Every measurement has error
Three students measure the same pencil with the same ruler and get 15.32 cm, 15.33 cm, and 15.31 cm. Every measurement carries a little error误差. Error doesn’t mean a mistake. It’s the uncertainty that comes with any measurement. Our job is to keep it small and report results honestly.
Accuracy and precision
- Accuracy准确度: how close a measurement is to the true value.
- Precision精确度: how close repeated measurements are to each other.
Think of a target: the bullseye is the true value, and each dot is one measurement. Systematic error (such as a balance that isn’t zeroed) shifts every dot the same way, so the results aren’t accurate. Random error scatters the dots, so the results aren’t precise. Drag the sliders and see:
Precise doesn’t mean accurate. If a balance isn’t zeroed and reads 0.5 g too high every time, the results agree closely with each other (precise) but are all off from the true value (not accurate).
Estimate one extra digit
When you read an instrument, read exactly to the smallest marked division, then estimate one more digit. With a ruler marked in millimeters, record to 0.01 cm: 2.35 cm, where 2.3 is read exactly and the final 5 is estimated.
🔧 Math tool card · Significant FiguresSkip it if you already know this
Significant figures are all the digits you know for certain, plus the one estimated digit. The rules for counting them:
- Nonzero digits always count: 2.35 has 3.
- Zeros between nonzero digits count: 1002 has 4.
- Leading zeros only locate the decimal point and don’t count: the first two zeros in 0.0520 don’t count.
- Trailing zeros after a decimal point count: the last zero in 0.0520 counts, so it has 3; 12.30 has 4.
- Trailing zeros in a whole number with no decimal point are ambiguous, so use scientific notation: 3.50 × 10³ has 3.
Example 1: Reading a ruler
You measure a pencil with a ruler marked in millimeters. The end falls between 15.3 cm and 15.4 cm, a little past halfway. How should you record it?
- The certain part. The smallest division is 0.1 cm, so you can read 15.3 cm exactly.
- Estimate. The end is about 6/10 of the way from 15.3 to 15.4, so estimate 0.06 cm.
- Record it. 15.36 cm, with 4 significant figures. The final 6 is the estimated digit.
- What not to write. Not 15.4 cm (you’ve lost the estimated digit) and not 15.362 cm (this ruler isn’t that precise).
Example 2: Accurate or precise?
An object’s true mass is 5.00 g. Student A gets 5.01 g, 4.99 g, and 5.00 g. Student B gets 5.31 g, 5.30 g, and 5.32 g. Evaluate each set.
- Student A. The average is 5.00 g, very close to the true value, so it’s accurate. The three results are very close together, so it’s precise.
- Student B. The three results are very close together, so it’s precise. But they’re all about 0.3 g too high, so it’s not accurate. There’s probably a systematic error, such as a balance that wasn’t zeroed.
- 10.0 g, 10.1 g, 9.9 g
The average is 10.0 g and the results are close together.
- 11.2 g, 11.3 g, 11.2 g
Close together, but all about 1.2 g too high.
- 9.0 g, 11.0 g, 10.1 g
The average is about 10.0 g, but the results are spread far apart.
- 12.0 g, 13.5 g, 11.1 g
The average is about 12.2 g, too high, and the results are scattered.
Common mistakes
Mistake 1: Treating precise as accurate. Results that agree closely are precise. Only results close to the true value are accurate.
Mistake 2: Not estimating, or estimating too many digits. With a ruler marked in millimeters, record to 0.01 cm: no more, no less.
Mistake 3: Counting leading zeros. 0.0520 has 3 significant figures, not 5.
Quick check
How many significant figures does 0.00340 have?
The three leading zeros don't count. The 3, the 4, and the trailing zero do, for a total of 3.
Using a ruler marked in millimeters, which recorded length is correct?
The smallest division is 0.1 cm, so estimate one more digit and record to 0.01 cm.
Summary
- Every measurement has error. Accuracy is closeness to the true value; precision is closeness of repeated results to each other.
- Systematic error shifts results off target (not accurate); random error scatters them (not precise).
- Read exactly to the smallest division, then estimate one more digit.
- Significant figures: leading zeros don’t count, zeros in between do, and trailing zeros after a decimal point do.
On to AP (for later)
AP calculations must be reported with the right number of significant figures, and lab questions often ask about sources of error and whether an error makes a result too high or too low. Stage 2 lesson 2.1.1 covers the rules for significant figures in calculations.