Statistics and Quantitative Analysis
Chemistry 321, Summer 2014
Statistics is the field of study that allows
you to understand the limitations of your
data; in other words, what reasonable
conclusion can you draw from your data?
Warning: What follows is
not meant to be a
complete course in
statistics. It will be enough,
though. to get you through
this course, but do not
apply it blindly to other
situations!
Quantitative measurements must be replicated
to establish the credibility of the data
Clearly, if your observations during the experiment suggest
that a procedural error occurred, then the data for that trial
may be safely omitted. In other words, be careful in lab.
But are there methods to detect socalled “outliers”? Yes, this is where
statistics is helpful. Remember, just
because you have a statistical outlier
does not mean that you should
necessarily throw out that data point!
But first, some slides about accuracy and
precision
Accuracy is the agreement between your measured
value and the published (sometimes called “true”) value.
Precision is the agreement between your repeated
measurements.
Thus, accuracy ≠ precision
An analogy with a dartboard
So why distinguish
between accuracy
and precision?
The terms allow us to
distinguish different
types of errors: those
that we can correct
easily, and those we
can correct with
difficulty or not at all.
Systematic versus random errors
Systematic (determinate) errors affect accuracy.
Because they bias the data one way (always too
high) or the other (always too low), they can
usually be corrected easily.
Random (indeterminate) errors affect precision.
Because they are the result of variability or
instrument uncertainty, they are much more
difficult to correct.
The arithmetic mean (aka the average)
The mean is simply the sum of the measurements
divided by the number of measurements;
symbolically, this is:
N
x = åx
i = 1
N
where
i
x
is the mean, xi is the i th
measurement and N is the
number of observations
Note that all of the measurements are equally important; in other
words, this is an unweighted mean. We will assume for the rest of
the course that all measurements are of equal weight.
The standard deviation is a measure of
the “spread” of the data set
To get a sense of whether
the data are closelyspaced or widely scattered
in the data space of all
possible measurements,
the standard deviation is
used.
N
å ( x - x )
2
i
s = i = 1
N - 1
The term (xi – x) is the residual for the i th measurement
Note that as N increases, s decreases – generally, more
measurements decreases the standard deviation
The variable s is used when the standard deviation is calculated
for a sample set of data from which you wish to generalize to a
population from which the sample was selected. In this case,
the denominator of the fraction inside the square root is N–1, as
shown.
The variable σ (little sigma) is used when the standard deviation
is calculated for the entire population (which is not going to
happen in this course). In this case, the denominator would
simply be N.
N
Point of fact:
2
Once N > 30, then
i
s ≈ σ.
i = 1
s = å ( x - x )
N - 1
The standard deviation is a useful measure
of spread when there are many
measurements
As a rule of thumb, you should have at least ten
repeated measurements, though you will violate this
rule often in this course. For instance, if N = 2 and the
difference between the two measurements is d, then s
= √2 d/2, which is not particularly meaningful.
Standard deviations allow you to
distinguish two distinct populations
Each graph shows two normal distributions — in each case, are
there two distinguishable populations?
Compare the means and standard deviations of the
distributions.
Compare the masses of two sets of ten pennies, one minted
before 1982, the other after 1982. The question: are the two sets
of pennies distinguishable by mass?
pre-1982 (g)
mean
std dev (σ)
post-1982 (g)
3.067
2.534
3.088
2.544
3.094
2.566
3.056
2.513
3.050
2.555
3.049
2.532
3.061
2.538
3.077
2.541
3.063
2.570
3.071
2.548
3.068
0.015
2.544
0.017
Compare the overlap of the two-sigma ranges of each set of pennies
Pre-1982 range: 3.068 ± 0.030 g
Post-1982 range: 2.544 ± 0.034 g
When using the “ ± “ notation to show one or two-sigma ranges,
report the precision of the standard deviation to match the
precision of the mean.
The pre-1982 penny mass range is therefore 3.038 to 3.098 g,
whereas the post-1982 penny mass range is 2.510 to 2.578 g. The
two ranges do not overlap, so at the two-sigma range, the two
sets of pennies are distinguishable!
There is a good reason for this distinguishability: in 1982, the US Mint
changed the composition of the penny from mostly copper to mostly zinc.
The relative standard deviation (RSD%) is
a measure of precision
s
RSD% = ´ 100
x
Guideline:
RSD% £ 3%
is good for this course
Note that other situations may have a larger or smaller cutoff percentage.
The percent deviation is a measure of accuracy
æ x - x
ö
published
÷÷ ´ 100
% deviation = çç
è x published ø
Guideline:
% deviation £ 3%
is good for this course
Note that other situations may have a larger or smaller cutoff percentage.
So how do you know when you can omit a
measurement in a set of measurements?
For this course, we
will assume that all
measureable
quantities will be
distributed
normally (in other
words, conform to
a Gaussian
distribution.
Note that the x-axis is marked in units of the standard deviation;
yes, they are using σ instead of s, but this is customary. For
instance, a measurement will be said to be “two-sigma” higher
than mean (rather than “two-ess”).
The normal distribution formula is:
1
f (x) = e
s 2p
( x - x )2
2s 2
where x is a given measurement and f(x) is the predicted
probability of that measurement.
The behavior of the normal distribution
In a normal distribution graph,
the y-axis is the number of
measurements with the value
along the x-axis, so to get a
smooth curve as shown, you
need literally hundreds of
measurements!
Fortunately, even though you will have few measurements, we
can use the behavior of the normal distribution to check the
quality of your data. For instance, we know that 95% of the data
points will be within two standard deviations of the mean.
The Q-test for excluding data
How do you know when
you can omit a data
point, even when you
have no observational
data to do so?
On a normal distribution
plot of the data, it is far
from the mean, and the
other points.
At this point apply the Qtest.
R. B. Dean and W. J. Dixon (1951) "Simplified
Statistics for Small Numbers of Observations".
Anal. Chem., 1951, 23 (4), 636–638.
The Q-test for excluding data
Consider the following data points. Note that 0.167 point
seems to be far off from the rest of the points; is it an outlier
that can be omitted?
Calculate the parameter Qcalculated by dividing the gap between
the mean and the test point and the nearest point to it by the
range between the high and low values of the data set. Note
that the test point will either be the high or low of the data set.
Table 3.3 (page 99) in text has Qtable against
which your Qcalculated can be compared
In this course, we’ll be using
the 90% confidence level (CL)
criterion, which means that if
Qcalculated > Qtable then the
outlier can be omitted. Thus, in
our example, N = 10 so Qtable =
0.412. Since 0.455 > 0.412, we
can omit the “0.167” point
• If we used a 95% CL, then the point would not be omitted;
higher confidence levels demand a higher cut-off criterion.
• If there were two fewer data points (N=8), then the point
would not be omitted.
Criteria for omitting a data point from your calculations
• If documented observations in your lab notebook
show a procedural error for a particular measurement.
• If the Q-test on a particular data point determines that
that point can be omitted.
Note: Do not keep applying the Q-test on the same data
set; in other words, after omitting one point, do
recalculate the mean and standard deviation but do not
apply the Q-test on another outlier.
Challenge problem
You collect the following data for an analysis:
12.4
12.1
11.8
13.8
What is the reportable average and RSD% for this data set?
Your observations made in your lab notebook do not allow
you to omit any data values; however, you suspect the 13.8
value can be omitted for statistically valid reasons. Apply the
Q-test at the 90% confidence level to follow up on this
suspicion, and determine whether the 13.8 value can or
cannot be omitted.