Significant Figures
The secret life of data digits
Measurements in Science have limited certainty. Our answers should reflect our certainty.
With most devices, you should measure to the finest mark on the scale and estimate the next
digit (“Doubtful Digit” marked in gray).
How long is the ant’s body?
How long is the ant’s body?
Ant 1 + Ant 2 = 5.2 cm
2.6 cm
+2.65 cm
5.25 cm= 5.2 cm
Between 2 and 3 cm
• 2.6 cm
The ant is a bit more than
halfway between 2 cm and 3
cm
With small scales, estimating
more than half a digit is hard.
Between 2.6 and 2.7 cm
• 2.65 cm
Ant 2 – Ant 1 = 0.0 cm
2.65 cm
- 2.6 cm
0.05 cm= 0.0 cm
Measuring: Keep the First Doubtful
Digit to show your certainty.
A field is 1.05km by
2.1km. How long is
the fence going
around it?
10.5 cm
x 21.35 cm
22.4175 cm2 = 22.4 cm2
1.05 km
1.05 km
2.1 km
+2.1 km
6.30 km = 6.3 km
Calculations:
Keep your digits until the end. Then:
Addition and Subtraction:
# Significant Figures:
• Same as the figure with fewest decimal
places
What is significant?
All nonzero digits ARE significant
All zeroes between significant figures ARE
significant
50035 has 5 sig figs
Leading zeroes ARE NOT significant
0.0000050035 has 5 sig figs
Trailing zeroes with a decimal point ARE
significant
10000 has 1 sig fig
10000. has 5 sig figs.
Multiply and Divide:
# Significant Figures:
• Same as the value with the fewest
significant figures
More discussion: http://cyberbridge.mcb.harvard.edu/math_2.html,
http://www.chem.tamu.edu/class/fyp/mathrev/mr-sigfg.html
http://academic.umf.maine.edu/magri/PUBLIC.acd/tools/SigFigsAndRounding.html,
Ant photo from Wikipedia Commons
Rounding Significant Figures: Not like usual.
•
•
•
•
Examine the digit after the last one to keep.
If below 5, round down.
If above 5, round up.
If 5, round so that the last digit to keep
becomes even.
• Why? Normal 5’s rounding always rounds up
• Normal rounding gives 1-20% error (3% shown)
Round as normal, except when rounding off 5.
Look at the last digit to keep:
• Even, round down
• Odd, round up.
Constants, exact counts, and conversions have infinite significant figures:
I have exactly 10 toes. A dozen has exactly 12. There are exactly 100 cm per meter. Pi is exact.
Zeroes left of the Decimal (5600 rpm = 2 figures) are not significant, but zeroes to the right are
(540.0 Hz = 4 figures)
Scientific Notation explicitly gives precision: 5.6 x 103 rpm vs 5.600 x 103 rpm
“Why do we round strangely?” http://www.chemteam.info/SigFigs/Rounding.html and
https://soinc.org/sites/default/files/uploaded_images/SO%20SIGNIFICANT%20FIGURES%20FINAL12-5-12.pdf