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Uncertainty in Measurements
SIGNIFICANT FIGURES
When a number is obtained from an instrument, two kinds of information are given
at the same time. One is the magnitude of the measurement. The other is the extent of its
reliability.
We would expect the measurement obtained from instrument I to be more reliable
because it has a smaller amount of uncertainty.
The number 5.37 implies that the experimenter using instrument I had to estimate
the hundredths place. We assume this measurement to be uncertain by about (0.01).
Instrument II, however, enables one to estimate only the tenths place (0.1).
Digits that result from measurement, such that only the digit furthest to the right is
not known with certainty are called SIGNIFICANT FIGURES. The number of significant
figures in a number is equal to the number of digits known for sure, plus one that is
uncertain.
same
5.37 g
(3 sig. fig) obtained from instrument I
object
5.4 g
(2 sig. fig) obtained from instrument II
Instrument I which provides more significant figures (for the same object) is more
precise than instrument II.
Expressing possible error:
Because no measurement is exact, measurements are always made to the "nearest
something", whether it is stated or not. The greatest possible error when measuring is
considered to be one half of that measuring unit. For example, you measure a length to be
5.3 cm with instrument II. Since the measuring unit is 1 cm on the ruler, the greatest
possible error will be half that value, or 0.5 cm.
Example: 5.3 ± 0.5 cm
The "plus-or-minus" (±) means "give-or-take" the possible error. The measured length could
be as little as 4.8 cm, or as much as 5.8 cm.
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ACCURACY AND PRECISION
Accuracy of a measurement reflects its nearness to the accepted value. Accurate
measurements depend on careful calibration and the skill of the experimenter in using that
instrument.
Precision of a measurement reflects the reproducibility of that measurement, or its
reliability. In general, the more significant figures there are in a number, the greater its
precision. That is, the more significant figures in a measurement, the more actual scale
markings were read and the more scale markings, the smaller the unit which has to be
estimated.
Analogy: Accuracy and Precision
Every measurement has some degree of uncertainty that depends upon the
instrument used. Although the actual or true value can never be known, some
measurements have been established with sufficient reliability to be the "accepted" value.
These values are found in scientific handbooks. The difference between a measured value
and the accepted value is the error. The percentage error is the ratio of error to accepted
value.
Absolute Error = [measured value – accepted value]
% Error = [measured value - accepted value]
accepted value
X 100%
Determining the number of significant figures.
The measurement mass of 5.37 g was obtained from instrument I. This number has 3
significant figures.
Convert 5.37 g to kg.
5.37 g = 0.005 37 kg
or
5.37 X 10-3 kg
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Since we have only changed the units, and not actually used another instrument to
obtain a more precise measurement, the conversion must maintain the same number of
significant figures.
Rules for counting significant figures
1. All nonzero digits are significant. 127.34 (5)
2. All zeros between two nonzero digits are significant. 40.07 (4)
3. Zeros which are merely used to fix a decimal place are NOT
significant. 0.005 37 (3) or 5.37 X 10 -3 (3)
4. Zeros after a decimal point and to the right of a nonzero digit are
significant. 32.0 (3), 0.0420 (3)
5. Zeros after a nonzero digit but to the left of an understood
decimal point are usually not significant.
186 000 (3)
52 000 000 000 (2)
150 (2)
1.5 X 102 (2)
1.50 X 102 (3)
Question: How many significant digits in the following examples?
1.
2.
3.
4.
5.
6.
7.
1032
3.30
3.300
0.040
0.040 5
0.040 50
1.004 0
(
(
(
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)
8.
9.
10.
11.
12.
13.
14.
50
500
500.0
5.00 X 102
2.030 X 10-5
10-5
25 000 000
(
(
(
(
(
(
(
)
)
)
)
)
)
)
Scientific Notation
Extremely large or small data numbers are frequently recorded in scientific notation
eg. 1 860 000 as 1.86 x 106
The scientific notation form is written as a number with a decimal after the first
digit times 10 to some power. It must correctly show the precision (ie. correct # of sig figs).
2 840 000 = 2.84 x 106
Practice: Write the following measurement numbers in scientific notation.
1) 780 =
3) 900 100 =
2) 0.0076 =
4) 75.4 =
Combining Numbers in Calculations
Measured values are usually combined through arithmetic calculations to give some
desired final answer. If one of these measurements is very precise but the other is not, we
can't expect too much precision in the calculated result. We follow certain rules to avoid
answers that may contain too many or too few significant figures.
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Multiplication and/or Division
Answer generally contains the same number of significant figures as the least
number of significant figures present in the original data.
60.34 (4) = 1.775 588 2 = 1.8 (2)
34 (2)
0.428 09 (5) = 0.037 (2) = 0.037 00 (4) or 3.700 X 10 -2 (4)
11.57 (4) (incorrect!)
(correct!)
Note: Calculators usually give answers with too many figures. An exception occurs when
the answer has zeros at the right that are significant. An answer of 0.037 00 is displayed
simply as 0.037 (which is incorrect). Don't forget zeros that may be significant.
Practice: Calculate the following:
1) 9.23 x 2.1 =
2) 96.38 / 0.3 =
Multiple Calculations:
4) 0.00080 x 7.1864 =
5) 4.82 / 3.3 =
(5.2 – 3.24) x 2.19 =
(2.43 + 7.8) / 4.165 =
Addition and/or Subtraction
The answer should have the same number of DECIMAL PLACES as the quantity
having the LEAST number of decimal places.
8.493
11.262
4 375.2
0.36
4 395.315
least precise measurement
Final answer: 4 395.3
Practice: Calculate the following:
1) 29.58 + 7.8 =
4) 0.040 + 0.002 =
2) 22 + 14 =
5) 765.0 + 0.08 =
*Intermediate Rounding in Multi-Step Calculations*
When doing multi-step calculations, keep at least one more significant digit in
intermediate results than needed in your final answer.
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