Significant Figures and Error Analysis Page 1
Illinois Central College
CHEMISTRY 130
Significant Figures and Error Analysis
Accuracy and Precision
The only kind of physical quantity that can be determined with perfect accuracy is a tally of
discrete objects, for example, dollars and cents or the number of objects in a museum case. In
measuring a quantity capable of continuous variation--for example, mass or length--there is
always some uncertainty because the answer, like an irrational number such as π , cannot be
expressed by any finite number of digits. Besides errors resulting from mistakes made by the
experimenter in the construction and use of measuring devices, other errors over which the
experimenter has no control are inherent in measurements. Therefore, at least two, preferably
three or more, determinations of any quantity should be made. The "true" value--more correctly
the "accepted" value--of a quantity is chosen by some competent group such as a committee of
experts as the most probable value from available data, examined critically for errors.
The precision of a measurement is a measure of the mutual agreement of repeated
determinations; it is a measure of the reproducibility of an experiment. The arithmetic average
of the series is usually taken as the "best" value. The simplest measure of precision is the
average deviation, calculated as follows.
1.
Determine the average of the series of measurements.
2.
Calculate the deviation of each individual measurement from the average.
3.
Average the deviations (treat each as a positive quantity).
EXAMPLE: In a series of determinations, the following values for the molarity of a
potassium permanganate solution were obtained: 0.1010, 0.1020, 0.1012, 0.1015
millimoles per milliliter (moles per liter). Calculate the average deviation.
Average of the individual
measurements
Average:
Individual deviations
from the average
0.1010
0.0004
0.1020
0.0006
0.1012
0.0002
0.1015
0.0001
0.4057 ÷ 4
0.0013 ÷ 4
0.1014 mmol/mL
Average deviation: 0.0003 mol /mL
These results would be reported as 0.1014 + 0.0003 mmol/mL.
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Significant Figures and Error Analysis
Precision: Relative Average Deviation
Frequently, precision is expressed as the relative average deviation, r.a.d., defined as the
average deviation divided by the average. Thus the r.a.d. for the series of measurements in the
above example is:
r.a.d. = 0.0003 ÷ 0.1014 = 0.003 (dimensionless)
Multiplication by 100 yields the r.a.d. on a percentage basis.
%r.a.d. = (0.0003 ÷ 0.1014) x 100 = 0.3% r.a.d.
The precision of an experiment varies with the method and apparatus used. Using the apparatus
commonly available for quantitative analytical work, an experienced chemist can attain a
precision of 1 part per 1000 or better in the gravimetric determination of the chloride in a
water-soluble sample; the average inexperienced student more frequently obtains a precision of
about 10 parts per 1000 parts. With more complex analyses, the precision may decrease sharply.
In planning an experiment, the experimenter should consider what overall precision is desirable
and then choose the appropriate methods and equipment.
Accuracy: Relative Error
Precise measurements, however, are not necessarily accurate. The accuracy expresses the
agreement of the measurement with the accepted value of the quantity. Accuracy is expressed in
terms of the error (also called the absolute error), the experimentally determined value minus the
accepted value. The relative error is the error divided by the accepted value. If the accepted
value is unknown, the accuracy cannot be ascertained.
% relative error =
exp erimental value - accepted value
accepted value
× 100
EXAMPLE: The accepted value for the molarity of the permanganate solution is 0.1024
mol/mL. Calculate the error and relative error for the determination of the
molarity in the previous sample.
0.1014,
- 0.1024,
- 0.0010,
the determined value
the accepted value
the error
from which the relative error is -0.98%.
- 0.0010
0.1024
x 100 = - 0.98%
Significant Figures and Error Analysis Page 3
Propagation of Errors
When measured quantities are used to calculate another quantity, errors in the measurements
introduce errors into the calculated result. The errors are said to be propagated through the
calculations. When the error in each measured quantity has been estimated, the error in the
result can be obtained, in simple cases, by the following rules.
1. The error in a sum or difference is the sum of the errors in the individual terms.
EXAMPLE:
Weight of container + contents 16.7193 ± 0.0005 g
Weight of container
9.8264 ± 0.0005 g
Weight of contents
6.8929 ± 0.0010 g
Since the uncertainty is now in the third decimal place, we should round off
the result to 6.893 ± 0.001 g.
2. The relative error in a product or quotient is the sum of the relative errors in the
individual factors. (A divisor also counts as a factor.)
EXAMPLE:
Relative error
Weight of object:
9.2152 ± 0.0003 g
Volume of object:
8.74 ± 0.07 mL
density = 9.2152 g = 1.05 g/mL
8.74 mL
0.0003(100) ≅ 0.003%
9.2152
≅
0.8%
0.8% + 0.003% ≅
0.8%
0.07(100)
8.74
Hence the error in the density = 0.008 x 1.05 ≅ 0.008. We may thus write the density as
1.05 ± 0.01 g/mL (0.008 may be rounded to 0.01).
Significant Figures in Measurements
The number of significant figures in a quantity is the number of digits--other than the zeros that
locate the decimal point--about which we have some knowledge. For example, the number of
significant figures in 16.7193 is six; in 6.023 x 1023, four; in 0.00780, three; in 6.8929 ± 0.0010,
four, the final 9 not being actually known. Therefore, we should write the last quantity
as 6.893 ± 0.001.
When the last digit to be discarded is a 5, the number is rounded up or down in order to make
the preceding digit even. For example,
6.85 → 6.8
and
6.935 → 6.94
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Significant Figures and Error Analysis
When the error in a measurement has been estimated, the number of significant figures is
immediately apparent. A digit that is uncertain by more than 6 or 7 should be discarded; for
example, 7.263 ± 0.006 covers the range 7.257 to 7.269, and therefore is better written
as 7.26 ± 0.01.
When numbers are used to calculate a result, the proper number of significant figures appearing
in the answer can be decided by the following rules.
1. For addition and subtraction, the number of figures to the right of the decimal point in
the sum (or difference) is equal to the number of figures to the right of the decimal point in
the term that has the fewest such figures.
EXAMPLE
0.784 + 15.16 - 9.6782 = 6.266 or 6.27 (rounded)
There are two figures to the right of the decimal point in 15.16, and there should likewise
be two in the answer. However, it is frequently desirable to carry one extra figure in
calculations to minimize rounding error. Then round off the answer to the correct number
of significant figures at the end of the calculation.
2. For multiplication and division, the number of significant figures (regardless of the
position of the decimal point) in the product or quotient is equal to the number of
significant figures in the factor that has the fewest significant figures.
EXAMPLE
6.27 x 0.08352 = 0.905
5.784
The answer contains three significant figures because that is the number of significant
figures in 6.27. The other two factors have four each.
Rules for Significant Figures
1. All nonzero digits are significant. example:
127.34
2. All zeros between nonzero digits are significant.
example: 120.0007
3. Unless specifically indicated by the context to be significant, all zeros to the left of an
understood decimal point but to the right of a nonzero digit are not significant.
example: 109,000
4. All zeros to the left of an expressed decimal point and to the right of a nonzero digit are
significant.
example: 109,000.
5. All zeros to the right of a decimal point but to the left of a nonzero digit are not significant.
example: 0.00476
6. All zeros to the right of a decimal point and to the right of a nonzero digit are significant.
example: 0.04060