Uncertainty in Measurement
- Because people can only measure something to
a certain degree of accuracy, it is important to realize that a measurement
always has some degree of uncertainty, which depends on the precision of the
measureing device.
For example, if you weigh two heads of lettuce on a bathroom scale, the bathroom
scale may show that both weigh 1.5 kg. But if you weigh the two on a balance,
the balance may show that one weighs 1.488 kg while the other weighs 1.521
kg. So do the two heads have the same mass? Your conclusion depends on the
certainty of those measurements. Therefore, it is important to indicate the
uncertainty in any measurement. This is done by using significant figures.
- Significant Figures
- Rules for counting significant figures:
-
- Nonzero Integers: Nonzero integers always count as significant
figures (i.e. three significant figures in the measurement 2.45
g).
- Zeros: There are three classes of zeros.
- Leading zeros are zeros
that preceed all nonzero digits. They are never significant (i.e.
0.00057 has only two significant figures).
- Captive zeros are zeros between
non zero digits. They are always significant (i.e. 90.08 has
four significant figures).
- Trailing zeros are zeros at
the right end of the number. They are significant only if the
number contains a decimal point (i.e. 100 has only one signfiicant
figure, but 1.00 x 102 has three significant figures).
- Exact Numbers: Numbers that are obtained by counting
rather than measuring. They are assumed to have infinite significant
figures. Numbers that arise from definitions are also exact (i.e.
one inch is defined as exactly 2.54 centimeters, so that when
2.54 cm/in is used in a calculation, it will not limit the number
of significant figures).
- Using Significant Figures
in Calculations:
- Multiplication and Division: The number of significant figures in
the result is the same as the number in the least precise measurement.
For example:
4.28 x 8.3 = 35.524 before correction; after correction of significant
figures, the result should be 36, since the limiting term (8.3)
has only two significant figures.
- Addition and Subtraction: The result has the same number of decimal
places as the least precise measurement used in the calculation.
For example: 53.984 + 2.5 = 56.484 before correction; after correction,
the result should be 56.5, since the limiting term has only one
decimal place.
|