Download Gaussian Distribution

Measurements and Noise
Significant Figures
• scientists convey information by the numbers
they report
• 4.21 mL means the 4.2 mL is certain and the
0.01 mL is uncertain
• it is important for you to convey the proper
information when reporting numerical values
Rounding
• 4.2051 is rounded to 4.21
• 4.2049 is rounded to 4.20
• 4.2050 is rounded to 4.2?
– from 4.201, 4.202, 4.203, 4.204 round to 4.20
– from 4.206, 4.207, 4.208, 4.209 round to 4.21
– rounding 4.205 to 4.21 would provide a higher
probability of rounding up (5 cases to 4 cases)
• since there are 4 cases of each
– round up to make an even number
– round down to make an even number
– 50% of the time you will round down (or up)
Rounding
• when doing a compound calculation, never
round until the very end of the calculation
– it is a good idea to record 2 or three extra sig figs
for future calculations
• notebook entry
1.4562 g NaCl  58.54 g/mol = 0.0248752 mol
0. 0248752 mol  0.500 mL = 0.0497504 = 0.04975 M
– (if the extra sig figs are left out...)
0.02487 mol  0.500 mL = 0.04974 M (-0.02% error)
– (if the numbers were rounded to begin with)
1.5g  59 g/mol  0.5 mL = 0.051 M (2.5% error)
Rounding
24  4.52  100.0 = 1.0848 = 1.1
24  4.02  100.0 = 0.9648 = 0.96
• should both answers have 2 sig figs?
• what if the number were 0.999 ?
• should you round up then drop a sig fig?
• if a number barely rolls over into the next
digit, add an extra digit to the sig figs (1.00)
1
Significant Figures
• the standard deviation conveys uncertainty
– round all types of error to 2 sig figs
– since uncertainty lies in the first digit, the second
digit is even more uncertain
– the second digit is useful to prevent rounding
errors
– the third, fourth, etc. digits are completely useless
(in the final answer, but may be useful in
propagation of error calculations)
Error Sig Figs
Error Sig Figs
• sig figs are meant to communicate uncertainty
in a number
• a buret reading of 20.14 mL
– certain of the 20.1 mL
– uncertain of the 0.04 mL
• if a measurement is 20.14 (±1.2) mL
– the 0.04 mL is meaningless (so is the 0.2 mL, really)
– it is OK to use 2 sig figs of uncertainty
– 20.1 (±1.2) mL or 20 (±1) mL is OK
Read About the Following
• 20.1 (±1.2) mL or 20 (±1) mL is OK
• keep the extra sig fig of uncertainty to
prevent rounding errors from increasing
your uncertainty
• guard digit  the extra sig fig in uncertainty
Measurement Statistics
(read on your own)
• the mean (
x)
Use the spreadsheet
– the center of the distribution function AVERAGE
– the number that would result from the
measurement if there were no errors
– represents accuracy
• the standard deviation (s)
– is a measure of the error in a single measurement
2
– has the same units as the mean
 N 
x


 i
N
– represents precision
2
xi   i 1 

Use the spreadsheet
N
s  i 1
function STDEV
N 1
Measurement Statistics
• population (or universe) distribution
– created when N   (infinity)
– yields a mean (), and a standard deviation ()
• since it is impossible to perform an infinite
number of measurements, we are stuck with
the sample distribution
– created when N is small
– yields a mean ( x ), and a standard deviation ( s )
• the sample statistics are usually the same as
the population statistics when N>20 or 30
2
Pooled Standard Deviation
• if data from an instrument or method has been
collected over a period of time, you have a
track-record of the random error
• instead of having to make many replicates in
one sitting (N>20), spooled can be used
 x  x    x  x 
2
s pooled 

What do you
call this?
5000 averaged scans
2
i
set1
Example:
i
set 2
12 averaged scans
What do you
call this?
N1  N 2  2
s ( N1  1)  s22 ( N 2  1)
N1  N 2  2
2
1
Systematic Errors (Bias)
• could be positive or negative (but not at the
same time for the same error)
• effects the accuracy of the measurement
• can sometimes be eliminated
• different types:
– instrumental - did not calibrate instrument
– method errors - problems w/ chemicals, etc.
• strong acid/strong base titration with phenolphthalein
– personal - color blindness, bias in reading scale
Detecting Systematic Errors
• calibrate your instrument
• use standard reference materials
– their purity and composition are certified
• use another method which has been validated
– it’s like a second opinion
• can eliminate with digital meters
Hypothesis Testing
Types of Errors (Summary)
• this is the scientific method
• random or indeterminate  a measure of
precision
• systematic (bias) or determinate a measure
of accuracy
• PIBCAK
• ID10T
–
–
–
–
–
make a hypothesis
make some measurements
do the results support the hypothesis?
if yes, then use the hypothesis again
if no, then abandon the hypothesis
• you will use this when you need to compare
your results to something
– the true value or theoretical value
– to another measurement (another method)
3
Hypothesis Testing (t-testing)
• Null Hypothesis  two numbers (means) are
the same
– is my mean the same as  within error?
– is the mean from my sample the same as the mean
from your sample?
– is the mean from instrument #1 the same as the
mean from instrument #2?
– is the standard deviation from instrument #1 the
same as the standard deviation from instrument
#2?
Comparing An Experimental Mean
and the True Value
• you assay an NIST antacid tablet for CaCO 3
– you get 535  12 mg (N = 4)
– NIST gets 550.0 mg
• are they the same? is there significant bias?
x
t
s
N
where  is the true value, s is the
std. dev. and N is the number of
replicates
• then compare tcalc with ttable from the Student’s t table
• if |tcalc |  ttable then reject the null hypothesis
Comparing An Experimental Mean
and the True Value
• if |tcalc |  ttable then there is a significant bias
(systematic error) in the method
• the systematic error needs to be tracked down and
eliminated
• NOTE: just because the null hypothesis is correct,
does not mean that your method is acceptable (std
error may be too large)
as N  (actually 20
or so), t  z
Comparing An Experimental Mean
and the True Value
• you assay an NIST antacid tablet for CaCO 3
– you get 535  12 mg (N = 4)
– NIST gets 550.0 mg
t
• are they the same?
t
535  550
  2.5
12 / 4
x
s
N
tcrit= ttable= 3.18
(@ 3 DOF and 95%)
• |tcalc |  ttable to reject the null hypothesis
• 2.5 < 3.18, so the numbers are the same
Comparing An Experimental Mean
and the True Value
• what if s is a good estimate
of ?
• then, t becomes z and you
look up ztable from the table
z
x

N
• if |zcalc |  ztable then there is a significant bias
(systematic error) in the method
• the systematic error needs to be tracked down
and eliminated
4
Comparing Experimental Means
• aspirin tablets from two different batches are
assayed for their aspirin content
– batch #1: 328.1  2.6 mg/tablet
– batch #2: 341.5  2.3 mg/tablet
(N=4)
(N=5)
Comparing Experimental Means
– batch #1: 328.1  2.6 mg/tablet
– batch #2: 341.5  2.3 mg/tablet
• are they the same? is there significant bias?
s1 = 2.6
s2 = 2.3
• are they the same?
 x  x    x  x 
2
• since both samples were collected under the
same conditions, use spooled to answer the
question
s pooled 

(N=4)
(N=5)
• are they the same? is there significant bias?
t
x1  x 2
s pooled
N1N2
328.1  341.5

N1  N2
2.433
45
 8.2098
45
• |tcalc |  ttable to reject the null hypothesis
• 8.21>2.36 (7 DOF and 95%), so the null hypothesis is
rejected
• there is a significant different between the two batches
2
i
set1
i
set 2
N1  N 2  2
s ( N1  1)  s22 ( N 2  1)
N1  N 2  2
2
1
spooled = 2.433
Summary
Comparing Experimental Means
– batch #1: 328.1  2.6 mg/tablet
– batch #2: 341.5  2.3 mg/tablet
(N=4)
(N=5)
• compare exp. mean with true
– use the eqn as is
– compare to tcrit from the table
t
x
s
N
• comparing two exp. means (same method)
– change xt to x 2 (bar)
x x
N1 N 2
t 1 2
– find spooled
s pooled N1  N 2
These equation change slightly when you move from
sample statistics to population statistics (s)
Definitions
Summary
• compare the calculated t to the t value in the
table (tcrit) for a given confidence interval and
degrees of freedom (DOF)
• blank  the solution that contains everything
except the analyte
• matrix  everything except the analyte
• |tcalc |  ttable to reject the null hypothesis
• |tcalc | < ttable to accept the null hypothesis
• difference?
– The matrix is in every solution that contains the
sample
– The blank is one solution of the calibration curve
made without the analyte
5
Figures of Merit
• Precision  standard deviation of a
measurement - a measure of random error
– example: white noise or static
• Bias  is measured value higher or lower than
the actual?
– example: analyte loss in processing
• Sensitivity  the slope of the calibration curve
• Limit of Detection (LOD)  how low can you
go? (S is the signal, NOT the stdev (rep. by s or )
why the
difference?
Figures of Merit
• Dynamic Range or Limit of Linearity (LOL) 
linear calibration curve from LOD to some
concentration
• Selectivity  how easy is it to ignore
contaminants? A selective method/instrument
can detect a group of related analytes.
Limit of Detection Example
LOD: sigm = sigbl + 3 bl
LOQ: sigm = sigbl + 10 bl
• examine the error in the y-int of the graph and blank
signal
• take the larger of the two values (blank and error)
– y-int: 0.0161470.0028
– blank: sigbl=0.02130.0018
• Specificity  how selective is the
method/instrument? The ultimate selectivity is
one that is specific for a single analyte.
Limit of Detection Example
0.16
Slope
Std. Dev. in Slope (s m )
Y-intercept
Std. Dev. in Y-int (s b )
0.015178
0.000581
0.016147
0.0028
0.14
R2
0.991293
Error Analysis
Standard error in y (s r )
N (count of standards)
S xx
ybar (avg absorbance)
0.005837
8
1.01E+02
6.56E-02
sigm = sigbl + 3 bl
sigm = 0.0213 + 30.0028 = 0.0297
• this is the minimum signal, so substitute it into the linear eqn:
y=mx+b (where y is the signal)
x = (y-b)/m = (0.0297- 0.016147)/0.015178 = 0.8929
• this is the minimum signal for the LOD
• Repeat for the LOQ (replace 3 with 10): 2.184
y = 0.0152x + 0.0161
R2 = 0.9914
0.12
0.1
0.08
0.06
0.04
0.02
0
0
2
4
6
8
10
12
Concentration (mg/L)
Limit of Detection Example
y-int: 0.0161470.0028
blank: sigbl=0.02130.0018
Calibration Curve
0.18
y = 0.0152x + 0.0161
R2 = 0.9913
0.16
0.14
0.12
Signal
LOD: sigm = sigbl + 3 bl
LOQ: sigm = sigbl + 10 bl
Calibration Curve
0.18
Regression Analysis
Signal
– Qualitative (LOD): Sm = Sbl + 3bl
– Quantitative (LOQ): Sm = Sbl + 10bl
– We’ll do an example in a minute
0.1
0.08
0.06
LOD
LOQ
0.04
0.02
0
0
2
4
6
8
10
12
Concentration (mg/L)
6
External Standards
Methods of Quantitation
• method of external standards - commonly
associated with calibration curves
• method of multiple standard additions - used
when the matrix of the sample is complicated
• produce as series of solutions with known analyte
concentration
• produce a calibration curve (signal vs. conc.)
• measure the signal of the unknown
• use the regression statistics to find unknown
concentration, uncertainty
• advantage - curve can be used for several unknowns
• disadvantage - since std’s are made w/ pure solvent,
matrix is not the same as unknown
External Standards
Calibration Curve
• IMPORTANT: the error in the unknown
concentration is minimized if it is centered in
the calibration curve
0.9
y = 0.0056x - 0.0076
R2 = 0.9998
0.8
unknown signal
(measured)
0.6
0.5
Uncertainty in Calc. x
1
0.4
0.3
unknown concentration
(calculated)
0.2
0.1
0
0
50
100
150
Observed Y
Absorbance
0.7
0.8
0.6
Data
Uncertainty
0.4
0.2
0
200
0
Concentration
50
100
150
200
Calculated x
• disadvantage - must repeat the entire procedure
for every unknown sample
Multiple Standard
Additions
Multiple Standard Additions
• advantage - the sample and standards have the
same matrix, so there is not systematic error
due to a matrix mismatch
step 1:
step 2:
step 3:
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Multiple Standard Additions
(graphing method #1)
1) measure signal of all solutions
2) plot signal vs. volume of standard added
3) find the and use x-intercept (gives the
equivalent volume of the standard that
represents the unknown)
Rephrase: gives the # of mL’s of the standard that, if
diluted to the volume of the original unknown,
would give the concentration of the unknown
(this method is the one discussed in your textbook)
Multiple Standard Additions
(graphing method #1)
1) measure signal of all solutions
2) plot signal vs. conc of diluted standard
3) find the and use x-intercept (gives the equivalent
conc. of the unknown in the diluted volume)
Internal Standard
• A compound that the analyte can be compared to
in quantitation
• example:
– you need to inject a sample into an instrument, but it
is difficult to get the injection volumes from one
injection to another to be exactly the same
Internal Standard
• Example (continued):
– if, after processing, 50% of the internal std. was lost,
then 50% of the analyte was also lost
– thus, the real conc. of analyte is twice the measured
conc.
– variations in the injection volume lead to variations in
the analyte signal
– add a known amount of an I.S. before the injections
– if the injection volume is large, then the I.S. signal is
large (vice versa for smaller injections)
8