Errors in measurement and instruments

Four students use the same ruler to measure the width of the laboratory 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'.

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 callipers? In the figure you can see a pair of callipers 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 centimetres; a school ruler, which marks millimetres; a calliper, which marks tenths of a millimetre.

Using a ruler that marks millimetres, 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.

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?

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

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.

Iratxe 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?

A group measures the perimetre of the school playground with a tape measure in four attempts: $24.8\ \text{m}$, $24.9\ \text{m}$, $25.1\ \text{m}$ and $25.2\ \text{m}$. a) Calculate the mean of the measurements. b) What is the difference between the highest and lowest measurements? c) Do you think these measurements were taken carefully? Justify your answer.

Compare two ways of measuring the length of a table: measuring once with a ruler with millimetre 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?

In the laboratory, a measuring cylinder is read with your eye level with the surface of the liquid, never from above or below, as shown in the diagram.

1. a) What do you think happens to the reading if you look from above or below? b) Why does this error always recur in the same direction if you always look incorrectly?

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.