Errors in measurement and instruments

Four students use the same ruler to measure the width of the lab table and get $80.2\ \text{cm}$, $80.3\ \text{cm}$, $80.2\ \text{cm}$, and $80.4\ \text{cm}$. a) Did any of them measure incorrectly? b) Why are the four measurements not exactly equal? Explain what the sentence "every measurement has an error" means.

Complete the sentence with the missing words: 'No measurement is $\dots$: there is always a small difference between the $\dots$ value and the measured value. We call this difference $\dots$'.

An observatory measures the distance from the Earth to the Moon with a laser: $384\,400\ \text{km}$, with an error of about $4\ \text{m}$. a) Does this measurement have an error? Is it a bad measurement because of that? b) Compare this measurement with measuring your desk, $1.20\ \text{m}$ long, with an error of $1\ \text{cm}$: which measurement is better relative to the size of the object being measured? c) Explain in your own words the phrase 'all measurements have error, but not all measurements have equal value'.

A carpenter cuts boards for a shelf and a watchmaker adjusts tiny parts of a watch. a) What resolution does each person's measuring instrument need? b) What would happen if the carpenter measured with the watchmaker's resolution and the watchmaker with the carpenter's?

What is the smallest division marked on a standard school ruler (its resolution)? Could you reliably measure the thickness of a sheet of paper with it? Explain your reasoning.

To measure the diameter of a screw, which instrument has the finer resolution: the $30\ \text{cm}$ ruler in your pencil case or a pair of calipers? In the figure you can see a pair of calipers measuring a screw:

Explain what it means for an instrument to have a "finer resolution" than another, and whether that alone makes it more precise.

Order these instruments from lowest to highest precision for measuring lengths: a sewing tape measure, which marks centimeters; a school ruler, which marks millimeters; a caliper, which marks tenths of a millimeter.

Using a ruler that marks millimeters, Omar measures his pen: $14.3\ \text{cm}$. His classmate says that using the same ruler she measured $14.32\ \text{cm}$. a) Is that second measurement possible with that ruler? b) Explain the relationship between the resolution of an instrument and the digits that make sense to record in the result.

To weigh $2\ \text{g}$ of yeast, which scale would you choose: a bathroom scale, which reads in $100$ $100\ \text{g}$ steps, or a kitchen scale, which reads in $1$ $1\ \text{g}$ steps? Justify your choice by considering the resolution of each.

The pressure gauge at the gas station reads the tire pressure in steps of $0.1$ $0.1\ \text{bar}$, and on Lara's bicycle tire it reads $2.3\ \text{bar}$, as shown in the diagram.

1. a) Can you be certain that the pressure is exactly $2.3\ \text{bar}$? b) What is the smallest change in pressure that this device can detect?

Five stopwatches measure the winner's time in a swimming final and record: $26.4\ \text{s}$, $26.5\ \text{s}$, $26.4\ \text{s}$, $26.6\ \text{s}$ and $26.6\ \text{s}$. Calculate the mean of the five measurements.

Emma weighs an apple four times with the same balance and gets: $150\ \text{g}$, $152\ \text{g}$, $149\ \text{g}$ and $151\ \text{g}$. a) What is the mean value of the mass? b) Why is the mean taken as the "best value" for a measurement repeated several times?

Measuring a pencil, four students record: $8.2\ \text{cm}$, $8.3\ \text{cm}$, $8.4\ \text{cm}$ and $9.5\ \text{cm}$. The diagram shows the four measurements on a line:

1. a) One of the measurements stands out as unusual: which one is it and what might have happened? b) Calculate the mean of the four measurements, and then the mean without the anomalous measurement. Which of the two do you trust more?

Liam uses his phone's stopwatch to measure the time it takes for a coin to fall from a height of one meter and records: $0.45\ \text{s}$, $0.51\ \text{s}$ and $0.48\ \text{s}$. a) Why are the times so different if the coin falls in the exact same way each time? Think about who presses the button. b) What would you do to obtain more reliable data?

Compare two ways of measuring the length of a table: measuring once with a ruler with millimeter resolution, or measuring ten times with the same ruler and calculating the mean. a) Does repeating the measurements improve the instrument's resolution? b) So, what is improved by calculating the mean? c) In which cases is it worth repeating many measurements, and in which cases is it not?

You want to find the length of the school hallway with the highest possible reliability, and you only have a $30\ \text{cm}$ ruler, a $5\ \text{m}$ tape measure and a stopwatch. a) Which instrument do you choose and why? b) How many measurements would you make and what would you do with them? c) What sources of error will you need to watch out for during the measurement?

The nurse measures Mateo's blood pressure three times in a row and records (in cmHg): $11.2$, $10.8$ and $11.0$. a) Why is a single measurement not sufficient? b) Calculate the mean of the three readings. c) Mateo says: 'So my blood pressure is exactly $11.0$ cmHg'. Correct his sentence.

The wooden ruler in the workshop has worn down at the zero end and now starts directly at the $5\ \text{mm}$ mark, as shown in the diagram.

1. a) If you use it without noticing, aligning the worn end with the edge of the object, what will happen to all your measurements? b) Is this an error that changes with each measurement, or does it recur in the exact same way? Do you know what this type of error is called? c) How can you continue to measure accurately with this ruler without buying a new one?

Three groups measure the mass of the same stone. Group A: $45.2\ \text{g}$, $45.3\ \text{g}$ and $45.2\ \text{g}$. Group B: $44.1\ \text{g}$, $46.0\ \text{g}$ and $45.5\ \text{g}$. Group C: always $50.0\ \text{g}$, because their balance was incorrectly tared. The diagram shows the measurements of the three groups:

1. a) Calculate the mean for each group. b) Which group is most precise, that is, has the most closely clustered measurements? c) Given that the true value is $45\ \text{g}$, explain the difference between having precise (clustered) measurements and accurate (correct) measurements.

Two thermometers simultaneously measure the hot water in a pot: one reads $68\ \celsius$ and the other $72\ \celsius$, as shown in the diagram.

1. a) Can both thermometers be measuring accurately? b) Propose a way to find out which one is faulty without buying anything: think about the temperature at which ice melts. c) What type of error does the faulty thermometer have: a different error in each reading, or one that always recurs in the exact same way?