Uncertainty in Measurement: Significant Figures

All of the numbers in a scientific measurement are certain, except for the last digit. The certainty of measurement depends on two factors: the number of digits in the measurement and the precision of the instrument used. 

In a measured quantity, all of the digits, including the last uncertain digit, are called significant figures and can be determined using specific rules.

Any non-zero digits and all captive zeros, which lie between two non-zero digits, are significant. For example, 28 has two significant figures, while 26.25 has four, and 208 has three.

Leading zeros are never significant, as they just locate the decimal point. For example, 0.00208 has three significant figures. Such quantities can be expressed using exponential notations. Thus, 0.00208 can be written as 2.08 × 10⁻³.

Trailing zeros are only significant in decimal formatted numbers. 2200 has two trailing zeros and two significant figures, whereas 2200.0 and 2200.1 both have 5 significant figures. 

For quantities without decimal points, the significance of trailing zeros becomes ambiguous. Thus, 2200 can be written as 2.2 × 10³ with two significant figures or 2.20 × 10³ with three significant figures.

Significant figures help achieve certainty in mathematical operations, too. In addition or subtraction, the result should be rounded off to have the same number of decimal places as the measurement with the fewest decimal places. 

Rounding down should be performed when the last digit is below 5, and rounding up carried out when it is 5 or above. Other rounding methods are sometimes used when the last digit is 5. For instance, the sum of 2.052 and 1.2 is rounded off as 3.3.

However, while multiplying or dividing, the result should be rounded to have the same number of significant figures as the measurement with the fewest significant figures. Thus, the product of 2.052 and 1.2 is rounded off as 2.5.

Scientists often repeat experiments to achieve precision in their measurements. Standard deviation is the statistical expression of such precision and measures the dispersion from the expected value. If precision is high, the standard deviation is small, and vice versa. 

For example, two groups measured the thickness of a book in centimeters. They found the same average: 10.6 cm. However, the measurements by the first group are more precise, and thus have a lower standard deviation. The second group has more spread out measurements and a higher standard deviation.