Errors in measurement

Skills and methods in this topic, easy to hard

1. 1Upper and lower limitsLimits = value ± half the unit

   Method12 cm (nearest 1 cm): 11.5 ≤ x < 12.5

   Another way: On a number lineNumbers rounding to 12: 11.5 up to 12.5

2. 2Absolute, relative and percentage errorsGreatest absolute error = unit ÷ 2; relative error = that ÷ value

   Method25 cm: 0.5 ÷ 25 = 0.02 = 2%

   Another way: From the upper limit25.5 − 25 = 0.5

3. 3Errors in measurement at workMost pieces = upper ÷ lower; area: lower × lower to upper × upper

   Method10 bags of 5 kg: 45 ≤ total < 55

   Another way: Check with the measured values10 × 5 = 50, between 45 and 55

4. 4Planning a measurementA tool to suit the size; too thin or quick, measure many and divide

   Method100 sheets are 9 mm: one is 0.09 mm, its error 0.5 ÷ 100 = 0.005 mm

   Another way: Measure again and take the meanThree timings, 15.8, 16.2, 16.0 s: mean 16.0 s

Worked examples

From different skills in this topic. Have a go first, then open each one for the answer and the working.

1. Example 1Upper and lower limits →

   A bag of flour is weighed as 22 kg to the nearest 0.5 kg. What is the upper limit of its weight, in kg?

   ▶Show the answer and working

   Answer: 22.25 kg

   The unit is 0.5 kg, half of it 0.25 kg: upper limit = 22 + 0.25 = 22.25 kg.

2. Example 2Errors in measurement at work →

   A rectangular sheet is 9 cm by 23 cm, each to the nearest 1 cm. Its area A lies in cm² ≤ A < cm².

   Fill in each blank

   ▶Show the answer and working

   Answer: 191.25, 223.25

   Least: the lower limits multiplied, 8.5 × 22.5 = 191.25; greatest: the upper limits, 9.5 × 23.5 = 223.25 (the true area is less).

3. Example 3Planning a measurement →

   A pendulum is timed for 50 swings, taking 73 seconds in all. On average, how many seconds does one swing take?

   ▶Show the answer and working

   Answer: 1.46 s

   73 ÷ 50 = 1.46 seconds. Timing many and dividing shares the stopwatch's error among them.