Uncertainty and Error Propagation in Physics
A measurement is only worth the number of digits you can defend. Every instrument has a smallest readable division, every repeated reading scatters around its own mean, and every calculation built on those readings drags that scatter along with it. Two things must then be true of any number you report. You can say how wrong it might be, and you can say how much wronger it becomes once it passes through an equation.
Mean of repeated readings
Given
Deviation of one reading from the mean
Given
Sum of all the deviations
Given
Range of a set of readings
Given
Variance of a set of readings
Given
Sample standard deviation of a set of readings
Given
Population standard deviation of a whole set of values
Given
Uncertainty of the mean from the spread of the readings
Given
Uncertainty of the mean from the uncertainty of one reading
Given
Weighted mean of readings of different precision
Given
Uncertainty of the weighted mean
Given
Absolute uncertainty from the instrument scale
Given
Fractional uncertainty of a measurement
Given
Percentage uncertainty of a measurement
Given
Absolute uncertainty from a percentage uncertainty
Given
- (%)
Reporting a measured value with its uncertainty
Given
Largest and smallest possible values of a reported measurement
Given
Absolute error against an accepted value
Given
Relative error against an accepted value
Given
Percentage error against an accepted value
Given
Percentage difference between two measured values
Given
Uncertainty in a sum or a difference
Given
Uncertainty in a sum or a difference, worst case
Given
Uncertainty in the difference of two readings from the same instrument
Given
Constant used
Uncertainty after multiplying by an exact number
Given
Uncertainty in a product from the uncertainties of the factors
Given
Uncertainty in a quotient from the uncertainties of the quantities
Given
Percentage uncertainty in a product or a quotient, worst case
Given
- (%)
- (%)
Uncertainty in a product, in absolute form
Given
Uncertainty in a quotient, in absolute form
Given
Uncertainty in a power
Given
Uncertainty in a general power law
Given
Uncertainty in a linear combination
Given
Uncertainty in a function of one variable
Given
Uncertainty in a function of several variables
Given
Uncertainty in a natural logarithm
Given
Uncertainty in a base-10 logarithm
Given
Constant used
Uncertainty in an exponential
Given
Uncertainty in a power of ten
Given
Constant used
Uncertainty in a sine
Given
- (rad)
- (rad)
Uncertainty in a cosine
Given
- (rad)
- (rad)
Uncertainty in a tangent
Given
- (rad)
- (rad)