2.4 Uncertainty in Measurement
1. There are two kinds of numbers in a measurement made with a measuring device (e.g., ruler,
graduated cylinder, balance, etc.).
Certain numbers – are within the accuracy of the measuring device.
Uncertain number – a single estimated number after the last certain number. There can only be one
uncertain number.
2. When making a measurement record all the certain numbers and the first uncertain number.
(See fig 2.5 on page 23 in text.)
3.
The certain and uncertain numbers in a measurement are called significant figures (sig figs).
4. The number of significant figures 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.
5. 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
Rules for Counting Significant Figures (see page 24, text, and below)
1.
Nonzero integers (1 to 9) are always significant.
Ex: in the number 1457, all integers are significant.
2.
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 do not limit the number of significant figures in a calculation.
a.
These are numbers obtained by counting, not by measurement.
Ex: 10 cars, 8 molecules, 3 apples
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.
c.
All metric equivalencies are exact numbers. Ex: 1 m = 1000 mm
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.
Ex: 12340000 in scientific notation is 1.234 x 107. There are 4 sig figs in the number and four sig
figs in the coefficient.
Ex: 0.000060 has 2 sig figs. In scientific notation it would be written as 6.0 x 10-5
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)
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 of significant figures.
1.
2.
same.
Identify the last significant figure in the calculated answer (see rules above).
If the number next to the last significant figure is less than five, the preceding digit remains the
Ex: 1.33 rounds to 1.3
3. If the number next to the last significant figure is greater than five, the preceding digit is
increased by 1.
Ex: 1.36 rounds to 1.4.
4. 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.