Errors in measurement · Ages 13–14
Planning a measurement
A tool to suit the size; too thin or quick, measure many and divide.
How to do it
Method
A tool to suit the size; too thin or quick, measure many and divide
100 sheets are 9 mm: one is 0.09 mm, its error 0.5 ÷ 100 = 0.005 mm
Another way
Measure again and take the mean
Three timings, 15.8, 16.2, 16.0 s: mean 16.0 s
In full
Before measuring, think of three things. First, the tool: long enough to measure in one go (not end over end), with a smallest unit that suits the size of the thing — a tape for a corridor, a ruler or calipers for a coin's diameter, kitchen scales to 1 g for a letter, not bathroom scales. To judge whether it will do, look at the percentage error: half the smallest unit ÷ the value × 100%. Second, ways to cut the error: anything too thin, too light or too quick, measure many together and divide. One sheet of paper is too thin, so measure 100 sheets and divide by 100: the ruler's 0.5 mm error is shared among the 100, only 0.005 mm each. Time 20 swings of a pendulum and divide by 20, and the error of pressing the stopwatch is shared out too. Measuring a few times and taking the mean cuts chance errors as well. Third, measure an awkward shape another way: put a stone in a measuring cylinder, and the rise in the water is the stone's volume. One job can have more than one sensible way, as long as you can say which has the smaller error.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
A pendulum is timed for 10 swings, taking 21.9 seconds in all. On average, how many seconds does one swing take?
Show the answer and working
Answer: 2.19 s
21.9 ÷ 10 = 2.19 seconds. Timing many and dividing shares the stopwatch's error among them.
Example 2
A measuring cylinder holds 54 mL of water; with a stone dropped in (right under the water), the level rises to 83 mL. What is the stone's volume, in cm³?
Show the answer and working
Answer: 29 cm³
An odd shape has no length, width and height to measure, so measure the water it pushes aside: 83 − 54 = 29 mL = 29 cm³.
Example 3
With a ruler correct to the nearest 1 mm, a stack of identical sheets is measured and divided by the number of sheets to find one sheet's thickness. Which way gives the most accurate thickness?
- A. Measure 25 sheets
- B. Measure 200 sheets
- C. Measure 5 sheets
- D. Measure 80 sheets
Choose the right answer
Show the answer and working
Answer: B. Measure 200 sheets
The stack's error is always 0.5 mm, so the more sheets, the smaller each one's share: 0.5 ÷ 200 = 0.0025 mm is the least.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.A stack of 50 identical cards is measured as 31 mm thick. How thick is one card, in mm?
Answer:
2.A pendulum is timed for 50 swings, taking 115 seconds in all. On average, how many seconds does one swing take?
Answer:
3.A measuring cylinder holds 36 mL of water; with a stone dropped in (right under the water), the level rises to 64 mL. What is the stone's volume, in cm³?
Answer:
4.A ruler correct to the nearest 1 mm measures 100 identical sheets together, and the result is divided by 100. The greatest absolute error of one sheet's thickness found this way is less than 0.005 mm.
True or false?
Answer:
5.50 identical coins are stacked, and a ruler correct to the nearest 1 mm measures the stack as 33 mm tall. One coin is mm thick, with a greatest absolute error of mm.
Fill in each blank
Answer:
All the answers
- 1. 0.62 mm
- 2. 2.3 s
- 3. 28 cm³
- 4. False
- 5. 0.66, 0.01
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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