Measurement Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Measurement.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Measurement is the process of assigning numerical values to attributes of objects or events according to a defined rule or scale.

To measure is to quantify—turning 'how much' or 'how many' into a number.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: All measurements have uncertainty; precision and accuracy are different things.

Common stuck point: Measured value \neq true value. There's always some error or approximation.

Sense of Study hint: Try measuring the same thing three times and compare results. If they vary, that variation is your measurement uncertainty.

Worked Examples

Example 1

easy
A ruler measures a pencil as 14.7 cm. The ruler has markings to the nearest 0.1 cm. Explain measurement error and determine the range of true values.

Solution

  1. 1
    The ruler reads 14.7 cm with precision to 0.1 cm
  2. 2
    Measurement error is at most half the precision unit: \pm 0.05 cm
  3. 3
    True value range: 14.7 - 0.05 \leq \text{true length} \leq 14.7 + 0.05
  4. 4
    So the true length is between 14.65 cm and 14.75 cm

Answer

True length is in [14.65, 14.75] cm due to \pm 0.05 cm measurement error.
All measurements have inherent error bounded by the instrument's precision. The reported value is the best estimate; the true value lies within half a precision unit on either side. Reporting appropriate significant figures communicates this uncertainty.

Example 2

medium
A scale consistently reads 0.5 kg too high (systematic error). A second scale gives variable readings due to vibration (random error). Explain the difference and how each affects measurements.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Five measurements of the same object yield: \{10.2, 10.3, 10.1, 10.4, 10.2\} grams. Calculate the mean and comment on the precision of the measurement process.

Example 2

hard
A thermometer reads the boiling point of water as 99.2°C at sea level (true value: 100°C). If 10 repeated readings give mean 99.2°C with SD=0.3°C, identify the type(s) of error and suggest how to address each.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

quantity