Chapters 3 Uncertainty

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Chapters 3Uncertainty

• January 30, 2007
• Lec_3

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Outline

• Homework Chapter 1
• Chapter 3
• Experimental Error
• keeping track of uncertainty
• Start Chapter 4
• Statistics

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Homework

• Chapter 1 Solutions and Dilutions
• Questions 15, 16, 19, 20, 29, 31, 34

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Chapter 3

• Experimental Error
• And propagation of uncertainty

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Keeping track of uncertainty
Significant Figures
Propagation of Error
35.21 ml
35.21 ( 0.04) ml
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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

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Significant Figures (contd)

• The last measured digit always has some
  uncertainty.

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3-1 Significant Figures

• What is meant by significant figures?
• Significant figures

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Examples

• How many sig. figs in
• 3.0130 meters
• 6.8 days
• 0.00104 pounds
• 350 miles
• 9 students

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Rules

• All non-zero digits are significant
• Zeros
• Leading Zeros are not significant
• Captive Zeros are significant
• Trailing Zeros are significant
• Exact numbers have no uncertainty
• (e.g. counting numbers)

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Reading a scale
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What is the value?
When reading the scale of any apparatus, try to
estimate to the nearest tenth of a division.
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3-2Significant 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!

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3-2Significant 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

?
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Try this one

• 1.632 x 105
• 4.107 x 103
• 0.984 x 106

0.1632 x 106 0.004107 x 106 0.984 x
106


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3-2Significant 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
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3-2Logarithms and Antilogarithms

• From math class

log(100) 2 Or log(102) 2 But what about
significant figures?
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3-2Logarithms and Antilogarithms
Lets 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)

• log (3.39 x 10-5)

• log (3.39 x 10-5)

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3-2Logarithms and Antilogarithms

• Try the following
• Antilog 4.37

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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.

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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.

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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

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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

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3-5. Estimating Random Error (absolute
uncertainty)

• Consider the summation
• 0.50 ( 0.02)
• 4.10 ( 0.03)
• -1.97 ( 0.05)

2.63 ( ?)
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3-5. Estimating Random Error

• Consider the following operation

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Try this one
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3-5. Estimating Random Error

• For exponents

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3-5. Estimating Random Error

• Logarithms antilogs

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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

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Question

• Calculate the absolute value for the hydronium
  ion concentration for a solution that has a pH of
  7.02 ( 0.02)
• H 0.954992 ( ?) x 10-7

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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

What is its uncertainty?
4.1 0.2 g/ml
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The minute paper

• Please answer each question in 1 or 2 sentences
• 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?

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Chapter 4

• Statistics

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General Statistics Principles

• Descriptive Statistics
• Used to describe a data set.
• Inductive Statistics
• 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.

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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.

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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.

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Random Error

• Precision describes the closeness of data
  obtained in exactly the same way.
• Standard deviation is usually used to describe
  precision

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Standard Deviation

• Sample Standard deviation (for use with small
  samples nlt 25)
• Population Standard deviation (for use with
  samples n gt 25)
• U population mean
• IN the absence of systematic error, the
  population mean approaches the true value for the
  measured quantity.

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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.

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Standard deviation

• 0.752, 0.756, 0.752, 0.760 ppm lead.

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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.

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Replicate data on the calibration of a 10-ml
pipet.
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Frequency distribution
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The minute paper

• Please answer each question in 1 or 2 sentences
• 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?