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Physical quantities & measurement

Measuring multiples and averages

Measuring time intervals Core 3 min read

Why measure multiples?

When an individual measurement is very small or difficult to time precisely, measuring a large number of identical events and then dividing gives a much more accurate result for a single event.

The technique of measuring multiples reduces the effect of random errors (including human reaction time) by averaging over many repetitions.
\[ \text{Value of one} = \frac{\text{Total measurement}}{n} \] where \( n \) is the number of repetitions.

Common applications

Measuring the period of a pendulum

This is the most common IGCSE exam example. The procedure was covered in the previous subtopic. The key idea is:

  • Measure the time for 20 (or more) complete oscillations.
  • \( T = \dfrac{\text{total time}}{20} \) — this averages out reaction time errors across 20 attempts.
  • Repeat the timing at least three times and find the mean, then divide by 20.

Measuring the diameter of a thin wire

A single turn of wire is too thin to measure reliably with a ruler. Winding many turns into a tight coil and measuring the total length of the coil is more precise.

Coil method for wire diameter
  1. Wind the wire tightly in closely packed turns around a pencil (so coils do not overlap).
  2. Count the number of turns \( n \) carefully.
  3. Measure the total length \( L \) of the coil with a ruler.
  4. Diameter of wire: \( d = \dfrac{L}{n} \)
L = total width of 5 turns → d = L / 5
Worked example — wire diameter

40 turns of a copper wire are wound tightly on a pencil. The total length of the coil is 24 mm. Calculate the diameter of the wire.

\( d = \dfrac{24}{40} = 0.60 \text{ mm} \)

Converting: 0.60 mm = 6.0 × 10−4 m

Measuring the thickness of a sheet of paper

Similarly, a single sheet is too thin to measure accurately with a ruler.

  1. Stack exactly 100 sheets of paper (count carefully).
  2. Measure the total thickness of the stack with a ruler.
  3. Thickness of one sheet = total ÷ 100

Taking the mean — reducing random error

Repeating any measurement multiple times and averaging removes random errors. A random error is one that varies unpredictably from one reading to the next (e.g. reaction time, parallax from a slightly different angle each time).

Mean (average) \[ \bar{x} = \frac{x_1 + x_2 + x_3 + \ldots + x_n}{n} \] Take at least three readings; discard any obvious outlier before averaging.
Taking more readings does not reduce a systematic error — one that is always in the same direction (e.g. a ruler that has its zero mark worn away so it reads 1 mm too high on every measurement). Systematic errors must be found and corrected by inspecting the instrument or method.
Worked example — mean value of reaction time

A student measures the time for 20 oscillations of a pendulum three times: 31.8 s, 32.2 s, 32.0 s. Find the period.

Mean time for 20 oscillations: \( \bar{t} = \dfrac{31.8 + 32.2 + 32.0}{3} = \dfrac{96.0}{3} = 32.0 \text{ s} \)

Period: \( T = \dfrac{32.0}{20} = 1.60 \text{ s} \)

Exam questions may ask "explain how your method reduces error." The expected answer is: "measuring a large number of [repetitions] reduces the effect of [reaction time / random error] by averaging."