Unit 3C
Dealing with Uncertainty
• Significant Digits
• Understanding Error
○ Type – Random and Systematic
○ Size – Absolute and Relative
○ Accuracy and Precision
• Combining Measured Numbers
Significant Digits
Ex. lf you measure your weight to be 132 lb on a scale that can be read only to the nearest pound, the statement that you weigh
132.00 lb would be misleading. Why?
132.00 incorrectly implies that you measured (and therefore know) your weight to the nearest one hundredth of a pound
and you don’t!
The digits in a number that represent actual measurement and therefore have meaning are called significant digits.
When Are Digits Significant?
Type of Digit
Significance
Nonzero digit
Always significant
Zeros that follow a nonzero digit and
lie to the right of the dec. pt. (4.20 or 3.00).
Always significant
Zeros between nonzero digits (4002 or 3.06) or
other significant zeros (the 1st zero in 30.0).
Always significant
Zeros to the left of the first nonzero digit
(as in 0.006 or 0.00052).
Never significant
Zeros to the right of the last nonzero digit but
before the dec. pt. (as in 40,000 or 210).
Not significant unless
stated otherwise
Ex. Count the significant digits:
a.) t = 11.90 s
b.) l = 0.000067m
c.) population of 240,000
d.) population of 2.40000 X 10^5
More Examples:
96.2 km/hr
= 9.62×10 km/hr
3 significant digits
(implies a measurement to the nearest .1 km/hr)
100.020 seconds
= 1.00020 x 102 seconds
6 significant digits
(implies a measurement to the nearest .001 sec.)
0.00098 mm
=9.8×10(-4)
2 significant digits
(implies a measurement to the nearest .00001 mm)
0.0002020 meter
=2.020 x 10(-4)
4 significant digits
(implies a measurement to the nearest .0000001 m)
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Ex.
a.) 7.7 mm X 9.92 mm; give your answer with 2 significant digits.
b.) 24,000 X 72,706; give your answer with 4 significant digits.
Understanding Errors
Two Types of Measurement Error
○
Errors can occur in many ways, but generally can be classified as one of two basic types: random or systematic errors.
○
Whatever the source of an error, its size can be described in two different ways: as an absolute error, or as a relative error.
○
Once a measurement is reported, we can evaluate it in terms of its accuracy and its precision.
Random Errors occur because of random and inherently unpredictable events in the measurement process.
Systematic Errors occur when there is a problem in the measurement system that affects all measurements in the same way, such as
making them all too low or too high by the same amount.
weighing babies in a pediatricians office
Shaking and crying baby introduces random error because a measurement could be “shaky” and easily misread.
A miscalibrated scale introduces systematic error because all measurements would be off by the same amount. (adjustable)
Example: (Errors in Global Warming Data; The Census)
Size of Errors
absolute error = measured value - true value.
absolute error
relative error =
measured value - true value
=
true value
true value
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Example
You ask for 6 lbs of hamburger, you get 4 lbs VS you buy a car which the owner's manual says weighs 3132 lbs, but you find th at it really
weighs 3130 lbs.
Absolute Error = Measured Value – True Value
Relative Error = Absolute Error
True Value
Example
Ex5a/178 My true weight is 125 pounds, but the scale says I weight 130 pounds.
absolute error = measured value – true value
relative error
Ex5b/178 The government claims that a program costs $49.0 billion, but an audit shows that the true cost is $50.0 billion
absolute error = measured value – true value
relative error
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Describing Results: Accuracy and Precision
Accuracy describes how closely a measurement approximates a true value. An accurate measurement is very close to the true value.
Precision describes the amount of detail in a measurement.
ex. Your true height is 70.50 inches.
A tape measure that can be read to the nearest ⅛ inch gives your height as 70⅜ inches.
A new laser device at the doctor’s office that gives reading to the nearest 0.05 inches gives your height as 70.90 inches.
Which device is more accurate? Which is more precise?
1/8 inch = .125 inch, so 70 3/8 = 70.375 inches
70.5 –70.375 = -.125 VS. 70.5 – 70.9 = -.4 inches so the tape measure is more accurate.
But the laser device is more precise.
Pg180 The population of your city is reported as 300,000 people. Your best friend moves to your city to share an apartment.
Is the new population 300,001?
300001 = 300000 + 1
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Combining Measured Numbers
Rounding rule for addition or subtraction: Round your answer to the same precision as the least precise number in the problem.
Rounding rule for multiplication or division: Round your answer to the same number of significant digits as the measurement with the
fewest significant digits.
Note: You should do the rounding only after completing all the operations – NOT during the intermediate steps!!!
We round 300,001 to the same precision as 300,000.
So, we round to the hundred thousands to get 300,000.
Examples
ex. Subtract 1.45 hours from 60 hours
ex. Multiply 62.5 km/hr by 2.4 hours.
ex. A freeway sign tells you that it is 36 miles to downtown. Your destination is 2.2 miles beyond downtown. How much
farther do you have to drive?
ex. What is the per capita cost of $2.1 million recreation center in a city with 120,342 people?
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