Measurements

1
Measurements Quantitative Reasoning
2
Performing Experiments

• Experiments must be repeatable requires careful
  control over variables
• Possible outcomes of an experiment
• The experiment may support the theory
• We then continue to make predictions and test
  them
• The experiment may falsify the theory
• We need a new theory that describes both the
  original data and the results of the new
  experiment
• Since we cannot do every possible experiment, a
  theory can never be proven true it can only be
  proven false

3
Making Measurements

• Errors
• Random
• Systematic
• With every measurement, it is essential to
  provide an estimate of the uncertainty the
  likely range of errors
• Example
• Using a ruler marked in mm, we round to the
  nearest marking at most off by half a division,
  or 0.5 mm
• Cite a measurement of 15 mm as 15 ? 0.5 mm to
  indicate that the real value of the length is
  likely to be anywhere between
• 14.5 mm and 15.5 mm
• If a theory predicts a value of 15. 2 mm, then a
  reading of
• 15 ? 0.5 mm is in agreement with the theory
  but a reading of
• 15 ? 0.1 mm is probably not

4
Activity Analyzing the Results

• Why arent all the results the same?
• How do we compare results?
• What kind of errors occurred?
• Is our error small or big?
• Is our result precise, accurate or what?
• So, do our results agree?

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Why arent all the results the same?

• Useful questions to ask, if results dont agree
• Which object did you measure?
• What units where you using?
• What is your estimated error?

6
How do we compare results?

• Dont compare apples with oranges!
• Need to use the same units
• 1 inch 2.54 cm
• To get inches from centimeters, divide by 2.54
• To get centimeters from inches, multiply by 2.54

7
What kind of errors occurred?

• There are systematic and random errors
• To beat down random errors, measure the same
  thing many times, and the errors will even out,
  i.e. the overall error will be smaller
• The systematic error can be reduced by doing a
  better experiment, or understanding your
  instruments better (miscalibrations etc.)
• Human error is not an acceptable error source in
  science! It just means you are a bad experimenter.

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Is our error small or big?

• It depends!
• If you have a small error and the measured length
  is also small, you might have a huge error!
• Use percentages
• Percent error (estimated error)/(result) x 100
• Example 51.3 cm 0.2 cm gives
• Percent error (0.2 cm)/(51.3cm) x 100 0.4
• 
  (This is a pretty small error)

9
Is our result precise or accurate or what?

• Two different concepts precision and accuracy!
• High precision means small error
• High accuracy means close to an accepted value
• Examples

• high
  precision, high accuracy
• high
  precision, low accuracy
• low
  precision, high accuracy
• low
  precision, low accuracy

accepted value
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So, do our results agree?

• Results agree, if they are within the error
  margins of each other
• Examples
• O O
  
• values very different, but errors large
  agreement!
• O O
• values closer, but errors smaller no agreement!

11
Quantitative Reasoning

• Amazingly powerful tool to understand the world
  around us
• Fundamentals
• Ratios
• Graphs
• Area Volume
• Scaling
• Arithmetical statements

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Achieving Scientific Literacy(Arons Article)

• Two types of knowledge
• Declarative (Learned Facts, book knowledge)
• Operative (actually knowing how to solve
  problems)
• Trouble with GenEd courses
• Too much in too little time
• Getting a feeling for the subject doesnt work
• Need to understand the underpinnings first (area,
  volume, scaling, energy, atoms,)

13
Scaling

• Often one is interested in how quantities change
  when an object or a system is enlarged or
  shortened
• Different quantities will change by different
  factors!
• Typical example how does the circumference,
  surface, volume of a sphere change when its
  radius changes?