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2.4 Uncertainty in Measurement
When a measurement is made with a tool (ruler, graduated cylinder, balance, etc.) there are certain
numbers and an uncertain number.
Certain numbers – are within the accuracy of the measurement tool and will be the same for anyone
doing the measurement
Uncertain number – an estimate defined by the inherent uncertainty of the measuring device
In making a measurement, it is customary to record all the certain numbers and the first uncertain number.
(See fig 2.5 and example in text.) The numbers recorded in a measurement (both certain and estimated)
are called significant figures (sig figs).
The number of sig figs is determined by the inherent uncertainty of the measuring device. In the example
on page 23, the ruler can only give results to one-hundredths of a centimeter. The uncertainty of the last
number (the estimated number) is assumed to be ± 1 (unless otherwise noted). Therefore, a measurement
of length recorded as 2.85 cm ± 0.01 cm means the length could actually be 2.84 cm to 2.86 cm.
Our laboratory balances have an uncertainty of ± 0.01 g. If an item on the balance has a mass of
10.56 grams, how is this interpreted?
2.5 Significant Figures
How to determine the number of significant figures in a calculated result?
Remember, significant figures include all the certain numbers and the first estimated number of a
measurement.
The degree of uncertainty is signified by the number of significant figures recorded.
Rules for Counting Significant Figures (see page 24, text, and below)
1.
2.
Nonzero integers (1 to 9) are always significant.
Ex: in the number 1457, all integers are significant.
Zeros
a. Leading zeros that precede all nonzero digits never count as significant figures.
Ex: In the number 0.0025, the zeros only indicate the position of the decimal point
and are not significant. Only 2 and 5 are significant.
b.
Captive zeros that fall between nonzero digits are always significant.
Ex: 1.008 has four significant figures
c. Trailing zeros are zeros at the right end of a number. They are significant only if the
number is written with a decimal point.
Ex: 100 has one sig fig, 100. has three sig figs, 100.0 has four sig figs
3.
Exact Numbers.
a. These are numbers obtained by counting. They have unlimited sig figs.
b. They can also arise from definitions. Ex: 1 inch is defined as exactly 2.54 cm. Therefore,
neither 1 nor 2.54 limits the number of significant figures when used in a calculation.
Rules for Significant Figures Applied to Scientific Notation (page 24, bottom)
When writing numbers in scientific notation, the number of sig figs is shown in the coefficient. For
example, the number 12340000 in sci notation is 1.234 X 107. There are 4 sig figs in the number and four
sig figs in the coefficient.
For example, 0.000060 has 2 sig figs. In sci notation it would be written as 6.0 X 10-5
Rules for Rounding Off Numbers on a Calculator (page 25)
When performing calculations on a calculator the answer displayed is usually greater than the number of
significant figures the result should possess. Rounding off reduces the number of digits to the correct
number.
1.
If the digit to be removed
a.
b.
is less than 5, the preceding digit remains the same. Ex: 1.33 rounds to 1.3.
is 5 or greater, the preceding digit is increased by 1. Ex: 1.36 rounds to 1.4.
2. In a series of calculations, carry the extra digits through to the final result and then round off. In
other words, do not round off as you go.
Rules for Using Significant Figures in Calculations (page 26)
1. For multiplication or division, the number of significant figures in the result is the same as the
measurement with the smallest number of sig figs. This is the limiting measurement. (See examples)
2. For addition or subtraction, the limiting term is the one with the smallest number of decimal
places. (See examples)