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Uncertainty Analysis

This page will familiarize you with commonly used methods to estimate the accuracy of experimental data and quantities derived from this data.

Errors in experimental data consist of two components:

The following technique for estimating the propagation of errors/uncertainties into the result is that proposed by Kline and Mclintock. We use specified uncertainties for primary experimental measurements e.g. p = 500Pa ± 10Pa, where ±10Pa is the uncertainty.


Take measurements with estimated uncertainty for each measurement. Then estimate the uncertainty in the result R as a function of independent variables xi. Thus


Let wR be the uncertainty in the result and wi the uncertainty in the independent variables. Then the uncertainty in the result is

General uncertainty


The speed indicated by a Pitot static tube is given by:

Speed value

where Δp is the dynamic pressure indicated by the Pitot tube and r is the gas density at the measurement point. Thus the uncertainty in the speed may be estimated as:

Speed uncertainty

As an example, assume the measured independent values and their estimated uncertainties are:

Δp=600Pa ± 0.5%

ρ=1.18kg/m3 ± 0.03 kg/m3

Estimate the uncertainty in the speed wv

∂v/∂Δp = 0.5(2/ρΔp)0.5

∂v/∂ρ = -0.5(2Δp)0.51.5

thus the uncertainty in the speed wv is:

Speed uncertainty example

wv = 0.413m/s


wv / v = 0.413 / 31.89 = 1.29%

How do we estimate the uncertainty?

1) By repeating experimental readings. Then estimate uncertainty within a certain confidence interval. Pertains to random errors as for a sufficient number of readings random errors tend to approach a Gaussian distribution allowing statistical analysis.

2) Experience.

Expanding on point 1), suppose we have repeated a set readings. Let n be the number of readings and xi a particular reading.

then the arithmetic mean is given by


the deviation from the mean for each reading is given by

di = xi-xm

the average of the absolute value of the deviations is given by

Mean deviation

and the root mean square deviation or standard deviation is given by

Standard deviation (large n)

for a large number of readings or

Standard deviation (small n)

for a small data set (n < 20)

From statistical analysis define

Mean standard deviation

Then the uncertainties are

Thus the repeated data may be expressed as


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