Download Analysis of Experimental Data

Analysis of
Experimental Data
65441597.120479 ± 0.000005 g
“Quantitative Uncertainty”
For Example:
Two students, Raffaella and Barbara, measured
the temperature of boiling water, which by definition
should be 100°C under 1 atmosphere of pressure. Each
student made 10 temperature measurements, shown
below as red (Raffaella) and blue (Barbara) dots.
The average of
Raffaella's temperature
measurements is
100.1°C and the
average of Barbara's is
also 100.2°C. Given the
“actual” value for b.p.
both students had good
accuracy
Experimental Errors
There are three types of experimental errors
affecting values:
1. Systematic error – errors affecting the accuracy in
measurements which have a definite value that can,
in principle, be measured and accounted for.
Systematic errors can be corrected for but only after
the cause is determined. Examples may include an
incorrect calibration of a balance that always reads
.0001 g higher than the actual mass or the presence of
an interfering substance in calorimic studies.
In our example, systematic error is why the students did
not get exactly 100.0 oC for their measurements.
Accuracy vs. Precision
When we make a measurement in
the laboratory, we need to know how
good it is. We want our measurements
to be both accurate and precise.
• Accuracy refers to the proximity of a
measurement to the true value of a
quantity.
• Precision refers to the proximity of
several measurements to each other, that
is, the reproducibility of a measurement
or set of measurements.
On the other hand, you
can see from the figure that
the precision of Raffaella's
measurements was far better
than Barbara's.
So, what caused the two students to get values that
were not equal to the true boiling point of water. And
even if they didn’t get the true value, why didn’t they get
the same number every time they took the measurement
of the same sample?
The answer:
Experimental Errors
2. Random error – errors affecting precision in every
measurement which fluctuate randomly and do not
have a definite value; They cannot be positively
identified. To further understand random errors,
consider the weight of an object obtained by doing
five different weighings on a four place analytical
balance.
trial
trial
trial
trial
trial
1: 0.7952 g
2: 0.7950 g
3: 0.7951 g
4: 0.7953 g
5: 0.7951 g
The first three digits are the same in all cases. The last
digit has an uncertainty associated with it. This
uncertainty is a function of the type of sample, the
conditions under which it is being weighed, the balance,
and the person doing the weighing.
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Even when all factors are
optimized, there will still be
some variation in the weight.
This variation or uncertainty is
the result of pushing the
balance to its limit.
We could cut the last figure off; then all the weights
would be the same, but the weight would be known only
to the nearest milligram. We obtain more information if
we keep that last figure but remain aware of its
uncertainty. That uncertainty arises because of random
error; and is indicative of the precision of the
measurement.
In our example, random error is why the students did not
get the same measurement every time.
It is impossible to perform a chemical
analysis in such a way that the results are totally
free of errors. All one can hope is to minimize
these errors and to estimate their size with
acceptable accuracy.
Rarely is it easy to estimate the errors of
experimental data; However, we must make such
estimates because data of unknown precision and
accuracy are worthless.
The question now
becomes, how do we
describe the accuracy and
precision quantitatively?
Estimating Accuracy in
Measurement
• Absolute error (AE)
– The difference between an experimental value
and accepted value
A.E. = exp – known
• % Absolute Error (%AE)
%AE 
exp  known
 100
known
• % Accuracy
– Percentage your value differs from 100.
% Accuracy = 100 - |%AE|
3. Gross error – errors producing values that are
drastically different from all other data. These
errors are the result of a mistake in the procedure,
either by the experimenter or by an instrument. An
example would be misreading the numbers or
miscounting the scale divisions on a buret or
instrument display. An instrument might produce
a gross error if a poor electrical connection causes
the display to read an occasional incorrect value. If
you are aware of a mistake at the time of the
procedure, the experimental result should be
discounted and the experiment repeated correctly.
Gross errors result in values that make no sense
and greatly affect the overall precision of the
experiment.
Estimating Accuracy in
Measurement
Accuracy can be estimated easily when a
“true” value is known. For example the density
of iron is 7.87 g/cm3. In the laboratory, you find
the mass and volume of an iron block to
calculate the density as 7.32 g/cm3. There is
obvious systematic error in accuracy since you
did not derive the theoretical density of pure
iron.
Note: Since all values are limited by the technology to
describe them a better term for “true” value is
“accepted” value.
Back to our example. There are two ways we
could describe the accuracy of the data collected. 1) we
could calculate the accuracy for every data point
individually. 2) we could calculate the accuracy of each
experimenter by treating the data sets as single values,
or average values.
In our example, the
average value for each
students’ results were 100.1oC
and 100.2oC.
1.Calculate the absolute error, % absolute
error and % accuracy for each experimenter.
2
Often, you will need the average for a data set to
gain a better estimate of experimental error. There
are two common ways of expressing an average: the
mean and the median.
• The mean (χ) is the arithmetic average of the
results, or:
n
Mean  x  
i 1
x i x1  x 2  ...  x n

n
n
2. If the analysis of acetaminophen tablets
resulted in the presence of 428mg, 479mg,
442mg, and 435mg active ingredient in four
different trials, what was the mean value?
When Should a Value be Omitted as an
Outlier When Finding an Average?
There are many different ways to
determine whether or not a value is an outlier
due to gross error. Some methods include:
• Q test
• 10% assumption
• Standard deviation (4σ approximation)
• Five number summary (Box Plot)
We will utilize the five number summary.
V. Multiply IQR by 1.5
VI.Subtract modified IQR from Q1. Anything
less than this value is an outlier.
VII.Add the modified IQR to Q3. Anything
greater than the result is an outlier.
5. Determine if outliers exist in the
following set of data. Then find both
the mean and median of the
appropriate data.
• 12, 42, 56, 61, 62, 69, 73, 107.
• The median is the value that lies in the
middle among the results. Half of the
measurements are above the median and
half below. If there are and even number of
values, the median is the average of the two
middle results.
3. What then is the median for the values found in
analyzing the acetaminophen tablets?
There are often advantages for using the median
in place of the mean when an average is desired. If a
small number of measurements are made, one value
can greatly affect the mean.
4. Compare the mean to median in our sample set.
Which one would be a more realistic average?
The Five Number Summary
For any group of numbers you believe may
contain an outlier:
I. Arrange numbers in ascending or descending
order and find the median.
II. Calculate Q1 by finding the middle value for
the numbers left of the median.
III.Calculate Q3 by finding the middle value for
the numbers right of the median.
IV.Calculate the Inner Quartile Range (IQR) by
subtracting Q1 from Q3.
Estimating Precision in
Measurement
In chemistry you are often looking for an
unknown value and have nothing to compare your
experimental values to.
In quantitative work, precision is often used
as an indication of accuracy; we assume that the
average of a series of precise measurements (which
should “average out” the random errors because
of their equal probability of being high or low), is
accurate, or close to the “true” value. Again, you
may be trying to determine the true/accepted
value.
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Estimating Precision in
Measurement
• Relative error (RE)
– The difference between an experimental value
and the average (mean or median) for a set of
experimental values
R.E. = exp – avg
• % Relative Error (%RE)
exp  avg
%RE 
 100
avg
• % Precision
– The closeness of a value to a set of values in
terms of percentage.
% Precision = 100 - |%RE|
Again, to our example. When looking at the
precision of the experiment we could describe the
precision in two ways. 1) the precision of each data
point collected in respect to the data for each student
using the previous treatments. 2) estimate the precision
of each students’ experiment by investigating the
uncertainty of their measurements (discussed later)
6. Calculate the relative
error, % relative error
and % precision for the
third measurement taken
by each student.
When dealing with precision, the greater the number
of trials, the better the estimation of the overall range.
We have estimated values for accuracy
and precision but what confidence is
there in these values obtained
experimentally?
You can see the results above are spread over a range of
values. The width of this spread is a measure of the
uncertainty caused by random errors. The element of
uncertainty in experimental data can be quantified and should
be reported along with the actual experimental value itself
written as the experimental value ± some degree of
uncertainty. In this case Raffaella would report a boiling point
of 100.1 ± 0.3°C, and Barbara would report 100.2 ± 1.4°C.
The value of the uncertainty gives one an idea of the precision
inherent in a measurement of an experimental quantity; here,
Raffaella is more certain of her values than Barbara.
There are many ways to quantify
uncertainty, ranging from very simple
techniques to highly sophisticated methods.
The method used will depend upon how many
measurements of a single quantity are made
and on how crucial the reporting of the value
of uncertainty is with regard to the
interpretation of the experimental data.
We will consider:
The Graduation Method
You must use your own judgment in choosing the
precision using the Graduation Method. If in doubt,
always be conservative; i.e. report the largest of possible
uncertainties (50% vs. 20%).
When a measurement is made directly by the
student in lab, the uncertainty must be approximated.
Using the graduation the uncertainty in a single
measurement is estimated by a value one-half of the
smallest level of graduation in the measuring
instrument.
For example, if a single measurement of the length
of an object is to be made using a meter stick marked
with millimeter graduations, the length should be
reported ± .5 mm (or ± .005 m). [some professors
prefer a 20% rule]
We will use ½ the lowest graduation in our lab.
1. The Graduation Method
2. Range
3. Sample Standard Deviation
4. Confidence Limits
7. Use a ruler to measure the width of your text. Report
your measurement with the correct uncertainty
according to the Graduation Method.
8. Make a temperature reading and report with the
correct uncertainty.
9. Repeat for the volume of water in an accurately read
50 mL volumetric flask.
This is how all measurements must be recorded in your lab reports.
Some instruments or glassware may have the uncertainty printed
on the tool and should be used in place of the graduation method.
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Range
The graduation method adequately estimates the
uncertainty of a single value but can not be used when a
series of values exists for a single observable. In this case
the uncertainty can be crudely approximated by the
range. The range is given as the difference between the
maximum and minimum values of the measured
quantity. In the case of the set of five weights given:
trial 1: 0.7952 ± .0001 g
trial 2: 0.7950 ± .0001 g
trial 3: 0.7951 ± .0001 g
trial 4: 0.7953 g
trial 5: 0.7951 g
…the range is 0.7953 g - 0.7950 g = 0.0003 g.
Sample Standard Deviation
The most common way to
1
2
describe the uncertainty, or
 n
2
x

x
precision, for a set of data is by
i


the sample standard deviation s   i 1

(s). The standard deviation is
n 1 

used to describe the likelihood


that a value will fall near the
mean for a normal set of data
For normal data sets, 68.3 % of
experimental values have
statistical probability of falling
within one standard deviation
(1σ) of the mean, 95.5% within
2σ and 99.7% within 3σ.

So in our example, the value should be recorded as the
average (mean or median) ± range/2, rounded to the
correct sig. figs.
Or, 0.7951 ± 0.0002 g (R)
If you remember our previous example, we did not
have a way to describe the precision of the two students’
data. Errors cause uncertainty, uncertainty affects
precision; therefore, uncertainty can quantify precision.
The greater the uncertainty, or range, the less precise the
values. Here we see that, crudely, the mass of the sample
should be somewhere between 0.7949 and 0.7953 g.
We could use range to estimate the precision of
each student but range tends to be to much of an over
estimation. We should seek a better estimation.
let’s determine the uncertainty of our
weights using standard deviation:
trial
trial
trial
trial
trial

Our precision, or uncertainty, for the weights
measured has been found as:
Range 0.7951 ± 0.00015 g, or 0.7951 ± 0.0002 g (R)
SD
0.7951 ± 0.00014 g, or 0.7951 ± 0.0001 g (s)
Notice that the standard deviation method gives
a smaller margin for error, however, it only describes
a 68.3% confidence. In other words, there is a 68.3
% probability that the “true” mass of the weights is
between 0.7950 and 0.7952 g.
68.3% is a reasonable description of
uncertainty but a 95 % confidence interval is the
minimum standard for reporting uncertainty in
chemistry and physics.
1: 0.7952 g
2: 0.7950 g
3: 0.7951 g
4: 0.7953 g
5: 0.7951 g
 n
2
   xi  x  

s   i 1
n 1 



1
2
 0.7952  0.79512  0.7950  0.79512  0.7951  0.79512  0.7953  0.79512  0.7951  0.79512 
s

4


1
2
= 0.00014
So the value should be recorded as the average (mean
or median) ± the sample deviation:
0.7951 ± 0.00014 g, or 0.7951 ± 0.0001 g (s)
This tells us that there is a 68.3 % chance that the mass
is between 0.7950 and 0.7952 g.
Confidence Limits
The sample standard deviation can be adjusted to
any given confidence interval by utilizing the following
equation:
ts
Confidence interval =  N
Where s is the sample standard deviation, t is a
statistical constant (found on the following slides) and N is
the number of samples in the data set.
Let us calculate the uncertainty of our weight
measurements with a 95 % confidence
C.I. = ± (2.571)(0.0001)/√5 = ± 0.00011
So our uncertainty is 0.7951 ± 0.0001 (95%,N=5)
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Confidence Limit T values
50%
t value
60%
70%
80%
90%
95%
96%
98%
99%
99.5%
99.8%
99.9%
Confidence Level
10.In lab, you have titrated 8 equal volumes of an
unknown acid with a standard base. You have
dispensed the following volumes of titrant:
43.82mL, 43.30mL, 44.10mL, 43.90mL,
52.69mL, 43.87mL, 43.88mL and 43.70mL. Do
the following analysis:
A. Determine if any gross errors occurred.
B. Find the mean and median.
C. Choose the best value for the average; justify.
D. Estimate the uncertainty of the measurements using the
Graduation Method (assume correct measurements).
11.If the actual amount of titrant added in
problem 10 should have been 43.85mL,
what was the absolute error and %
accuracy?
12.On a further titration, you obtained a
volume of 44.02mL. What is the relative
error and % precision in relationship to
the new average volume added?
E. Calculate the precision of your measurements in terms of
range and sample standard deviation.
F. Adjust the uncertainty to a 99% confidence level.
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