Welcome! How Trustworthy Are Your Measurements?
You and a friend measure the same desk. You get 120.3 cm; your friend gets 119.8 cm. Who is right? Maybe neither of you is exactly right! In this chapter you will learn why every measurement has some error, what scientists mean by accuracy and precision, and how to take readings you can trust.
Note: apparatus names and how to heat, mix and transfer solutions are in the chapter "Laboratory equipment and basic practical skills". Here we focus on measurement quality.
Part 1: Choosing the Right Apparatus and Reading Its Scale
Every measuring instrument has a range (the smallest to the largest value it can measure) and a smallest division (the gap between two neighbouring marks, or the last digit on a digital display).
Five common measuring instruments
1. Stopwatch – measures time in seconds (s). A digital stopwatch usually shows readings to 0.01 s.
2. Metre rule – measures length up to 1 m (100 cm). Its smallest division is usually 1 mm (0.1 cm).
3. Measuring cylinder – measures the volume of a liquid in millilitres (mL). Remember that \(1\ \text{mL} = 1\ \text{cm}^3\).
4. Electronic balance – measures mass in grams (g). School balances often read to 0.1 g or 0.01 g.
5. Thermometer – measures temperature in degrees Celsius (°C). A common laboratory thermometer has 1 °C divisions.
Matching the instrument to the job
Ask two questions: Is the range big enough? and Is the scale fine enough?
For example, to measure 8 mL of water, use a 10 mL measuring cylinder, not a 100 mL one. Its finer divisions give a reading closer to the true value.
Reading tips for each instrument
Stopwatch: Reset it to zero first, then start and stop exactly when the event starts and ends.
Metre rule: Lay the rule flat with its scale touching the object. Line up one end with the zero mark and read the other end with your eye directly above it.
Measuring cylinder: Stand it on a flat bench. The surface of water curves, forming a meniscus. Bend down so your eye is level with the bottom of the meniscus, and read there.
Electronic balance: Place it on a flat, steady bench. Press the zero/tare button so it reads 0.0 g. With a container, put the empty container on first, press tare, then add the substance.
Thermometer: Keep the bulb fully in the liquid, not touching the container. Stir gently, wait until the reading is steady, then read at eye level.
Finding the volume of an irregular object (water displacement)
You cannot use a ruler on a stone or a key! Instead, use the displacement method: when an object sinks in water, it pushes aside (displaces) its own volume of water.
Step 1: Pour enough water into a measuring cylinder to cover the object. Record the reading, \(V_1\).
Step 2: Tie the object to a thread and lower it gently until it is fully under water. (Never drop it in – it may crack the cylinder.)
Step 3: Record the new reading, \(V_2\).
Step 4: Calculate: \(\text{volume of object} = V_2 - V_1\)
Example: The water level rises from 50.0 mL to 57.5 mL. Volume of the stone \(= 57.5 - 50.0 = 7.5\ \text{mL} = 7.5\ \text{cm}^3\).
This works for objects that sink and do not dissolve in or soak up water.
Key takeaway: Pick an instrument with a suitable range and a fine enough scale, and learn the correct way to read it.
Part 2: Error, Accuracy and Precision
What is error?
Every quantity has a true value (its real value). When we measure it, we get a measured value. The two are almost never exactly the same.
Error is the difference between the true value and the measured value:
\(\text{error} = \text{true value} - \text{measured value}\)
We usually care about the size of the error, so we can ignore any minus sign. For example, a book is truly 25.0 cm long and you measure 24.6 cm. The error is \(25.0 - 24.6 = 0.4\ \text{cm}\).
Accuracy: close to the true value
A measurement is accurate if its error is small, that is, the measured value is close to the true value. For the 25.0 cm book, a reading of 24.9 cm (error 0.1 cm) is more accurate than 24.6 cm (error 0.4 cm).
Precision: close to each other
Precision describes the spread between repeated measurements of the same thing. If repeated readings are close together, they are precise. You can only find out about precision by taking repeated measurements.
Example: Two groups time a toy car rolling down the same ramp three times.
Group A: 2.31 s, 2.33 s, 2.32 s – the spread is only 0.02 s, so the readings are precise.
Group B: 2.10 s, 2.60 s, 2.40 s – the spread is 0.50 s, so the readings are not precise.
The dartboard analogy
Think of the true value as the bullseye, and each reading as a dart.
Accurate and precise: the darts are close together, right in the bullseye.
Precise but not accurate: the darts are close together, but far from the bullseye.
Accurate but not precise: the darts are spread out around the bullseye, but their average position is near the centre.
Neither: the darts are spread out and away from the bullseye.
Memory aid: Accuracy = Aim (how close to the bullseye). Precision = Packed together (how close the darts are to each other).
A data example
A mixture of pure ice and water is at 0 °C, so the true value is 0 °C. Four students each measure its temperature three times:
Ann: 0.0 °C, 0.5 °C, 0.0 °C – accurate and precise.
Ben: 3.0 °C, 3.5 °C, 3.0 °C – precise but not accurate. (His thermometer may have a zero error.)
Cara: −2.0 °C, 2.5 °C, 0.0 °C – the average is about 0.2 °C, so accurate on average, but not precise.
Dan: 4.0 °C, 7.0 °C, 1.5 °C – neither accurate nor precise.
Key takeaway: Error is the difference between the true value and the measured value. A small error means accurate. A small spread between repeated readings means precise.
Part 3: Sources of Error
Three common sources of error are zero error, parallax error and reading error.
1. Zero error
A zero error happens when an instrument does not read zero when it should. Every reading is then pushed up (or down) by the same amount.
Balance not tared: A balance shows 0.3 g with nothing on it. You weigh an eraser and get 25.6 g. The true mass is \(25.6 - 0.3 = 25.3\ \text{g}\).
Ruler with a worn end: If the zero mark is worn away, start from a clear mark such as 1.0 cm, then subtract it. If the other end of the object is at 13.4 cm, its length is \(13.4 - 1.0 = 12.4\ \text{cm}\).
How to avoid it: Always check the zero before you measure. A zero error shifts every reading the same way, so repeating does not remove it – readings can be precise but not accurate!
2. Parallax error
Parallax error happens when your eye is not directly in line with the mark you are reading. The reading seems to shift as you move your head.
Try this: look at your finger with one eye closed, then the other. It seems to jump – that is parallax!
Measuring cylinder: If you look down from above, the reading appears too high. If you look up from below, it appears too low. Always read with your eye level with the bottom of the meniscus.
Ruler and thermometer: Keep your eye directly in front of (or above) the mark you are reading.
Digital displays (stopwatch, electronic balance) have no parallax error.
3. Reading error
Every scale has a smallest division, so you can only read it so finely. When the end of an object falls between two marks, you have to estimate. This is reading error.
For example, a pencil ends between 12.3 cm and 12.4 cm on a ruler with 1 mm divisions. If you read to the nearest mark, your reading can be off by up to about half a division.
A scale with finer divisions gives a smaller reading error.
Did you know?
For 130 years, the kilogram was defined by one metal cylinder kept near Paris. Compared with its copies, its mass seemed to drift by tiny amounts, so in 2019 scientists redefined the kilogram using a constant of nature. Even top measurement experts worry about error!
Key takeaway: Zero error shifts every reading by the same amount. Parallax error comes from reading with your eye in the wrong position. Reading error comes from the limit of the scale.
Part 4: Getting Reliable Data
Repeat and take the average
Repeat each measurement at least three times. This shows how precise your readings are, and the average (mean) is usually closer to the true value than one reading.
\(\text{average} = \dfrac{\text{sum of readings}}{\text{number of readings}}\)
Example: Times of 2.31 s, 2.35 s and 2.33 s give an average of \(\dfrac{2.31 + 2.35 + 2.33}{3} = 2.33\ \text{s}\).
Spotting outliers
An outlier is a reading that is far away from the others. It is often caused by a slip, such as pressing the stopwatch late.
Example: 2.31 s, 2.35 s, 2.33 s, 2.95 s. The last reading clearly does not fit. You should:
1. Circle it in your table – do not erase it.
2. Repeat the measurement if you can.
3. Leave it out when calculating the average, and say why in your report. Here the average is still 2.33 s.
Recording and reporting data honestly
Record every reading straight away in a table with clear headings and units, e.g. "Time (s)". Write readings to the same number of decimal places as the instrument shows.
Scientists must report data honestly. Never change, invent or delete results to make them "look right". If a result is unexpected, report it and suggest possible sources of error. Honest data lets others check and repeat your work (see the chapter "Learning about science").
Key takeaway: Repeat, average, look out for outliers, and always report what you actually measured.
Common Mistakes
Mistake 1: Thinking "error" means "mistake". In science, error is the difference between the true value and the measured value, and every measurement has some.
Mistake 2: "My readings are all the same, so they must be correct." Not true! Precise readings can still be far from the true value, like Ben's.
Mistake 3: Thinking that repeating and averaging will fix a zero error. It will not, because every reading is shifted by the same amount. Check the zero first!
Quick Review
1. Stopwatch = time, metre rule = length, measuring cylinder = volume of liquid, electronic balance = mass, thermometer = temperature.
2. Volume of an irregular object by water displacement \(= V_2 - V_1\).
3. Error = true value − measured value.
4. Accurate = small error (close to the true value).
5. Precise = small spread between repeated readings; find it by repeating.
6. Three sources of error: zero error, parallax error and reading error.
7. Repeat, average, spot outliers and report data honestly.
Great job! Now you can take measurements that you – and other scientists – can trust.