Uncertainty in a measurement: how sure is your reading?

Between two marks

Count the candies in a jar and you know the number exactly. Measure anything else and some doubt remains. Every instrument stops at its smallest division.

Suppose a line ends between the 10.3 cm and 10.4 cm marks of a millimetre rule. If the end does not touch or cross the midpoint, record the previous mark, 10.3 cm. If it touches or crosses it, record 10.4 cm.

Either way, the true end is at most half a division away. Half of the 0.1 cm least count is 0.05 cm, so the maximum uncertainty is ±0.05 cm: half above and half below.

Uncertainty in a tiny length or a short time shrinks if you take many readings and average them. That is why one pendulum swing is timed through thirty.

Maximum uncertainty = least count ÷ 2. For a millimetre rule: 0.1 cm ÷ 2 = ±0.05 cm. Only counting is exact.

Key terms

Uncertainty

How far the true value may be from the recorded reading.

Smallest division

The finest mark of an instrument; it limits the accuracy.

Midpoint rule

Below the midpoint: previous mark. At or past it: next mark.

Counting

The one kind of measurement with no uncertainty.

Short questions with model answers

1. Q1. A line ends past the midpoint between 10.3 cm and 10.4 cm. Record it with its uncertainty.

   • Past the midpoint, so extend to the next mark: 10.4 cm.
   • 0.1 cm ÷ 2 = 0.05 cm

   10.4 cm ± 0.05 cm.

2. Q2. Which has no uncertainty: the number of students in your class, or your height?

   • Students are counted in whole numbers.
   • Height is read off a scale with a smallest division.

   The number of students: counting is exact.

Common mistakes

• ✗ Writing the uncertainty of a millimetre rule as ±0.1 cm.

  ✓ The least count is split in two, half above and half below: ±0.05 cm.

• ✗ Always taking the next mark when the line ends between two marks.

  ✓ Only if the end touches or crosses the midpoint; otherwise keep the previous mark.

• ✗ Saying every number in science has some uncertainty.

  ✓ Counting is the exception: candies in a jar are known exactly.

MCQs

1. 1. The maximum uncertainty of a metre rule calibrated in millimetres is:

   1. (a) ±0.1 cm
   2. (b) ±1 cm
   3. (c) ±0.05 cm
   4. (d) ±0.5 cm
   Show answer

   (c) Half of 0.1 cm (p. 20).

2. 2. Which has no uncertainty?

   1. (a) Counting candies in a jar
   2. (b) Timing a race
   3. (c) Measuring a wire
   4. (d) Weighing a stone
   Show answer

   (a) Every measurement except counting has uncertainty.

3. 3. A line ends just past the midpoint between 10.3 cm and 10.4 cm. Record:

   1. (a) 10.3 cm
   2. (b) 10.35 cm
   3. (c) 10.0 cm
   4. (d) 10.4 cm
   Show answer

   (d) It crossed the midpoint, so extend to the next division.

4. 4. On a digital instrument, uncertainty shows up as:

   1. (a) A blank screen
   2. (b) A fluctuating last digit
   3. (c) A negative reading
   4. (d) A zero error
   Show answer

   (b) The side box on p. 20.

5. 5. To reduce the uncertainty in the time of one pendulum swing, you should:

   1. (a) Time one swing very carefully
   2. (b) Use a longer string
   3. (c) Time 30 swings and find the average
   4. (d) Round the time to whole seconds
   Show answer

   (c) Your book's own example on p. 20.

Quick revision

• Every measurement except counting has uncertainty, set by the smallest division.
• Below the midpoint keep the mark; at or past it move up. Uncertainty is half the least count.
• Many readings and their average shrink the uncertainty.