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11.1 Uncertainty and error in measurement
11.2 Uncertainties in calculated results
11.3 Graphical techniques
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o In practical science, the results of
experiments are never completely
reliable as there are always experimental
errors and uncertainties involved.
o It is therefore important, especially in
quantitative work, to be able to assess
the magnitude of these and their effect
on the reliability of the final result.
o It is important to differentiate between
the accuracy of a result and the precision
of a result.
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o What can you tell about
the precision and
accuracy in the adjacent
figure?
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o The accuracy of a result is a measure of
how close the result is to some accepted
or literature value
o Accuracy is a measure of the systematic
error. If an experiment is accurate then
the systematic error is very small.
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o The precision is a measure of how
close the repetitions will be to each
other.
o The precision or reliability of an
experiment is a measure of the
random error. If the precision is high
then the random error is small.
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o Random uncertainties (or errors) arise
mostly from inadequacy or limitation in
the instrument or the way a measurement
is made.
o Random errors make a measurement less
precise, but not in any particular direction.
These are written as an uncertainty range,
such as 44.20 ± 0.05 cm3.
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o In the adjacent burette, We would
o
probably take the reading as 43.6, but in
so doing we are saying that it is nearer to
43.6 than it is to 43.5 or 43.7, and
smaller than 43.65 ,hence we should
record this value as 43.6 ± 0.05.
if we are reading a digital instrument,
such as a balance, then we should record
the uncertainty as being half of the last
digit. For example the reading 37.361 on
a digital readout should be recorded as
37.361 ± 0.0005.
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o Large random uncertainties obviously
decrease the precision of the values obtained
and can also lead to inconsistent results if the
procedure is repeated.
o If only one reading is taken, then a large
random uncertainty can lead to an inaccurate
result.
o Repeating experimental determinations
should, however, increase the precision of the
Final result as the random variations
statistically cancel out.
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o Systematic errors are due to identifiable
causes, and arise from flow in the instrument
or errors made in taking a measurement such
as an incorrect calibration of a pH meter or
reading the top rather than the bottom of the
meniscus.
o Systematic errors always affect a result in a
particular direction (always smaller or larger)
unlike random errors.
o Random uncertainties can be reduced by
repeating readings; systematic errors can not
reduced by repeating readings.
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o All non zero numbers are significant
• 613 has three sig figs
• 123456 has six sig figs
o Zeros located between non-zero digits are significant
• 5004 has four sig figs
• 602 has three sig figs
o Trailing zeros (those at the end) are significant only if the
number contains a decimal point; otherwise they are
insignificant
• 5.640 has four sig figs
• 120000 has two sig figs
o Zeros to left of the first nonzero digit are insignificant they are
only placeholders!
• 0.000456 has three sig figs
• 0.052 has two sig figs
• 0.000000000000000000000000000000000052 also has two sig figs!
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o Rules for addition/subtraction problems
o Your calculated value cannot be more precise than
the least precise quantity used in the calculation.
The least precise quantity has the fewest digits to
the right of the decimal point. Your calculated
value will have the same number of digits to the
right of the decimal point as that of the least
precise quantity.
o 7.939 + 6.26 + 11.1 = 25.299 (this is what your
calculator will give) In this case, the final answer is
limited to one sig fig to the right of the decimal or
25.3 (rounded up)
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o Rules for multiplication/division problems
o The number of sig figs in the final calculated
value will be the same as that of the quantity
with the fewest number of sig figs used in the
calculation.
o (27.2 x 15.63) ¸ 1.846 = 230.3011918 in this
case, since the final answer is limited to three
sig figs, the answer is 230. (rounded down)
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o Rules for combined addition/subtraction
and multiplication/division problems
o First apply the rules for addition/subtraction
(determine the number of sig figs for that
step), then apply the rules for
multiplication/division.
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o Absolute and Percentage Uncertainties
o The absolute uncertainty is the actual
uncertainty in the value, for example
±0.05 for a quantity that has the value
28.5 ± 0.05.
o The percentage uncertainty is the
absolute uncertainty expressed as a
percentage of the value. For example the
percentage uncertainty of 28.5 ± 0.05 is
±0.18% (100 × 0.05/28.5)
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o The uncertainties in individual measurements
can be combined to calculate the uncertainty
in the final value of the quantity being
determined.
o In addition and subtraction: Add absolute
uncertainties
o In multiplication, division and powers: Add
percentage uncertainties
o If one uncertainty is much larger than the others,
ignore the other uncertainties and estimate the
uncertainty based on the larger one using the
rules above.
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o calculate the value and uncertainty of
X, where
X = A (B – C)
given the values:
A = 123 ± 0.5;
B = 12.7 ± 0.2;
C = 4.3 ± 0.1
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o X = 123 × (12.7 – 4.3)
= 1033.2
o (note that this has not yet been rounded to an appropriate
precision as the uncertainty has not been calculated)
o Actual uncertainty in (B – C) = ± 0.3 (add actual uncertainties,
0.2 + 0.1 = 0.3)
o % uncertainty in A = 0.407% (100 × 0.5/123 )
o % uncertainty in (B – C) = 3.571% (100 × 0.3/8.4 )
o % uncertainty in X = 3.978% (add percentage uncertainties,
0.407 + 3.571)
o Actual uncertainty in X = 41.1 (1033.2 × 3.978/100 )
o Therefore X = 1033.2 ± 41.1
o The usual practice is to only give the uncertainty to one significant
figure and then to round of the value to a similar number of
decimal places; hence the final; result should be quoted as
o X = 1030 ± 40
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o Graphs are one of the most useful ways for interpreting
scientific data because they allow for direct visual
correlation between the data obtained and a particular
scientific model or hypothesis
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o When drawing graphs it is usual to choose the axes so that the
independent variable (frequently time) is plotted along the
horizontal axis and the dependent variable on the vertical axis.
The scale should be chosen so as to maximize the use of the
graph area, taking into account any extrapolation of the data
that may be required.
o Care must also be taken to have enough data points to ensure
that the graph really is linear. For example it is questionable
whether the data in Figure 1108 (a) really do represent a
straight line rather than a curve.
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