Every measurement has a degree of uncertainty associated with it. The uncertainty derives from the measuring device and from the skill of the person doing the measuring.
Let's use volume measurement as an example. Say you are in a chemistry lab and need 7 mL of water. You could take an unmarked coffee cup and add water until you think you have about 7 milliliters. In this case, the majority of the measurement error is associated with the skill of the person doing the measuring. You could use a beaker, marked in 5 mL increments. With the beaker, you could easily obtain a volume between 5 and 10 mL, probably close to 7 mL, give or take 1 mL. If you used a pipette marked to with 0.1 mL, you could get a volume between 6.99 and 7.01 mL pretty reliably. It would be untrue to report that you measured 7.000 mL using any of these devices, because you didn't measure the volume to the nearest microliter. You would report your measurement using significant figures. These include all of the digits you know for certain plus the last digit, which contains some uncertainty.
Significant Figure Rules
Measured quantities are often used in calculations. The precision of the calculation is limited by the precision of the measurements on which it is based.
Addition and Subtraction
When measured quantities are used in addition or subtraction, the uncertainty is determined by the absolute uncertainty in the least precise measurement (not by the number of significant figures). Sometimes this is considered to be the number of digits after the decimal point. Example
32.01 m
5.325 m
12 m
Let's use volume measurement as an example. Say you are in a chemistry lab and need 7 mL of water. You could take an unmarked coffee cup and add water until you think you have about 7 milliliters. In this case, the majority of the measurement error is associated with the skill of the person doing the measuring. You could use a beaker, marked in 5 mL increments. With the beaker, you could easily obtain a volume between 5 and 10 mL, probably close to 7 mL, give or take 1 mL. If you used a pipette marked to with 0.1 mL, you could get a volume between 6.99 and 7.01 mL pretty reliably. It would be untrue to report that you measured 7.000 mL using any of these devices, because you didn't measure the volume to the nearest microliter. You would report your measurement using significant figures. These include all of the digits you know for certain plus the last digit, which contains some uncertainty.
Significant Figure Rules
- Non-zero digits are always significant.
- All zeros between other significant digits are significant.
- The number of significant figures is determined starting with the leftmost non-zero digit. The leftmost non-zero digit is sometimes called the most significant digit or the most significant figure. For example, in the number 0.004205 the '4' is the most significant figure. The lefthand '0's are not significant. The zero between the '2' and the '5' is significant.
- The rightmost digit of a decimal number is the least significant digit or least significant figure. Another way to look at the least significant figure is to consider it to be the rightmost digit when the number is written in scientific notation. Least significant figures are still significant! In the number 0.004205 (which may be written as 4.205 x 10-3), the '5' is the least significant figure. In the number 43.120 (which may be written as 4.3210 x 101), the '0' is the least significant figure.
- If no decimal point is present, the rightmost non-zero digit is the least significant figure. In the number 5800, the least significant figure is '8'.
Measured quantities are often used in calculations. The precision of the calculation is limited by the precision of the measurements on which it is based.
Addition and Subtraction
When measured quantities are used in addition or subtraction, the uncertainty is determined by the absolute uncertainty in the least precise measurement (not by the number of significant figures). Sometimes this is considered to be the number of digits after the decimal point. Example
32.01 m
5.325 m
12 m
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