Uncertainty in Measured Quantities

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Uncertainty in Measured Quantities

• Measured values are not exact
• Uncertainty must be estimated
• simple method is based upon the size of the
  gauges gradations and your estimate of how much
  more you can reliably interpolate
• statistical method uses several repeated
  measurements
• calculate the average and the variance
• choose a confidence level (95 recommended)
• use t-table to find uncertainty limits
• Propagation of uncertainty
• Uncertainty in calculated values
• when you use a measured value in a calculation,
  how does the uncertainty propagate through the
  calculation
• Uncertainty in values from graphs and tables

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Comparing a Measured Value (x) to aTheoretical
or Known Value (Y)

• Compute ?, the uncertainty in x, as already
  described
• If (x - ?) lt Y lt (x ?)
• there is no significant difference between x and
  Y at the confidence level used to find ?.
• Otherwise
• x and Y are not equal at the confidence level
  used.

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Comparing Two Measured Values

• Suppose x was measured using two different
  instruments as an example
• Approach 1
• find ??1 and ?2 as previously described
• suppose x1 gt x2
• then if (x2 ?2) gt (x1 -??1) there is no
  significant difference between the two at the
  confidence level used to find the uncertainties

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Comparing Two Measured ValuesA Second Approach

• Calculate tcalc
• Look up ttable for (N1 N2 - 2) degrees of
  freedom at the desired confidence level
• There is no significant difference between the
  two values if

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Uncertainty in Values Read from Graphs

• Suppose x was measured or calculated and now y is
  being determined
• Note that ?low and ?high are not equal
• My personal preference is to take whichever is
  larger and use it as both the low and high
  uncertainty

6
Uncertainty from Charts with Parameters

• Suppose x and p were measured or calculated and y
  is now desired
• Method shown is a worst case uncertainty
• assumes maximum of both errors
• errors often offset each other

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Uncertainty in Values from a Table

• As before, ?low and ?high are not equal

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Error Propagation in Calculations

• Suppose x, y, and z were measured (or calculated
  from other measured values)
• this means the uncertainty for each is known
• Now want to calculate A which is a function of
  these measured values
• also want to know the uncertainty in the
  calculated value of A
• A f(x,y,z)

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Error Propagation

• For infinitesimal errors (dx, dy, and dz)
• Assuming the errors are small enough that the
  partials of f are not affected, the actual errors
  (?A, ?x, ?y and ?z) can be substituted for the
  infinitesimal ones (dA, dx, dy, and dz)

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Errors arent known Uncertainties are known

• The uncertainties are the maximum values of the
  errors, not the actual errors
• It is likely that some errors will cancel each
  other out
• and that most errors will be smaller than the
  maximum
• Square both sides of the previous equation, then
  average over all possible errors (assuming a
  normal distribution)

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Formula for Uncertainty in a Calculated Value

• Resulting equation for Uncertainty
• Example Suppose a cylinder has a radius of 3.3
  0.1 cm and a length of 10.8 0.2 cm. What is its
  volume and what is the uncertainty in that volume?

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Solution