Transcript January 30, 2007 - Lecture 3 Notes

Chapters 3
Uncertainty
January 30, 2007
Lec_3
Outline


Homework Chapter 1
Chapter 3


Experimental Error
“keeping track of uncertainty”
Start Chapter 4

Statistics
Homework
Chapter 1 – “Solutions and Dilutions”
Questions: 15, 16, 19, 20, 29, 31, 34
Chapter 3
Experimental Error
And propagation of
uncertainty
Keeping track of uncertainty
Significant Figures
35.21 ml
Propagation of Error
35.21 (+ 0.04) ml
Suppose
You determine the density of some mineral by
measuring its mass

4.635 + 0.002 g
And then measured its volume

1.13 + 0.05 ml
g
mass( g )
 4.1018

ml
volume(ml )
Significant Figures (cont’d)

The last measured digit always has
some uncertainty.
3-1 Significant Figures

What is meant by significant figures?
Significant figures:
Examples
How many sig. figs in:

a.
b.
c.
d.
e.
3.0130 meters
6.8 days
0.00104 pounds
350 miles
9 students
“Rules”
All non-zero digits are significant
Zeros:
1.
2.
a.
b.
c.
3.
Leading Zeros are not significant
Captive Zeros are significant
Trailing Zeros are significant
Exact numbers have no uncertainty
(e.g. counting numbers)
Reading a “scale”
What is the “value”?
When reading the scale of any apparatus, try to estimate to
the nearest tenth of a division.
3-2
Significant Figures in Arithmetic

We often need to estimate the uncertainty of
a result that has been computed from two or
more experimental data, each of which has a
known sample uncertainty.
Significant figures can provide a marginally
good way to express uncertainty!
3-2
Significant Figures in Arithmetic

Summations:

When performing addition and subtraction report
the answer to the same number of decimal places
as the term with the fewest decimal places
+10.001
+ 5.32
+ 6.130
?
Try this one
+
1.632 x 105
4.107 x 103
0.984 x 106
0.1632
x 106
0.004107 x 106
6
0.984
x
10
+
3-2
Significant Figures in Arithmetic

Multiplication/Division:

When performing multiplication or division report
the answer to the same number of sig figs as the
least precise term in the operation
16.315 x 0.031 = 0
? .505765
0.51
3-2
Logarithms and Antilogarithms
From math class:
log(100) = 2
Or log(102) = 2
But what about significant figures?
3-2
Logarithms and Antilogarithms
Let’s consider the following:
An operation requires that you take the log of
0.0000339. What is the log of this number?
log (3.39 x 10-5) =
3-2
Logarithms and Antilogarithms
Try the following:
Antilog 4.37 =

“Rules”
Logarithms and antilogs
1. In a logarithm, keep as many digits to the
right of the decimal point as there are
sig figs in the original number.
2. In an anti-log, keep as many digits are
there are digits to the right of the
decimal point in the original number.

3-4. Types of error

Error – difference between your answer and the
‘true’ one. Generally, all errors are of one of three
types.



Systematic (aka determinate) – problem with the
method, all errors are of the same magnitude and
direction (affect accuracy)
Random – (aka indeterminate) causes data to be
scattered more or less symmetrically around a mean
value. (affect precision)
Gross. – occur only occasionally, and are often large.
Absolute and Relative Uncertainty

Absolute uncertainty expresses the margin of
uncertainty associated with a measurement.
Consider a calibrated buret which has an
uncertainty + 0.02 ml. Then, we say that the
absolute uncertainty is + 0.02 ml
Absolute and Relative Uncertainty

Relative uncertainty compares the size of the
absolute uncertainty with its associated
measurement.
Consider a calibrated buret which has an
uncertainty is + 0.02 ml. Find the relative
uncertainty is 12.35 + 0.02, we say that the
relative uncertainty is
absolute uncertaint y
Relative Uncertain ty 
magnitude of measuremen t
3-5. Estimating Random Error
(absolute uncertainty)

Consider the summation:
+ 0.50 (+ 0.02)
+4.10 (+ 0.03)
-1.97 (+ 0.05)
2.63 (+ ?)
s y  s  s  s  ...
2
a
2
b
2
c
3-5. Estimating Random Error

Consider the following operation:
4.10(0.02)  0.0050(0.0001)
 0.010406( ?)
1.97(0.04)
2
2
2
 sa   sb   sc 
          ...
y
a b c
sy
Try this one
14.3(0.2)  11.6(0.2) 0.050(0.001)
820(10)  1030(5) 42.3(0.4)
3-5. Estimating Random Error

For exponents
For
ya
uncertaint y in a is Sa
x
 sa 
 x 
y
a
sy
3-5. Estimating Random Error

Logarithms
antilogs
For
For
y  log a
y  anti log a
uncertaint y in a is Sa
uncertaint y in a is Sa
 sa 
s y  0.434 
a
sy
y
 2.303sa
Question
Calculate the absolute
standard deviation for a
the pH of a solutions
whose hydronium ion
concentration is
2.00 (+ 0.02) x 10-4

y  log a
uncertaint y in a is Sa
 sa 
s y  0.434 
a
Question

y  anti log a
Calculate the absolute
value for the hydronium uncertaint y in a is Sa
ion concentration for a
sy
 2.303sa
solution that has a pH of
y
7.02 (+ 0.02)
[H+] = 0.954992 (+ ?) x 10-7
Suppose
You determine the density of some mineral by
measuring its mass
s
s  s  s 
2
2
2
  a    b    c   ...
y
a b c
y

4.635 + 0.002 g
And then measured its volume

1.13 + 0.05 ml
mass( g )

volume(ml )
What is its uncertainty?
 4.1018
=4.1 +0.2 g/ml
The minute paper
Please answer each question in 1 or 2
sentences
1)
2)
What was the most useful or meaningful thing
you learned during this session?
What question(s) remain uppermost in your
mind as we end this session?
Chapter 4
Statistics
General Statistics Principles

Descriptive Statistics


Inductive Statistics


Used to describe a data set.
The use of descriptive statistics to accept or reject
your hypothesis, or to make a statement or
prediction
Descriptive statistics are commonly reported
but BOTH are needed to interpret results.
Error and Uncertainty

Error – difference between your answer and
the ‘true’ one. Generally, all errors are of one
of three types.



Systematic (aka determinate) – problem with the
method, all errors are of the same magnitude and
direction (affect accuracy).
Random – (aka indeterminate) causes data to be
scattered more or less symmetrically around a
mean value. (affect precision)
Gross. – occur only occasionally, and are often
large. Can be treated statistically.
The Nature of Random Errors



Random errors arise when a system of
measurement is extended to its maximum
sensitivity.
Caused by many uncontrollable variables that
are an are an inevitable part of every physical
or chemical measurement.
Many contributors – none can be positively
identified or measured because most are so
small that they cannot be measured.
Random Error


Precision describes the closeness of data
obtained in exactly the same way.
Standard deviation is usually used to
describe precision
Standard Deviation

Sample Standard deviation

Population Standard
(for use with small samples
n< ~25)
deviation (for use with
samples n > 25)


U = population mean
IN the absence of
systematic error, the
population mean
approaches the true value
for the measured quantity.
( xi  x )
s
n 1
2
( xi   )

N
2
Example

The following results were obtained in
the replicate analysis of a blood sample
for its lead content: 0.752, 0.756,
0.752, 0.760 ppm lead. Calculate the
mean and standard deviation for the
data set.
Standard deviation

0.752, 0.756, 0.752, 0.760 ppm lead.
x  0.755
Distributions of Experimental
Data


We find that the distribution of replicate
data from most quantitative analytical
measurements approaches a Gaussian
curve.
Example – Consider the calibration of a
pipet.
Replicate data on the
calibration of a 10-ml pipet.
Trial
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
Volume
9.988
9.973
9.986
9.980
9.975
9.982
9.986
9.982
9.981
9.990
9.980
9.989
9.978
9.971
9.982
9.983
9.988
Mean
9.982 ml
median
9.982 ml
spread
0.025 ml
Standard Deviation
Trial
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
0.0056 ml
Volume
9.975
9.980
9.994
9.992
9.984
9.981
9.987
9.978
9.983
9.982
9.991
9.981
9.969
9.985
9.977
9.976
9.983
Trial
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
Volume
9.976
9.990
9.988
9.971
9.986
9.978
9.986
9.982
9.977
9.977
9.986
9.978
9.983
9.980
9.983
9.979
Frequency distribution
Volume
Range, mL
Number in Range
% in range
9.969 to 9.971
3
9.982 to 9.974
1
9.975 to 9.977
7
9.978 to 9.980
9
9.981 to 9.983
13
9.984 to 9.986
7
9.987 to 9.989
5
9.990 to 9.992
4
9.993 to 9.995
1
6
2
14
18
26
14
10
8
2
1 ( x )2 / 2 2
y
e
 2
14
Number of measurements
12
10
8
6
4
2
0
9.965
9.970
9.975
9.980
9.985
9.990
Range of measured values
9.995
The minute paper
Please answer each question in 1 or 2
sentences
1)
2)
What was the most useful or meaningful thing
you learned during this session?
What question(s) remain uppermost in your
mind as we end this session?