Coaching Centre
Measurement error and limits of accuracy
Precision of an instrument, absolute error, upper and lower limits of accuracy, percentage error, sources of error, and how errors grow when measurements are used in calculations.
No measurement is exact. A ruler can only be read to its smallest mark, a bathroom scale wobbles, and your eye might not be level with the line. Good measurers don't pretend the error isn't there. They say how big it could be.
Precision and absolute error
The precision (or limit of reading) of an instrument is the smallest unit marked on it. A school ruler marked in millimetres has a precision of $1$ mm. A kitchen jug marked every $50$ mL has a precision of $50$ mL.
When you read to the nearest mark, the true value could be up to half a mark either side and you would still have written down the same number. That half-mark is the absolute error.
A bolt is measured with a ruler marked in millimetres as $234$ mm. Find the absolute error and the limits of accuracy.
The way a number is written tells you its precision. "$4.0$ kg" was measured to the nearest $0.1$ kg, so its limits are $3.95$ kg to $4.05$ kg. "$4$ kg" only claims the nearest kilogram: $3.5$ kg to $4.5$ kg.
- With mm marks, type your reading of the blue bar to the nearest mark, then press Check. Read the limits of accuracy.
- Press Show true length. Is the real length inside your limits? Try five new objects. Is it always?
- Switch to cm marks with the same object. What happens to the absolute error and the percentage error?
Percentage error
An error of $0.5$ mm is tiny on a $2$ m plank but huge on a $2$ mm screw. To judge how good a measurement is, compare the error with the size of the measurement.
Mia weighs herself on digital scales that show one decimal place. The reading is $62.4$ kg. Find the limits of accuracy and the percentage error, correct to two decimal places.
You may also meet a percentage error worked out from a known true value: $\dfrac{|\text{measured} - \text{actual}|}{\text{actual}} \times 100\%$. Science pracs often use this one. If a question gives you the actual value, use it. If it only gives the instrument, use the absolute error.
Where errors come from
- Reading error: the scale can only be read to its smallest mark. This is what the absolute error measures.
- Parallax error: reading a scale from an angle instead of straight on. Measuring cylinders and car speedometers are classic examples.
- Calibration and zero error: the instrument itself is off, such as kitchen scales that show $20$ g with nothing on them, or a tape measure with a bent hook.
- Human and method error: starting the stopwatch late, or measuring a bent thing as if it were straight.
Taking several readings and averaging them reduces the effect of random slips, but it can't fix an instrument that is off in the same direction every time. Re-zero the scales instead.
Errors in calculations
When measurements are multiplied, their errors grow. To find the largest possible answer, use every upper limit. For the smallest, use every lower limit.
A rectangular garden bed is measured as $12$ m by $8$ m, each to the nearest metre. Find the smallest and largest possible area.
Your turn
A nurse draws up $2.5$ mL of medicine in a syringe marked every $0.1$ mL. Find the percentage error of the reading.
Show the working
Absolute error $= \tfrac12 \times 0.1 = 0.05$ mL. Percentage error $= \dfrac{0.05}{2.5} \times 100\% = 2\%$. For a child's dose, that 2% matters, which is why small doses use a smaller syringe.
A thermometer is marked every $0.2$°C. What is the absolute error of a reading, in °C?
Absolute error $= \tfrac12 \times 0.2 = 0.1$°C.
A length is measured as $15.6$ cm, to the nearest millimetre. What is the upper limit of accuracy, in cm?
Precision $1$ mm $= 0.1$ cm, so the absolute error is $0.05$ cm. Upper limit $= 15.6 + 0.05 = 15.65$ cm.
A measuring cylinder marked every $1$ mL reads $40$ mL. Find the percentage error.
Absolute error $= 0.5$ mL. $\dfrac{0.5}{40} \times 100\% = 1.25\%$.
Which measurement has the smallest percentage error?
The percentage errors are $\dfrac{0.05}{2} = 2.5\%$, $\dfrac{0.05}{20} = 0.25\%$, $\dfrac{0.5}{20} = 2.5\%$ and $\dfrac{0.5}{50} = 1\%$. The second is smallest.
Reading a measuring cylinder from above the liquid line instead of at eye level causes:
Reading a scale from an angle is parallax error. Get your eye level with the mark.
A parcel is weighed as $3.5$ kg on scales that go up in steps of $0.5$ kg. What is the lower limit of its mass, in kg?
Absolute error $= \tfrac12 \times 0.5 = 0.25$ kg. Lower limit $= 3.5 - 0.25 = 3.25$ kg.
Each side of a square is measured as $8$ m, to the nearest metre. What is the greatest possible perimeter, in metres?
The largest possible side is $8.5$ m, so the largest perimeter is $4 \times 8.5 = 34$ m.
A label says a bag of rice is "$4.0$ kg". Compared with "$4$ kg", this tells you:
Writing the zero shows the mass was measured to one decimal place, so the limits are $3.95$ to $4.05$ kg.
Exam style: A nurse measures $2.5$ mL of liquid in a syringe marked every $0.1$ mL. Calculate the percentage error.
Absolute error $= \tfrac12 \times 0.1 = 0.05$ mL. Percentage error $= \dfrac{0.05}{2.5} \times 100\% = 2\%$.
Exam style: A bedroom is measured as $4.2$ m by $3.6$ m, each to the nearest $0.1$ m. Find the greatest possible floor area, in m², correct to two decimal places.
Upper limits: $4.25$ m and $3.65$ m. Greatest area $= 4.25 \times 3.65 = 15.5125 \approx 15.51$ m².