Download Significant Figures Rounding Rounding Rounding Significant

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)
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)
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)
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
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Error Sig Figs
• 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
i 1
s
function STDEV
N 1
Read About the Following
• Accuracy, precision
– How sample statistics relate to accuracy & precision
•
•
•
•
average: μ or x
standard deviation: σ or s
median
range
– How the following errors relate to accuracy &
precision
• random errors
• systematic errors
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
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
2
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)
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?
as N  (actually 20
or so), t  z
3
Comparing An Experimental Mean
and the True Value
• you assay an NIST antacid tablet for CaCO3
– 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)
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 An Experimental Mean
and the True Value
• you assay an NIST antacid tablet for CaCO3
– you get 535  12 mg (N = 4)
– NIST gets 550.0 mg
• 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
Comparing Experimental Means
– batch #1: 328.1  2.6 mg/tablet
– batch #2: 341.5  2.3 mg/tablet
(N=4)
(N=5)
• 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
t
s pooled 

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
4
Summary
Comparing Experimental Means
– batch #1: 328.1  2.6 mg/tablet
– batch #2: 341.5  2.3 mg/tablet
• compare exp. mean with true
(N=4)
(N=5)
• are they the same? is there significant bias?
t
x1  x 2
s pooled
328.1  341.5
N1N 2

2.433
N1  N 2
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
– 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
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 )
– Qualitative (LOD): Sm = Sbl + 3bl
– Quantitative (LOQ): Sm = Sbl + 10bl
– We’ll do an example in a minute
why the
difference?
5
Figures of Merit
• Selectivity  how easy is it to ignore
contaminants? A selective method/instrument
can detect a group of related analytes.
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
• Dynamic Range or Limit of Linearity (LOL) 
linear calibration curve from LOD to some
concentration
Limit of Detection Example
0.1
0.08
0.06
LOD
LOQ
0.04
0.02
0
0
2
4
6
8
10
12
Concentration (mg/L)
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
External Standards
• 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
6
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
2
R = 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
• advantage - the sample and standards have the
same matrix, so there is not systematic error
due to a matrix mismatch
• disadvantage - must repeat the entire procedure
for every unknown sample
Multiple Standard
Additions
Multiple Standard Additions
step 1:
step 2:
step 3:
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)
7
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)
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