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: Measurement assigns a number to one attribute of an object using a chosen unit and scale.

Common stuck point: The procedure for measurement is the easy part; the trap is writing a number without a unit. Asking "Have I named the one attribute and the unit before writing the number?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Have I named the one attribute and the unit before writing the number?

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.

Answer

True length is in [14.65,14.75][14.65, 14.75] cm due to ±0.05\pm 0.05 cm measurement error.

First step

1
The ruler reads 14.7 cm with precision to 0.1 cm

Full solution

  1. 2
    Measurement error is at most half the precision unit: ±0.05\pm 0.05 cm
  2. 3
    True value range: 14.7−0.05≤true length≤14.7+0.0514.7 - 0.05 \leq \text{true length} \leq 14.7 + 0.05
  3. 4
    So the true length is between 14.65 cm and 14.75 cm
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.

Example 3

medium
Three students measure a desk: 75.1,75.0,75.275.1, 75.0, 75.2 cm. The true length is 80.080.0 cm. Describe the precision and the accuracy.

Example 4

medium
Four GPS distance readings for a hike are 4.81,4.79,4.82,4.804.81, 4.79, 4.82, 4.80 km. Estimate the best single value and explain.

Example 5

medium
A ruler reads to the nearest 0.50.5 cm. A pencil measures 13.513.5 cm. State the implied range of true lengths.

Example 6

hard
A balance has a +0.3+0.3 g systematic bias and ±0.1\pm 0.1 g random scatter. You take 25 readings of a true 20.020.0 g mass and average them. What value do you converge to, and what is the leftover bias?

Example 7

hard
A reaction time test is repeated 9 times by one student, giving SD =0.06= 0.06 s. What is the standard error of the mean of these 9 trials?

Example 8

challenge
Two independent labs measure the same metal bar. Lab A reports 12.50±0.1012.50 \pm 0.10 cm, Lab B reports 12.70±0.0512.70 \pm 0.05 cm. Is the difference between their readings consistent with measurement uncertainty alone?

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}\{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.

Example 3

easy
Convert 250250 centimeters to meters.

Example 4

easy
A ruler measures a pencil as 14.214.2 cm. Which attribute is being measured?

Example 5

easy
A scale reads 3.03.0 kg, 3.13.1 kg, 2.92.9 kg for the same object. The true mass is 5.05.0 kg. Is the scale precise or accurate?

Example 6

easy
To measure how heavy an object is, which attribute and unit are appropriate?

Example 7

easy
A measurement is recorded as 12.012.0 cm with the note 'plus or minus 0.10.1 cm.' What does the ±0.1\pm0.1 represent?

Example 8

easy
Convert 22 kilometers to meters.

Example 9

easy
Is a measured value ever perfectly exact?

Example 10

easy
To compare a length in inches with a length in centimeters, what must you do first?

Example 11

medium
Add 1.51.5 m and 3030 cm, giving the result in meters.

Example 12

medium
Three students measure a table's length: 120.1,120.0,119.9120.1, 120.0, 119.9 cm. The true length is 120.0120.0 cm. Describe the measurements' precision and accuracy.

Example 13

medium
A digital thermometer always reads 1.01.0 degree too high. Is this a precision problem or an accuracy problem?

Example 14

medium
A recipe needs 500500 g of flour, but the only scale reads in ounces (11 oz ≈28\approx28 g). About how many ounces are needed?

Example 15

medium
A measurement of 4.504.50 cm has more significant figures than 4.54.5 cm. What does the extra digit communicate?

Example 16

medium
A speedometer is checked against GPS five times, reading consistently 22 mph below the true speed each time. Is the issue random or systematic?

Example 17

medium
Why must units accompany a measured value like '5'?

Example 18

medium
A lab averages 10 repeated mass readings to report a single value. What measurement benefit does averaging provide?

Example 19

challenge
A balance has a systematic +0.2+0.2 g bias and random scatter of about ±0.1\pm0.1 g. If you average many readings of a true 10.010.0 g mass, what value do you approach, and why does averaging not fix it?

Example 20

challenge
A cube's side is measured as 2.02.0 cm with ±0.1\pm0.1 cm uncertainty. Roughly, is the relative uncertainty in the side 5%5\%, and why does volume amplify it?

Example 21

challenge
Two thermometers read a true 00 degrees C as: A gives −2,−2,−2-2,-2,-2; B gives −3,0,3-3,0,3. Which has better accuracy and which has better precision?

Example 22

medium
A track is measured as 400400 m with ±0.5\pm0.5 m uncertainty. Express the range of plausible true lengths.

Example 23

easy
Convert 4.54.5 meters to centimeters.

Example 24

easy
A pitcher holds 2.52.5 liters of water. How many milliliters is this?

Example 25

easy
A scale displays a child's mass as 25.025.0 kg with ±0.1\pm 0.1 kg uncertainty. Give the range of plausible true masses.

Example 26

easy
To compare a path that is 33 km with a path that is 25002500 m, which is longer?

Example 27

medium
A track meet timer is consistently 0.30.3 s slow because of a calibration drift. Is this random or systematic error, and does averaging many race times remove it?

Example 28

medium
A rectangle's sides are measured as 55 cm and 88 cm. Convert the area to square millimeters.

Example 29

medium
A jug has 2.02.0 L of water; you pour out 750750 mL. How much remains, in liters?

Example 30

medium
A road sign reads 3030 miles. About how many kilometers is this? (Use 11 mile ≈1.6\approx 1.6 km.)

Example 31

medium
A student adds 2020 cm and 0.150.15 m and reports 20.1520.15. What unit error did they make, and what is the correct sum in meters?

Example 32

medium
A bottle holds 750750 mL. How many bottles are needed to hold 66 L?

Example 33

hard
A square's side is measured as 4.04.0 cm with ±0.1\pm 0.1 cm uncertainty. Estimate the relative uncertainty in the area.

Example 34

hard
A protractor reads angles to the nearest 1°1°. You measure an angle as 48°48°. What is the range of true angles, and what is the relative uncertainty?

Example 35

hard
You measure a rectangle as 10.010.0 cm by 5.05.0 cm, each ±0.1\pm 0.1 cm. Estimate the absolute uncertainty in the area, using the rule that relative uncertainties add for a product.

Example 36

challenge
A cube's side is measured as 5.005.00 cm with ±0.05\pm 0.05 cm uncertainty. Estimate the relative uncertainty in volume.
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Background Knowledge

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

quantity