to random measurement effects are distinct from systematic errors introduced by defective equipment or procedures or caused by unknown influences on the equipment. Also we will treat the uncertainty differently if it arises from an independent source than when it is related to a source already considered.

1. Random error.
The error will be different with each repetition of the measurement.
2. Systematic error.
Repeating the measurement will reproduce the same error.
3. Independent sources of error.
Uncertainty is compounded.

Last Updated August 4, 2015