In the last column we discussed the use of pooling to get a be

In the last column we discussed the use of pooling to get a
better
estimate of the standard deviation of the measurement method,
es-
sentially the standard deviation of the raw data. But as the last
column
implied, most of the time individual measurements are averaged
and
decisions must take into account another standard deviation, the
stan-
dard deviation of the mean, sometimes called the “standard
error” of the
mean. It’s helpful to explore this statistic in more detail: fi rst,
to under-
stand why statisticians often recommend a “sledgehammer”
approach
to data collection methods; and, second, to see that there might
be a
better alternative to this crude tactic. We’ll also see how to
answer the
question, “How big should my sample size be?”

For the next few columns, we need to discuss in more detail the
ways
statisticians do their theoretical work and the ways we use their
results.
I often say that theoretical statisticians live on another planet
(they don’t,
of course, but let’s say Saturn), while those of us who apply
their results
live on Earth. Why do I say that? Because a lot of theoretical
statistics
makes the unrealistic assumption that there is an infi nite
amount of data
available to us (statisticians call it an infi nite population of
data). When we
have to pay for each measurement, that’s a laughable
assumption. We’re
often delighted if we have a random sample of that data,
perhaps as many
as three replicate measurements from which we can calculate a
mean.
That last sentence contains a telling phrase: “a random sample
of that
data.” Statisticians imagine that the infi nite population of data
contains

all possible values we might get when we make measurements.
Statisti-
cians view our results as a random draw from that infi nite
population of
possible results that have been sitting there waiting for us. If we
were
to make another set of measurements on the same sample, we’d
get
a different set of results. That doesn’t surprise the statisticians
(and it
shouldn’t surprise us if we adopt their view)—it’s just another
random
draw of all the results that are just waiting to appear.
On Saturn they talk about a mean, but they call it a “true” mean.
They
don’t intend to imply that they have a pipeline to the National
Institute
of Standards and Technology and thus know the absolutely
correct value
for what the mean represents. When they call it a “true mean,”
they’re
just saying that it’s based on the infi nite amount of data in the
popula-
tion, that’s all.

Statisticians generally use Greek letters for true values—μ for a
true
mean, σ for a true standard deviation, δ for a true diff erence,
etc.
The technical name for these descriptors (μ, σ, δ) is parameters.
You’ve
probably been casual about your use of this word, employing it
to refer to,
Statistics in the Laboratory:
Standard Deviation of the Mean
say, the pH you’re varying in your experiments, or the yield you
get from
those experiments, or maybe even constraints (“We have to stay
within
out budgetary parameters”). You can’t be sloppy like that when
you work
with a statistician: the word parameter has a very strict
meaning.
Because parameters are based on an infi nite amount of data,
there is no
uncertainty in their values. (We’ll see why in a minute.)
So, you’re saying to yourself, “I’m confused. And why would I
even worry

about what to call these things if I don’t have that infi nite
amount of data
and can’t calculate them, anyway?”
Good point. Here’s a key thing, though. Even though we’ll
never know
the values of these parameters, we can still use a limited sample
of data
to guess at their true values. It’s a process called estimation, so
the results
are called parameter estimates, also called sample statistics.
We use a Roman letter to represent individual measurements
(e.g., x1 =
3.6), and we put a “bar” above the letter when we want to
indicate an
arithmetic average (a mean). For example, if x2 = 4.8, and x3 =
4.5, we would
write the mean of x1 through x3 as x
_
= 4.3. Thus,we say that the statistic x
_
is
an estimate of the parameter μ. Because there is uncertainty in
the mea-

sured values that have been “drawn from the population at
random,”
there is uncertainty in these parameter estimates.
Backing up a bit, how do we measure the uncertainty in
measured
values? As we discussed in the last column, the estimate s of the
true
standard deviation σ is given by the familiar equation:
where the Greek capital letter sigma (Σ) is the summation
operator, and
its index i indicates the measurement number from 1 to n. For
x1 through
x3, s = 0.6807.
Now, let’s go to Saturn for a few minutes. On Saturn we can
play with
the infi nite population of data. Let’s suppose that for the
measurements
we’ve been making, μ = 4.76 (exactly) and σ = 0.30 (exactly).
The estimate
of s = 0.6807 seems a bit high in comparison to σ = 0.30, but
parameter
estimates can be quite variable when n is small (and to a
statistician n = 3

is small), so it isn’t anything to worry about.
We won’t live long enough to look at all of the data in the infi
nite popu-
lation, so let’s look at only one million pieces of data and say
that’s
by Stanley N. Deming
AL
20 AMERICAN LABORATORY MARCH 2019
AL
21
exhibit less variability than the raw data. The relationship
between sx-, s,
and n is a “reciprocal square-root” function, the statistician’s
“one-over-
the-square-root-of-n” effect:
Clearly, as n increases, the uncertainty in the mean decreases.
This relationship holds on Saturn, as well, and shows why on
Saturn
there is no uncertainty in the mean—if n = ∞ then σ x- = 0:

This equation can be rearranged to show in general how the
ratio of the
standard deviation of the mean to the standard deviation of the
raw data
decreases as 1/√n :
Figure 4 illustrates this 1/√n effect. Clearly, as n increases, σx -
decreases.
Doing a few replicates can reduce the uncertainty in the mean
by quite
representative enough. The Gaussian distribution in Figure 1
was ob-
tained by drawing at random one million data points (statistical
samples
of size n = 1) from the infinite population with μ = 4.76 and σ =
0.30.
The data have been “binned” to generate the “histogram
distribution”
shown in Figure 1. The bin size is 0.04 on the horizontal axis.
There are
100 bins from 3 to 7. If a sample mean had a value between 4.00
and
4.04, for example, it would be placed in bin number 26. The
height of

each contiguous histogram bar represents the number of data
points
that end up in that bin. Note that the mean of the one million
data points
is 4.760 (to three decimal places), and the standard deviation of
the one
million data points is 0.300 (to three decimal places). Figure 1
is what we
would expect to see for the individual measurements. No
surprises here.
Figure 2 is a little bit diff erent. For this fi gure, we didn’t pull
out only one
data point, but we pulled out two data points at a time and
binned their
means. So, Figure 2 is based on two million data points, or one
million
means for which n = 2. The “grand mean,” the “average of the
averages”
(represented by the symbol x with two bars above it) is equal to
4.760, as
expected, but now we see that the “standard deviation of the
means” sx - =
0.212, less than 0.30. Interesting.
For Figure 3 we pulled out four data points at a time and binned

their
means. The grand mean is again 4.760, but sx- = 0.150, exactly
half of
σ = 0.30 for the raw data. What’s going on here?
When data points are averaged, the negative deviations of some
of the
data points cancel the positive deviations of other data points.
Thus,
the estimated means tend to be closer to the true mean and
therefore
Figure 1 – The distribution of 1,000,000 individual pieces of
data (n = 1)
drawn at random from an infi nite population with μ = 4.76 and
σ = 0.30.
See text for discussion.
Figure 2 – Yellow: the distribution of 1,000,000 means, each
estimated
from two pieces of data (n = 2) drawn at random from an infi
nite popu-
lation with μ = 4.76 and σ = 0.30. Green in background: the
underlying
distribution of raw data. See text for discussion.
STATISTICS IN THE LABORATORY continued

marginal improvement in σx - decreases. Stated differently, the
first few
replicates give a lot of bang for the buck; after that, it gets more
and more
expensive to decrease σx -.
Many researchers want to know how big their sample size
should be (a
legitimate request). Suppose a researcher asks a statistician this
ques-
tion, expecting to get a simple answer: e.g., n = 3. Instead, the
statistician
turns around and silently walks off in disgust. Why do
statisticians be-
have this way? Because they know there is no simple answer to
this
question, and they’re going to have to work with the researcher
to try
to get information that the researcher might not have.
Experience has
taught them that the best time to get out of a bad deal is at the
begin-
ning. They don’t want to go through this excruciating process
again.

The researcher might have a pooled estimate of σ for the
measurement
process, but the researcher’s mean is probably going to be used
to make
a decision. The question then becomes, “How uncertain can the
reported
mean be and still make a good decision?” That is, how small
does σx- have
to be? It’s my opinion that because of the ways companies
compart-
mentalize their functions, the researcher making the
measurements is
often not aware of this last piece of information. It then
becomes the
statistician’s task to move across the company to discover this
piece of
information so the sample size can be determined. If you know
σ and σx -,
you can calculate the sample size n yourself. At this point, you
don’t need
the statistician.
Here’s an example. The percentage of toluene in 500 chemical
samples
of gasoline is to be estimated by making multiple gas

chromatographic
measurements for each gasoline sample and using the sample
mean as
an estimate of the toluene percentage. Each measurement costs
$50.
Previous experience has indicated that individual measurements
have
a standard deviation of 0.10% toluene (this is σ, the method
standard
deviation). However, the client requires a standard deviation of
0.025%
toluene (this will be σx-). How big should your sample size be?
You can almost calculate n in your head. If the ratio of σ x- to
σ is
0.025%/0.10% = 1/4, then √n = 4 and n = 16. You must make
16 replicate
measurements on each of the 500 chemical samples for a total
of 8,000
measurements. But this will cost $400,000. Your client is going
to balk at
this. They’ll ask, “Isn’t there a cheaper way to get the results
we need?”
Of course there’s a cheaper way. To get there, let’s look at an
assumption

statisticians usually make when they solve sample size
questions like
this. They assume σ is what it is, and it can’t be changed. They
then apply
the 1/√n sledgehammer to come up with a sample size, as we
did above.
But statisticians are often wrong about their assumption, and σ
can be
changed. Suppose we bought a better chromatograph that gave
mea-
surements with σ = 0.025% toluene instead of 0.10% toluene.
With that
new chromatograph, the calculation of sample size would be n =
1. Only
500 measurements would be needed, and the cost running the
samples
would be only $25,000.
Figure 5 illustrates the idea. Suppose you start out making 16
measure-
ments per sample ($800/sample) using the old chromatograph
and
you suddenly realize you could save money if you bought a
better

a bit. For example, when n = 4, σx - is decreased by a factor of
2. But to
decrease σx - by another factor of 2, the number of experiments
must be
quadrupled to 16. Clearly, as the number of replicates is
increased, the
Figure 4 – Illustration of the “one-over-the-square-root-of-n”
effect. The
ratio σ x - / σ decreases as 1/√n.
Figure 3 – Yellow: the distribution of 1,000,000 means, each
estimated
from four pieces of data (n = 4) drawn at random from an
infinite popu-
lation with μ = 4.76 and σ = 0.30. Green in background: the
underlying
distribution of raw data. See text for discussion.
AL
samples 101 through 220 (the yellow rectangle labeled
RECOVER). After
that, it’s pure SAVINGS, spending only $50 per sample rather
than $800
per sample (the green rectangle). The total cost of the project

(red area)
will be $190,000 ($90,000 for the chromatograph and $100,000
for the
measurements). This is a lot better than the $400,000 it was
going to
cost. (The total cost would have been only $115,000 if you’d
realized the
benefits of a better chromatograph at the beginning of the
project.)
Don’t try to do with statistics what you can do cheaper with an
improved
measurement method. The 1/√n sledgehammer isn’t always the
best
way to solve sample size problems.
In conclusion: a) σx- is important for most decision-making, b)
you can
make σx- as small as you want by using a large enough sample
size,
c) you can calculate your sample size yourself, and d)
sometimes it’s less
expensive to make σx- small just by using a better measurement
method
with a smaller σ.

I n t h e n e x t m o d u l e w e ’ l l s e e h o w σ x- c a n b
e u s e d t o c a l c u l a t e a
confidence interval for the mean.
Stanley N. Deming, Ph.D., is an analytical chemist
masquerading as a stat-
istician at Statistical Designs, El Paso, Texas, U.S.A.; e-mail:
[email protected]
statisticaldesigns.com; www.statisticaldesigns.com
chromatograph. By the time you’ve finished your 100th sample
(1600
measurements up to this point, an integrated COST of $80,000),
you’ve
put together the funding (the upper yellow rectangle in the
figure,
$90,000) and the new chromatograph you’re purchased has just
arrived.
Starting with sample 101 you use the new chromatograph and
start sav-
ing 15 measurements × $50 per measurement = $750 per
sample, which
you can use to recover the $90,000 cost of the new
chromatograph from
Figure 5 – An illustration of financial considerations when
deciding

whether or not to use a more precise measurement method. See
text
for discussion.
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