Summary of Significant Figures and Uncertainty

Number of sig figs is the number of digits reported, not including any zeroes to the left of the first non-zero digit. For example, 3.00 has 3 sig figs, 0.0045 has 2 sig figs and 3.0400 has 5 sig figs

Ambiguity

With no decimal point, the number of significant figures in the number 100,000 is ambiguous. For numbers without a decimal, zeros to the right of the last non-zero digit are assumed to be not significant. However, significance can be indicated in a number of ways:

1. Using a final decimal point. For example, "100,000." would indicate 6 sig figs.
2. Underlining the last significant digit. For example, 100,000 would indicate 1 sig fig, whereas 100,000 would indicate 4 sig figs.
3. Scientific notation. 1.00 x 10⁵ would indicate 3 sig figs.

Precision

The precision of a number is the decimal place of the furthest-right digit.  For example, 3.0 is precise to the tenths place.  3.0x10 is precise to the ones place.

Calculations

Keep all digits in your calculator throughout your calculation.  Then go back to determine significant figures at each stage.

Addition and Subtraction

Round your answer to the same decimal place as the number with the least precision.  (e.g. 4.1 + 7.00092 = 11.1)  Be careful when using scientific notation. (e.g. 4.1 + 1E-3 = 4.1 + 0.001 = 4.1)

Multiplication and Division

Round your answer so that it has the same number of significant figures as the number that has the least significant figures.  (i.e. 5 * 1.000 = 5)

Rounding

If digit to be dropped is > 5(00…) increase last digit by one (round up).
If digit to be dropped is < 5(00…) no change in last digit (round down).

Rounding 0.5

If digit to be dropped is exactly 5, with no nonzero numbers afterward, round last digit to the nearest even number.  (i.e. 2.5 rounds to 2; 3.5 rounds to 4)

x ± AU = x ± δx
Example: 54 ± 2 seconds

Absolute uncertainty

AU = δx
Example from above: AU = 2 seconds

Relative uncertainty
(Fractional uncertainty)

Example from above: RU = 2/|54| = 0.04

Percent uncertainty =
RU x 100

Example from above: % uncertainty = 0.04 x 100 = 4%

Standard deviation

When multiple measurements are collected and averaged, AU for the average = standard deviation of the measured values. In Excel, find the standard deviation using "=stdev(select cells)"

Reporting uncertainty

Uncertainty is always reported as a positive value, with only one significant figure. Measured (or calculated) value and uncertainty should be reported with the same precision. Use the precision of whichever is least precise.
Example 1: 564.45 ± 2.34 = 564 ± 2
Example 2: 435 ± 0.0434 = 435 ± 0

AU total = SUM of AUs

First calculate AU then calculate RU = AU/|q|.

Example:
(54 ± 2) + (83 ± 3)
= 137 ± 5 or 137 ± 4%

q = x +....+ y - z -....- u

q = 54 + 83 = 137

δq ≤ δx + Κ δy + δz + Κ + δu

AU < 2 + 3 = 5
RU < 5/137 = 0.04

Multiplication and Division

RU total = SUM of RUs

First calculate RU then calculate AU = RU|q|

Example:
(2.7 ± 0.3) x (4.5 ± 0.5)
= 12 ± 3 or 12 ± 20%

q = 2.7 x 4.5 = 12.15

RU < 0.3/2.7 + 0.5/4.5 = 0.222
AU < 0.222 x 12.15 = 2.7