Exercises: Choose a Level of Accuracy Appropriate to Limitations on Measurement When Reporting Quantities
Work through each section in order. Show your work where indicated. Remember: the goal is to report the right number of digits, not the most digits.
Recall / Warm-Up
A scale always reads 0.5 kg higher than the true weight. Five measurements of a 10.0 kg object give: 10.5, 10.5, 10.5, 10.5, 10.5. Which statement correctly describes these measurements?
Precise and accurate — the readings are consistent and correct.
Precise but inaccurate — the readings are perfectly consistent (range = 0) but systematically 0.5 kg above the true value.
Accurate but imprecise — the readings are close to the true value on average.
Neither precise nor accurate — the readings are all different.
Five measurements of a 5.000 cm rod give the values: 5.12, 4.87, 5.23, 4.96, 5.08. The average is 5.05 cm. Which statement correctly classifies these measurements?
Precise and accurate — the average is close to the true value.
Precise but inaccurate — the readings cluster tightly but average above 5.000 cm.
Accurate but imprecise — the readings vary widely but average near the true value.
Neither — the readings are too spread out to classify.
Apply the significant figures counting rules. How many significant figures does each measurement have?
m has ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ significant figures. kg has ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ significant figures. L has ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ significant figures.
Fluency Practice
A scientist measures the same length three times and gets: 3.70 cm, 3.70 cm, 3.70 cm. A colleague measures the same length once and reports: 3.7 cm. Which measurement communicates greater precision, and why?
3.7 cm — fewer digits means simpler and cleaner.
3.70 cm — the trailing zero after the decimal indicates measurement to the nearest 0.01 cm (three significant figures vs. two).
Both are equally precise — they represent the same number.
3.7 cm — shorter numbers are always preferred in science.
A rectangle is measured as m wide (2 significant figures) and m long (3 significant figures). A calculator gives the area as m². To how many significant figures should you report the area?
A student rounds the following measured sum: g g. She applies the "round to the same number of significant figures as the least precise input" rule and reports g (1 sig fig). What error did she make, and what is the correct answer?
No error — 1,000 g (1 sig fig) is the correct answer.
She applied the multiplication rule to an addition problem. The correct rule for addition is to round to the same decimal place as the least precise addend. Since 1,000 g is precise to the ones place, the answer is 1,001 g.
She should have kept all digits: 1,001.23 g.
She should have converted both to the same number of decimal places first.
Apply the correct significant-figure rule for each calculation.
Multiplication: ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ m² (report to correct sig figs).
Addition: ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ cm (report to correct sig figs).
A ruler is marked in millimeters. A student measures a piece of wood and reports its length as mm. What is the problem with this report?
No problem — more decimal places always improve accuracy.
The reported precision (0.001 mm) exceeds the ruler resolution (about 0.5 mm by estimation). The wood cannot be known to the nearest 0.001 mm from a millimeter ruler.
The measurement should use centimeters instead of millimeters.
The problem is that the measurement is too long — the wood must be shorter.
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