Choose a Level of Accuracy

Lesson 1 of 2: Accuracy, Precision, and Significant Figures

In this lesson:

• Distinguish accuracy from precision — two independent qualities
• Use significant figures to communicate how precise a measurement is

Deck 1 Learning Objectives Today

By the end of this lesson:

1. Distinguish accuracy and precision, and explain how each can fail independently
2. Count significant figures in a reported measurement and explain what the count signals
3. Apply significant-figure rules correctly when reporting calculation results

How Many of These Digits Are Real?

A student measures a sample and computes:

Density = 8.937246 g/cm³

The calculator produced this number exactly.

But how precisely were the mass and volume measured?

If the mass was measured to the nearest gram — none of those decimal digits are real.

Accuracy and Precision Can Fail Independently

Precision Without Accuracy: Systematic Error

Thermometer always reads 1°C too high:

Precise (consistent offset) but not accurate (always wrong).

Systematic error causes inaccuracy, not imprecision.

Accuracy Without Precision: Random Error

Example: A scale with ±0.5 kg random variation.

Five weighings of a 10.0 kg object: 9.7, 10.3, 9.8, 10.2, 9.9

Average ≈ 10.0 kg ✓ — accurate on average

Range: 9.7 to 10.3 — imprecise (0.6 kg spread)

Random variation causes imprecision but not systematic inaccuracy.

Which Failure Mode Matters More?

The context determines which failure mode is more dangerous.

Misconception: Precise Always Means Accurate

Five readings of a 10.000 cm rod: 10.312, 10.315, 10.314, 10.313, 10.315

Range: 0.003 cm — extremely precise. Average: ~10.314 cm — off by 0.314 cm.

Precise but not accurate. The 3% systematic error is hidden by the consistent readings.

Precision is visible. Accuracy requires knowing the true value.

From Measuring to Reporting Precision

Measurements have two quality dimensions: accuracy and precision.

Once you have a measurement, how do you communicate its precision?

Answer: significant figures — the digits that carry real precision information.

The number of digits you write is a claim about how carefully you measured.

Same Length, Three Different Measurements

Trailing zero in 3.70 is data — dropping it loses precision.

Significant Figures: A Promise to the Reader

Significant figures are the digits that carry real precision information.

They signal how carefully the measurement was made — and how many digits a calculated result can claim.

Reporting more digits than your measurement supports = false precision.

Counting Rule 1: Nonzero and Sandwiched Zeros

All nonzero digits are significant.

• 4.52 → 3 significant figures

Zeros between nonzero digits are significant.

• 4.052 → 4 significant figures
• 1,003 → 4 significant figures

These zeros carry information: something was measured between those nonzero digits.

Counting Rule 2: Leading and Trailing Zeros

Leading zeros: NOT significant

• 0.0045 → 2 sig figs (zeros just place the decimal)

Trailing zeros after decimal: significant

• 4.520 → 4 sig figs

Trailing zeros before decimal: ambiguous

• 4500 → use to show 2 sig figs clearly

Count Significant Figures: Four Cases

Significant Figures in Multiplication and Division

Rule: Round result to the fewest sig figs among inputs.

(2 sig figs from 4.2)

(2 sig figs from 3.4)

The least precise factor limits the result.

Significant Figures in Addition and Subtraction

Rule: Round to the same decimal place as the least precise input — not fewest sig figs.

(tenths place, from 12.5)

(ones place, from 1,000)

Ask: which addend has the fewest decimal places?

Addition Rule vs. Multiplication Rule Differ

Wrong: Apply sig-fig count to addition.

"1 sig fig → round to 1,000" — erases real information!

Correct: Use decimal-place rule → (ones place)

Sig-fig count: × and ÷ only. Decimal place: + and − only.

Area Calculation: Tracking Precision Through

Calculator shows: 4.2 × 6.84 = 28.728

Report as: 29 m² (2 sig figs — limited by 4.2)

Your Turn: Report with Correct Precision

A student measures:

• Time: 4.5 s (2 sig figs)
• Distance: 112.3 m (4 sig figs)

Calculate speed = distance ÷ time.

1. What does the calculator show?
2. What should you report, and why?

Apply the correct rule — show your reasoning.

Significant Figures Are a Precision Promise

✓ Write digits you measured — no more, no fewer

✓ Trailing zeros after the decimal are data — keep them

✓ × and ÷: use fewest sig figs from inputs

✓ + and −: use least precise decimal place

More digits ≠ more accurate

Coming Up: Instruments and Context

You can now count significant figures and apply them in calculations.

Next lesson (Lesson 2):

• Instrument resolution — what limits the precision you can report
• Context-appropriate precision — sometimes fewer digits are more honest
• Precision propagation through multi-step calculations