Harris Chapter 3: Experimental Error

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Harris Chapter 3 Experimental Error

• Significant Figures
• Significant Figures in Calculations
• Types of Error (Random and Systematic)
• Mean and Standard Deviation (intro)
• Propagation of Random Error
• Propagation of Systematic Error

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Significant Figures

• The minimum number of digits needed to write a
  given value in scientific notation without loss
  of accuracy-textbook
• Accuracy is how close one is to the true or known
  value
• The minimum number of digits that are known via
  a measurement-JCS. This excludes extraneous
  zeros.

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• Significant Figure (digits) Rules
• All known digits are significant
• Numbers should be written in scientific notation
  whenever possible
• Zeros
• To the left of the first non-zero digit are not
  significant
• Decimal placeholders
• To the right of the last non-zero digit are not
  significant (extra zeros)
• UNLESS you know the value is zero-say from a
  balance.
• Zeros in-between non-zero digits are significant
• Extra zeros are usually reported by calculators
  or spreadsheets.

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• 0.000345
• 3.4065E10
• 3.253200000
• 345634000000
• 0.0067865003

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Significant Figures in Arithmetic

• Remember during the next few chapters
• Please Excuse My Dear Aunt Sally
• Addition Subtraction
• The answer contains the number of decimal places
  as the most restrictive value
• Write all numbers in scientific notation
• Convert each to equal powers of 10
• Line up and add or subtract

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Examples

• 54.245323E10 3.52456E8
• .00002325536 - .000009342323256

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• Multiplication and Division
• The answer has the number of total digits as the
  most restrictive term in the calculation.
• One term has 5 significant digits and another has
  9, then the answer must have a total of 5 digits
  in it.
• Examples
• 6.022E23 4.245063
• (4.00 6.022E23) / 423.5335

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Some Terminology Definitions

• Accuracy
• How close you are to the true value.
• How close is the average of a series of
  measurements to the true value.
• Bulls eye examples.
• How many wins do the Mets have?
• Precision
• How reproducible your results are.
• Represented by a small standard deviation.

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• Systematic Error
• Bias or Determinate Error
• Error that is usually introduced by the
  experimenter or apparatus.
• Alters a measurement by some amount that can be
  determined
• Can be identified and usually eliminated.
• Parallax
• Uncalibrated balance
• Always reading analog dials (e.g. speedometer)
  too high
• Dirty glassware (water adheres in the neck of a
  vol. flask above the mark.

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Classical sources of systematic error
(determinate error, bias)

• Not reading meniscus consistently
• Not weighing by difference
• Errors in making calibration standards and then
  not checking the standards against another source
• Errors in electronic circuits (CHEM 362)
• Estimating the position of a meter dial or
  display

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Example..

• A balance has not been calibrated (checked) for a
  while.
• The balance is tared (zeroed) and a sample
  weighed. It reads 0.5000 grams.
• How do you know it is 0.5000 grams? Are you going
  to trust your grade to the balance???
• What if the balance has some fault and it always
  reads 0.1000 g too high?

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• A better choice.
• Weigh a piece of weighing paper on the balance,
  record the weight (say it is 0.2000 grams
  according to the balance)
• Add your sample and weigh it (say it reads 0.7000
  grams).
• Subtract the two. The 0.1000 gram bias appears in
  both values and then is subtracted out!
• You have eliminated a key source of systematic
  error in weighing!

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Other ways to eliminate systematic error..

• Analyze a known standard, or standard reference
  material
• You should get a value close to the true or known
  value for a standard that is widely accepted
• Standard Reference Materials-NIST (almost
  anything)
• USGS Standards (rocks)
• Privately prepared reference standards
  (calibration checks, etc.)

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• Have a second lab or analyst check your results.
• They are unlikely to make the same mistakes or
  have the same bias as you.
• Analyze a blank.
• Blanks have little or no analyte, but otherwise
  all the same components as your sample. You
  should get a very low value for the analyte in
  your blank.
• Vary sample size. You should get the same
  concentration, even if larger samples have more
  analyte in them and smaller samples have less.

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• Random or Indeterminate Error
• Error that is uncontrollable and whose source is
  unidentifiable
• Sometimes results in higher values, sometimes
  lower values
• You saw this in the Ocean Optics spectrometer
  readings.
• Can not be eliminated
• Can be reduced by taking multiple measurements
  and calculating the mean.

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Absolute and Relative Uncertainty

• Absolute uncertainty is uncertainty or error
  expressed in the same units as the value.
• A buret reads 50.00 /- 0.02 mL
• Your car is traveling at 67 /- 2 mph
• A sample weighs 5.0024 /- 0.0001 g

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• Relative Uncertainty
• Uncertainty expressed as a fraction of the value
  itself (unitless)
• Relative Uncertainty
• Uncertainty expressed as a percentage of the
  value itself (units of percent)

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Lets work some examples..
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Propagation of Random Error

• We will work on this by example.
• Remember PEMDAS
• Order of operations will dictate order of error
  propagation steps
• Addition/Subtraction Multiplication/Division
  covers most operations
• Review propagation of uncertainty in logs and in
  systematic error yourself.

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Note the error in the square root symbols. The
book is correct.
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• Examples
• 4.305 ? 2.056 g 34235.03 ? 2.40 g
• 343.325 ? 23.456 s x 74.43563 ? 0.23564 s
• (54.345 ? 1.043 g - 12.001 ? .001 g) / 34.053
  ? 1.003 g/mole
• Dont forget units, and significant figures too!

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• The real rule on significant figures, etc.
• There is often fair arguments over the true
  number of significant figures
• Complex calculations
• Chemists experience
• Often there is one extra digit that is kept, to
  minimize rounding errors
• 82/80 example from text.
• I expect the correct number of significant
  figures.
• In more complex calculations you may have one
  more or less without penalty.