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Propagation of Variance

Any operations that we perform on a measured value that has uncertainty will require us to propagate the variance associated with the measurements. Propagation of Variance (POV) is technically defined as: "the determination of the value to be assigned as the uncertainty of a given quantity using mathematical formulas for the combination of errors from constituent contributors". This occurs when we perform any of the following operations:

  1. adding two measurements
  2. dividing by a constant
  3. subtracting off a background
  4. multiplication or division of measurements
  5. producing a mean value of multiple individual measurements
  6. combining independent measurements with unequal errors

The general error propagation formula when a quantity u is calculated using a set of variables ( x, y, z, etc.) is given by

propagation_equ_1

This formula is fairly complicated; however, for the standard operations that we commonly perform, it readily simplifies. Below is a set of simplified formulas for common operations.

Adding and Subtracting

When adding (or subtracting) two measurements, each with a given uncertainty, the uncertainty of the final quantity will be produced by summing the standard deviations of the individual parts in quadrature. Thus,

propagation_equ_2

Multiplying and Dividing by a Constant

 If we divide (or multiply) a quantity with an uncertainty by a constant (i.e., a number with no uncertainty), then the standard deviation of the product is simply equal to the standard deviation of the original value multiplied (or divided) by the constant. Thus,

propagation_equ_3

Multiplying and Dividing Two Quantities with Uncertainty

If we multiply two quantities together, the fractional standard deviation of the product is determine by summing the fractional standard deviations of the original quantities in quadrature. Thus,

propagation_equ_4

Summary

So using the POV formulas, we can calculate the uncertainty to any measured quantity. This uncertainty then tells us something about the quality of the measured quantity and might tell us something about how that quantity differs from what we expected. For example, whether what we have measured has a mass significantly smaller than what was declared. We might also use this to determine if we need to measure a sample again or measure more samples.


 

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