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Uncertainty in Measurements How to handle human measuring errors

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					Uncertainty in Measurements
How to handle human measuring errors
    Choosing the Correct Piece of
            Equipment
Every measuring
 device you use
 has a scale
 (lines) on it.
    Choosing the Correct Piece of
            Equipment
The scale
 determines how
 precise your
 measurement
 will be.
     Choosing the Correct Piece of
             Equipment
Your first step is
 always figuring
 out how much
 each line
 represents.
     Choosing the Correct Piece of
             Equipment
What volume
 does each line
 represent on
 this graduated
 cylinder?
     Choosing the Correct Piece of
             Equipment
1 mL is correct!
  Reading a Graduated Cylinder
Liquids in glass
 containers curve
 at the edges
 forming a…
  Reading a Graduated Cylinder
Liquids in glass
 containers curve
 at the edges
 forming a…
 MENISCUS
  Reading a Graduated Cylinder
Bend down to get
 eye level with
 and read the
 bottom of the
 MENISCUS
      Making a Measurement
If a quantity falls
   between the lines on
   the scale, you must
   estimate some of the
   measurement
      Making a Measurement
Estimating introduces
  human error into the
  measurement
  Reporting Your Measurement
It is common practice
   to report all the
   “certain” digits (the
   ones you have a line
   for)


36 mL
  Reporting Your Measurement
Plus one more decimal
  place beyond the
  scale




36.3 mL
  Reporting Your Measurement
It is meaningless to
   report this level to
   the hundredths digit
   because you are
   already estimating at
   the tenths digits.

36.35 mL
  What is the Volume Here?




36.5 mL
  What is the Volume Here?




42.9 mL
  What is the Volume Here?




47.0 mL
   Be Careful What You Write
Acceptable uncertainty is plus or minus “1” of
            the last digit recorded.

   Ex. 1.5 lbs. means the measurement is
           between 1.4 and 1.6 lbs.

  Ex. 1.500 lbs. means the measurement is
  between 1.499 and 1.501 lbs. This conveys
     a much more precise measurement!!
          Significant Figures
All of the certain digits plus the first uncertain
  digit (the estimated number) are referred to
        as the Significant Figures of the
                  measurement.
       Rules For Counting Significant
                  Figures
1.    Nonzero numbers are always significant.




     Ex. 32,127     has 5 sig.figs.

     Ex. 24         has 2 sig.figs.
       Rules For Counting Significant
                  Figures
1.    Nonzero numbers are always significant.
2.    All numbers in scientific notation are significant.



     Ex. 6.02x1023 has 3 sig.figs.

     Ex. 1.392x10-5 has 4 sig.figs.
      Rules For Counting Significant
                 Figures
 3. Leading zeros (zeros that precede nonzero digits)
   are never significant.



Ex. 0.0025 has 2 sig.figs.

Ex. 0.7624 has 4 sig.figs.
    Rules For Counting Significant
               Figures
4. Captive zeros (zeros between nonzero digits)
  are always significant.


Ex. 34,005 has 5 sig.figs.

Ex. 10.00305 has 7 sig.figs.

Ex. 0.002008 has 4 sig.figs.
    Rules For Counting Significant
               Figures
5. Trailing zeros (zeros at the end of the number) are
   sometimes significant.
    YES if you see a decimal point
   NO if you don’t see a decimal point.

Ex. 2,000. has 4 sig.figs.

Ex. 2,000 has 1 sig.fig.

Ex. 2,000.00 has 6 sig.figs
    Rules For Counting Significant
               Figures
6. “Counting” numbers have an infinite number of
   significant figures

  Example 1: There are 20 students in the room.

  Example 2: Definitions…1 foot is exactly 12
    inches
    How many significant figures?
   0.0105
   0.050080
   8.050 X 10-3
   20,200


 Answers: 3, 5, 4,3
Handling Significant Figures in
  Mathematical Operations
Uncertainty accumulates as calculations are
 carried out so we must follow a set of rules
      to handle significant figures when
      multiplying, dividing, adding, and
             subtracting numbers.



          I’m uncertain!!
 When Multiplying or Dividing
  Your answer can NOT have more sig. figs.
   than the least precise measurement

Ex.      4.56       3 sig.figs.
       x 1.4        2 sig.figs.
         6.384      can only have 2 sig.figs
 When Multiplying or Dividing
  Your answer can NOT have more sig. figs.
   than the least precise measurement

Ex.        4.56          3 sig.figs.
         x 1.4           2 sig.figs.
           6.384         can only have 2 sig.figs

 You are limited to 2 sig.figs. by the measurement
       1.4, so you must round 6.384 to 6.4
 When Multiplying or Dividing
  Your answer can NOT have more sig. figs.
   than the least precise measurement

Ex.        4.56          3 sig.figs.
         x 1.4           2 sig.figs.
           6.4           can only have 2 sig.figs

 You are limited to 2 sig.figs. by the measurement
       1.4, so you must round 6.384 to 6.4
  When Adding or Subtracting
Your answer can’t have more places past the
 decimal than the least precise measurement


Ex.       12.11      2 places past decimal
        + 18.0       1 place past decimal
          1.013      3 places past decimal
          31.123     can only have 1 place
                     past decimal
     When Adding or Subtracting
   Your answer can’t have more places past the
    decimal than the least precise measurement


   Ex.       12.11       2 places past decimal
           + 18.0        1 place past decimal
             1.013       3 places past decimal
             31.123      can only have 1 place
                         past decimal
You must round answer to only 1 place past decimal
     When Adding or Subtracting
   Your answer can’t have more places past the
    decimal than the least precise measurement


   Ex.       12.11       2 places past decimal
           + 18.0        1 place past decimal
             1.013       3 places past decimal
             31.1        can only have 1 place
                         past decimal
You must round answer to only 1 place past decimal
   How to Round for Sig. Figs.
Only look at the digit immediately to the right
         of the last significant figure

  – if less than 5, preceding digit stays the same
  – if 5 or more, round preceding digit up by one
               You Try It
 Round   4.348 to two sig. figs.



    These are the 2 sig figs
               You Try It
 Round   4.348 to two sig. figs.



So look at the very next number for rounding
It’s less than 5 so the 3 stays like it is


  Final Answer is 4.3

				
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posted:1/5/2013
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