Notes on Statistics
By S.E. Van Bramer
March 8, 1995
Indeterminate (random) error: evaluate with statistics
Determinate (systematic) error: evaluate with reference standards.
Gross error: big mistake, like spilling everything on the floor.
One sided probability: Use one sided probability if comparing the size or
magnitude from two different data sets (ie. a is larger than b).
Two sided probability: Use two sided probability if comparing two different
data sets for a difference (ie. a is different than b).
Population: This refers to a set of all possible measurements. This is an ideal
that can only be approached. Greek letters are used to symbolize population
Sample: This refers to a set of actual measurements. The distinction between
sample and population statistics is most important for a small number of
measurements (less than 20).
t-Test: This is one of the most powerful and widely used statistical tests. The ttest (Student's t) is used to calculate the confidence intervals of a measurement
when the population standard deviation ()) is not know. Which is usually the
case. The t-test is also used to compare two averages. The t-test corrects for the
uncertainty of the sample standard deviation (s) caused by taking a small number
Action Limit; Lc 2), 97.7% certain that signal observed is not random
Detection Limit; LD 3), 93.3% certain to detect signal above the 2) action
limit when the analyte is at this concentration.
Quantitation Limit; LQ 10), Signal required for 10% RSD.
Type I Error; Type I error is identification of random noise as signal.
Type II Error; Type II error is not identifying signal that is present.
Descriptive Statistics. These statistics are used to describe a population or a
Population Mean (µ) and Sample Average (x, or x)
Standard Deviation: measurement of the spread in individual data points to
reflect the uncertainty of a single measurement.
Population standard deviation ()). For large sample sets (usually more
than 20 measurements) or when the population mean (µ) is known.
M (xi µ)
Sample standard deviation (s). For small sample sets (usually less than 20
measurements) when the sample average (x) is used.
M (xi x )
Pooled standard deviation (spooled ). When several small sets have the
same sources of indeterminate error (ie: the same type of measurement but
different samples) the standard deviations of the individual data sets may
be pooled to more accurately determine the standard deviation of the
x1) M (xj
Standard Error of the Mean ()m). The standard error of the mean is the
uncertainty in the average. This is different from the standard deviation ()),
which is the variation for each individual measurement. Notice that when N is 1
(a single measurement) )m = ).
If ) is known, the uncertainty in the mean is:
If ) is unknown, use the t-score to compensate for the uncertainty in s.
The value for t is obtained from a table for appropriate % confidence level
and for N-1 degrees of freedom. (N-1 because one degree of freedom is
used to calculate the mean.) Since the uncertainty is a range that could be
greater or less than the mean, a two-sided value should be used for t.
z-Score. Normalizes data points so that the average is 0 and the standard
deviation is 1. The cumulative normal distribution (z-score) shows what
percentage of a normal distribution is bounded by a given value for z. One sided
distributions, the distribution is the area from - and z. Two sided values give the
area between ± z. To illustrate this:
For a 1 sided distribution 97.72% of all data points will be less than 2
standard deviations above the average.
For a 2 sided distribution 68.28% of all data points will be between ± 1
standard deviation from the average.
Table 1. Cumulative Normal Distribution. The area under
a gaussian distribution where z is the population standard
Confidence Interval. The confidence interval is the preferred method for
describing the range of uncertainty in a value. The confidence interval is
expressed as a range of uncertainties at a stated percent confidence. This percent
confidence reflects the percent certainty that the value is within the stated range.
If the population standard deviation ()) is known. The standard error of
the mean ()m) combined with the z-score (from a table for the desired
Confidence Level) is used to express the uncertainty in the mean as a
range. This is the confidence interval at the stated certainty. The
percentage used should always be stated. This method is widely used to
report results with a percent certainty and is expressed as follows:
x ± z
Table 2. Values of z for given Confidence Level.
Confidence Level (%)
z (2 sided)
z (1 sided)
If the population standard deviation ()) is unknown, the sample standard
deviation (s) may be used to estimate the confidence interval. This is the
preferred method for reporting the uncertainty in experimental results. It
takes into account the number of measurements made, the variance in the
measurements, and expresses the range at the stated percent confidence
Based upon the confidence interval calculated above, an experimental
result should be expressed as:
5.3 ± 1.2 at the 95% confidence level
NOTE: Values for Student's t are given in Table 3.
Comparison Tests. These tests are used to compare averages to determine if there is
a significant difference between two values.
Comparing the sample to the true value. Method #1. The t-test is used to
determine if there is a significant difference between an experimental average and
the population mean (µ) or "true value". This method is used to compare
experimental results to quality control standards and standard reference materials.
This comparison is based upon the confidence interval for the sample mean
calculated above. If the difference between the measured value and the true value
is greater than the uncertainty in the measurement, there is a significant difference
between the two values at that confidence level. This may be expressed
mathematically that IF:
Then there is no significant difference at the stated confidence level. This could
be stated as "there is no significant difference between the experimental results
and the accepted value for the Standard Reference Materials at the 95%
Comparing the sample to the true value, Method #2. This is same test as
above, but it is often easier to understand the meaning of the test by calculating an
experimental value for t (t experimental). Then the experimental t-score (texperimental ) is
compared to t-critical (t c), the value of t found in a table. texperimental is calculated
There is a significant difference between the sample average and the true value if
texperimental is greater than tc. tc is chosen for N-1 degrees of freedom at the desired
percent confidence interval. If the experimental value may be greater or less than
the true value, use a two sided t-score. If specifically testing for a significant
increase or decrease (but not both) use a single sided value for t c.
Comparing two experimental averages. The t-test may also be used to compare
two experimental averages. This is most accurately done by using the pooled
standard deviation and calculating t experimental as:
If texperimental is greater than tcritical then there is a significant difference between the
two means. tcritical is determined at the appropriate confidence level from a table of
the t-statistic for N 1 + N2 - 2 degrees of freedom.
Table 3. tc for Normally Distributed Data.
1.325 1.725 2.086 2.528 2.845
1.282 1.645 1.960 2.326 2.576
Q-test. Use to identify statistical outliers in data. This test should be applied sparingly
and never more than once to a single data set.
R is the range of all data points, xa is the suspected outlier, and xb is the data point closest
to xa. At the 90% confidence interval Q for N replicate measurements is:
Table 4 Q test decision level at 90% confidence interval
Linear Regression. Fit the line y = mx + b to linear data where x is the dependent
variable, y is the independent variable, and x i is the i'th data point, and N different
standards are used. yave is the average of the y values for the standards, and xave is the
average of the x values for the standards. This method assumes that there is no variance
in the value for x.
( x i )2
x i yi
Assuming linear function and no replicates, the standard deviation about the
Uncertainty in ypredicted:
Uncertainty in xpredicted for an unknown with an average signal yunk from M
Error Analysis. The following techniques are used to determine how error
propagates through an experimental procedure. This method is based upon combining
the uncertainty for each step.
Table 5. Error Propagation.
Addition and Subtraction
Multiplication and Division
x = a*b/c
x = ab
x = log10 a
x = ln a
x = antilog10 a
x = ea
sx = (sa2 + sb2 + sc2 ...)½ .
sx = x * [(sa/a)2 + (sb/b)2 + (sc/c)2]½
sx = x * b * (sa/a) (no uncertainty in b)
sx = 0.434 * (sa/a)
sx = sa/a
sx = 2.303 * x * sa
sx = x * sa
From a calibration curve the concentration of an unknown is 16±2 ppm
The solution was prepared by:
dissolving 0.0452±0.0001 g of compound
in 250.0±0.1 ml of water
From this the weight of unknown in the compound is:
x = 16 ppm * 250 ml * (mg/l) * (1 g/1000 mg) * (1 l/1000 ml)
= 0.0040 g of unknown in the compound
The uncertainty in this weight is:
sx = x * [(sa/a)2 + (sb/b)2 + (sc/c)2]½
= 0.0040 * [(2/16)2 + (0.1/250)2]½
= 0.0040 * (0.0156 + 0.0004)½ (NOTE: concentration uncertainty is limiting)
Therefor the weight of the unknown is:
0.0040±0.0005 g (or 4.0±0.5 mg)
Howard, M.; Workman, J. Statistics in Spectroscopy; Academic: Boston, 1991.
Box, G.; Hunter, W.; Hunter, J. Statistics for Experimenters; Wiley: New York:
Akhnazarova, S.; Kafarov, V. Experiment Optinization in Chemistry and
Chemical Engineering; MIR: Moscow, 1982.
Skoog, D.; West, D.; Holler, F. Analtyical Chemistry, An Introduction; Saunders:
Statistics Problem Set #1
ENVR 303, 1995. Dr. Van Bramer
Use the following data sets for the calculations in this problem set. These data have a random normal
distribution. These are all concentrations (ppb) for replicate samples:
Set #1, a large sample set: 25.160, 25.227, 24.402, 23.924, 20.730, 23.615, 23.648, 23.747,
23.613, 22.910, 25.075, 24.301, 24.611, 25.133, 24.152, 24.196, 24.775, 23.841, 24.883,
Set #2, a small sample set: 22.143, 22.640, 24.084, 23.135, 24.967
A pair of small data sets
Set #3; 11.892, 10.491, 10.172, 12.480, 11.095
Set #4; 17.656, 16.874, 17.999, 17.825, 19.525, 17.712
Determine the average and standard deviation for each data set.
Determine the standard error of the mean for each data set.
Determine the following confidence intervals (2 sided) for each data set.
For the Set #1 and Set#2; compare the means, and determine if there is a significant
difference between the two sets at the 90, 95, and 99% Confidence Interval.
For Set #3 and Set #4; Compare the means, and determine if there is a significant
difference between the two sets at the 90, 95, and 99% Confidence Interval.
For the experiments with Set #3, and Set #4, I am really interested in finding if Set #4 is
significantly larger than Set #3. Determine if this is true at the 90, 95, and 99%
Confidence Interval. Think about what this means in terms of the t-score.
The true values for the data sets are:
Set #1 24.0
Set #2 24.0
Set #3 12.0
Set #4 17.5
Determine if there is a significant difference between the true value and the measured
value for each set at the 80, 90, 95, and 99% Confidence Interval.
z-Score. For set #1, normalize the data by z-scoring.
Rejection of outlayers. Use the Q-test to determine if 19.525 is an outlier in the second small data
Statistics Problem Set #2
ENVR 303, 1995. Dr. Van Bramer
Given the following data for mercury concentration in a soil sample, construct a
calibration curve, using linear regression determine the unknown concentration. The
sample was prepared from a 1.3452 g of a sample (weighed on an analytical balance).
Extracting the mercury, and diluting the sample to 100.0 mL in a class A volumetric
Based upon propogation of error, what is the concentration of mercury in the soil
How would you verify the accuracy of this determination.
Based on the replicat samples, what are the different LOD's. How many replicate
samples would be required to measure a 0.001 ppm sample with 10% RSD?
0.23143, 0.22492, 0.22656, 0.27000
A unknown sample was prepared for analysis by inductively coupled plasma emission.
Solution A; 0.0153±0.0001 g of unknown was weighed out on an analytical balance,
dissolved and diluted to 250.0±0.1 ml in a volumetric flask. Solution B; 100.0±0.2 ml of
solution A was transferred to a second flask and diluted to 250.0±0.1 ml. Solution C;
0.0021±0.0001 g of iron was dissolved, 100.0±0.2 ml of solution A was added, and the
solution diluted to 250 ml. Giving the following data, what is the weight percent of Fe in
the unknown sample? Based upon the replicate samples and propagation of error what is
the uncertainty in this value? Report your results as ± 90% confidence interval using the
t-test. (10 points)
S.E. Van Bramer
March 16, 1995
Problem Set #1, discussion
I have already posted some of the answers to the statistics problem set, but I also want you to
spend some time thinking about what those answers mean. In an effor to address that, I am
writing this document.
Data sets: What you see here is just a list of numbers. What do these numbers
correspond to. What are they a measure of. When you apply statistical tests to your own
data, you need to think about this. This is important, because it determins what you
measure the uncertainty of. For example, lets say that set #1 is measurement of lead in
If the set represents samples from 20 different locations in a lake, then you are
measuring the variability in the lake at the time you took the samples.
If the set represents samples of tap water taken on 20 consecutive days, then you
are measuring the day to day variablity of the tap water.
If the set represents samples of tap water taken one right after the next, then you
are measuring the variability of the tap water over this time frame.
If you took a single water sample and are measuring that sample 20 times, then
you are measuring the variablity of the analysis.
All of these measurements represent very different types of information. I want to make
two points. First, think carefully about what it is that you want to find out and design
your experiment accordingly. Each of the above experiments provides distinctly different
information. What are you interested in? Second, when you describe your experiment to
someone else be careful about what you say. Say what you did and what you measured.
Do not make a claim for a type of information that you did not obtain. ie: from
experiment d above, you can say that the concentration of lead in your sample was x +/- s.
but you do not know anything about how much that sample will vary from one taken a
month later at the same place.
Descriptive Statistics: These are just statistical tools that you use to represent your data
set. It is a shorthand to make life easier. Instead of having to list off all twenty values
from set #1, you can describe the set with statistics. When you do this, just be certain to
tell exactly which statistics you use so someone else can tell what you did.
POOR, Set #1 is 24.175 ± 1.066.
Better, Set #1 is 24.175 ± 1.066 (sample standard deviation from 20 replicat
Best, The average of Set #1 is 24.175 ± 0.467 at the 95% confidence interval.
Comparison Tests: These statistical tools are used to compare two things to see if there
is a significant difference. You can not tell if two sets are the same, only if they are
"significantly different". Some examples of what is found
Comparing set #1 to the "true value". t experimental = 0.735. Then compare this to t c
from the table. Since you are comparing for a difference, this is a two sided test.
You have 20 samples, so you have 19 degrees of freedom. Looking in the table
that I gave you, the closest that you can find is 20 degrees of freedom (pretty
close, if you want to do better interpolate between the values given, or go to a
bigger table in a different source). From the table, t c = 1.64. From this you can
say "Based upon this experiment there is no significant difference between Set #1
and the accepted value at the 90% confidence interval".
Comparing Set #3 and Set #4. Now you are comparing two experimental values.
First, you can improve your estimate of the population standard deviation by
pooling the standard deviation from each data set. Then you may used this pooled
standard deviation to determine t experimental = 12.146. From the table, find t c (for N1
+ N2 - 2 degrees of freedom) = 2.262 (95% CI, 2 sided). Based upon this you can
say "There is a significant difference between Set #3 and Set #4 at the 95%
Alternatively, if you expect Set # 4 to be larger, you may use a 1 sided t c = 1.833
(95% CI, 1 sided). Based upon this you can say "Set #4 is significantly larger than
Set #3 at the 95% confidence interval."
z-Score: Just to review from class, using Set #3 as an example, what I was looking for
here could be shown as:
Avg 11.226 0 ...
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