# Lab 01: Measurement, uncertainty, precision, and error

## Speaking on a soap box

### Equipment

• Boxes of hand soap
• Meter sticks
• Mass balances

### Introduction

[Class discussion of soap box volume and mass. How to count objects stacked in a cuboid.]

Fundamental units: Measurements of length, mass, and time using either the meter-kilogram-second or centimeter-gram-second system. Fundamental units measure space, time, and the amount of matter contained in an object. In this laboratory measurements will be made using centimeters and grams. The first four laboratories will use the "cgs" system of measurement.

Derived units: Measurements expressed using arithmetic combinations of standard units. For example, volume is derived from multiplying together three independent measurements of length. To avoid confusion these three measures are usually described as length × width × height. Density is a derived from the mass divided by the volume.

Another example of a derived unit is the metric measure of liquid volumes, the liter. A liter is defined to be 1000 cm³. The liter is defined using a fundamental unit of length.

Primary measurements: A quantity that is directly measured using an measuring device or instrument. Measuring devices might include rulers, stopwatches, mass balances, protractors, and thermometers.

Calculated measurement: A quantity calculated from a mathematical combination of primary measurements.

Uncertainty: the limit of our measuring tools for a single measurement, our uncertainty based on the smallest measurement our tools can accurately make. For a ruler marked in millimeters there is always at least a half a millimeter of uncertainty, often more.

Precision: the average variation in multiple measurements of an experiment usually expressed using the standard deviation.

Error: the difference between the measured result and the actual value. The actual value is usually unknown, hence the error is never truly known.

Density is the mass divided by the volume. In this laboratory the units of mass are grams, the units of volume are cubic centimeters, and the units of density are grams per cubic centimeter.

Both volume and density are "derived" quantities. Length and mass are fundamental quantities.

### Question

Given that there is uncertainty in any measurement, how can we report derived calculations such as volume or surface area in a way that expresses both the value of the calculation and the level of precision for that value?

### Introduction

Multiple measurements provides a series of checks on each other in our work. The tools of statistics can then be used to best report the volume and mass. The mean will capture the best estimate of the volume and surface area. The standard deviation will capture the best estimate of the level of precision in our estimate.

### Procedure

Measurements of the volume and mass will be conducted by the class using rulers and boxes of soap. Each student will measure their own soap box, calculate the volume and density, and then report their result to tables on the board.

### Data tables [d] [t]

#### Individual work

Box of soap

Use a ruler to measure the length, width, and height for a box. Determine the volume. Use a mass balance to determine the mass in grams.

BrandLength (cm)Width (cm)Height (cm) Volume (cm³)Mass (g)Density (g/cm³)

#### Class data [on the board]

Do not attempt to determine the correct significant digits at this point. Simply write your answers on the board as you have calculated them. The standard deviation will later tell us what digits are significant.

If multiple brands of soap are in use in the laboratory, use only one brand in your table. Use the brand of soap for which you performed measurements

Student Mass (g)Volume (cm3)Density (g /cm3)
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
Mean (average) x
Standard deviation sx

### Data Analysis and Results [a]

[Class presentation on calculator usage with handout]

The standard deviation provides the level of precision. The largest place value in the standard deviation is the smallest place value in the mean that can be kept. Another option that will used in many laboratories will be to count the non-zero digits in the original measurements and keep only that many non-zero digits in the derived calculations.

The data analysis for this laboratory is the calculation of the mean and standard deviation. These values are also sometimes written as the mean ± standard deviation.

### Data Display [none]

[No graphs required in this laboratory]

### Conclusions [c] [GVOC]

Discuss how the lab teams provided checks and balances on each other's data. Discuss any problems your team faced in the laboratory.

### Laboratory Report

Use a spreadsheet and/or word processing package to type up a report using the sections seen above. Include your tables. The class will spend the final portion of the period in A204 to learn how to do this. Lab reports are due a week later at the next laboratory period. The lab report can either be a hard copy from a printer or emailed to dleeling@comfsm.fm Put your name on the report!
[Cover save as... in A204]
[Cover using comfsm.fm or other external email account if lab report emailed]