Accuracy is closeness to the truth; precision is closeness of repeats to each other.
Precision How closely repeated measurements agree with one another.
Step 1: Something to Notice
Watch first. The explanation comes later.
Two measurements each precise to four significant figures are subtracted, and the answer has none.
Where did the precision go?
Step 2: Find Out
Isolate the operation and note which uncertainty — absolute or relative — is preserved.
Step 1 — Predict
Why do the addition and multiplication significant-figure rules differ?
Choose what you think will happen. You cannot see the experiment until you do — guessing first is what makes it worth watching.
Step 3: So Here Is Why
Now the explanation, after you have seen it happen.
Repetition cannot detect systematic error
A mis-calibrated balance gives precise wrong answers indefinitely.
Systematic error A consistent bias that repetition cannot reveal or reduce.
Adding: absolute uncertainties add
12.0 ± 0.1 plus 10.0 ± 0.1 gives 22.0 ± 0.2.
Multiplying: relative uncertainties add
0.8% and 1.0% give 1.8% in the product.
Subtracting close values destroys precision
The absolute uncertainty survives while the value collapses.
Significant figures approximate something exact
Proper error propagation combines uncertainties by defined rules. Significant figures are a rough substitute that works well enough for routine work and breaks down for careful measurement.
Random and systematic error differ
Random error scatters results around the true value and is reduced by repeating. Systematic error shifts them all one way and is not, which is why more trials cannot fix a miscalibrated instrument.
Step 4: A Common Mistake
Lots of people think
“The significant-figure rules for adding and for multiplying are two unrelated conventions to memorize.”
Step 5: Why the Mars Climate Orbiter Was Lost
The same idea somewhere new.
In 1999 a spacecraft was lost because one team supplied thrust data in pound-seconds while the receiving software expected newton-seconds. Nothing was imprecise: every number was correct to many figures in its own unit, and the trajectory was wrong by the ratio 4.45 throughout. It is a useful corrective to the impression that quantitative care means carrying more decimal places — the failure was a units error of the kind dimensional analysis exists to catch, and no amount of precision would have detected it.
Step 6: Think It Through
Practice makes it stick.
Where the Precision Went
Problem 1 of 2
Two four-figure measurements are subtracted and the answer has none. Why?
The Mis-Calibrated Balance
Problem 2 of 2
A balance reads 0.5 g high every time. Will averaging more readings help?
Carrying the Uncertainty
1 of 5
What propagates when quantities are added?
2 of 5
What propagates when quantities are multiplied?
3 of 5
Can repetition reduce systematic error?
4 of 5
How many significant figures does a counted quantity have?
5 of 5
When do the shortcut rules fail?
Step 7: Quick Check
Show what you know.
Question 1 of 1
Are the addition and multiplication significant-figure rules unrelated conventions?
Step 8: Explain It in Writing
Claim, evidence, then reasoning.
The question
Explain the significant-figure rules by deriving them from uncertainty propagation.
Fill in all three boxes. The reasoning box is the one that matters most.
What You Found Out
- Accuracy and precision are independent, and repetition improves only the second.
- Absolute uncertainties add for sums; relative uncertainties add for products.
- Both significant-figure rules are approximations to that propagation.
- Subtracting close values keeps the uncertainty and destroys the value.