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Here is how to estimate the probable error of a direct and indirect measurement.

Very often, in the test and measurement industry, a measurement is made by instrumentation, which is often subject to errors. It is difficult to estimate the true value of the measured quantity given a number of indeterminate, uncorrelated, and random factors.

This article describes a method to estimate the probable error of a measurement, specifically, the probable error of a direct measurement and an indirect measurement. A direct measurement is one in which the quantity sought is obtained directly by a measurement made by instruments indicating the sought quantity. An indirect measurement is the one in which the measurement is not directly made, but is calculated from other directly-measured quantities.

Such problems are industry-wide and are often overlooked. Instead, it’s often assumed that the displayed value is free of random errors, so the article describes a range of values where the true value probably lies. A brief description of the various possible types of errors is provided in the description below.

**Possible errors in measurements**

There are several errors that may occur when measurements are made with instrumentation. An analysis of the source of errors leads to the following classification of errors.

*Constant or systematic errors*

These errors, caused by the imperfections in the construction or adjustment of instruments, affect all measurements alike. They are deterministic and can be remedied by applying proper corrections.

*Mistakes*

These are large errors due to careless reading of measuring instruments or faulty recording of the readings. That includes reading the wrong scale, mis-recording, and so on. They do not follow any law and can be avoided by vigilance and careful checking.

*Accidental errors*

These are the errors whose causes are unknown and indeterminate. They are small and follow the laws of chaos. This article addresses a way to combat accidental errors, which are usually Gaussian.

*Precise and accurate measurements*

A precise measurement is one that is free of accidental errors. An accurate measurement is free of all types of errors.

**Probable error of direct measurements**

The probable error, r, of a single measurement of a series, is a quantity in which one half of the errors of the series are greater than it and the other half less than it. For a direct measurement, the probable error can be computed using the formula:

*r = 0.6745 √ ((v _{1 }^{2} + v_{2 }^{2 }+⋅⋅⋅ ….v_{n }^{2}) / n-1)*

Where *v*_{1}, *v*_{2}, *v*_{3} and *v*_{1n} are defined as residuals calculated as the difference between a measurement and the mean over a set of measurements. This derivation assumes that both the errors and the residuals follow the normal law.

The probable error of the mean, *r _{0}*, is given by

For example, consider the following series of 10 actual measurements made on an oscilloscope, measuring the frequency on channel 1.

M_{3} = 8.0029 |
v_{3 }=0.0051 |
v_{3 }^{2 }= 0.000026 |

M_{4} = 8.0048 |
v_{4 }=0.0032 |
v_{4} ^{2} = 0.00001024 |

M_{5} = 8.0043 |
v_{5 }=0.0037 |
v_{5} ^{2} = 0.00001369 |

M_{6} = 8.0056 |
v_{6 }= 0.0024 |
v_{6} ^{2} = 0.00000576 |

M_{7 }= 8.0030 |
v_{7 }= 0.0050 |
v_{7} ^{2} = 0.000025 |

M_{8 }= 8.0029 |
v_{8 }= 0.0051 |
v_{8} ^{2} =0.00002601 |

M_{9} =8.0076 |
v_{9 }= 0.0004 |
v_{9} ^{2} = 1.6e-7 |

M_{10} = 8.0083 |
v_{10}= -0.0003 |
v_{10} ^{2} = 9e-8 |

ΣM = 80.0804 |
Σv = –0.0004 |
Σv^{2 }= 0^{
} |

M = 80.0804/10 = 8.0080 |

*r=0.6745 √(0.00073195/9) **= 0.6745 (0.00901819149) =* 0.00608277016* (r = 0.6745 √ ((v _{1 }^{2} + v_{2 }^{2 }+⋅⋅⋅ ….v_{n }^{2}) / n-1)*

*r _{0 }=* 0.00608277016 /

Therefore, the frequency is M = 8.0080 GHz. The true value is between 8.009924 and 8.006076 GHz.

**Probable error of indirect measurements**

Let Q be a quantity that is measured as a function of other directly measured quantities.

Q = f (q_{1}, q_{2}, q_{3}….q_{n}), where q_{1}, q_{2}, q_{3}….q_{n} are directly measured quantities.

Then the probable error, r, of the indirect measurement, given that r_{1}, r_{2}….r_{n} are the relative errors of the individual directly measured quantities, is given as:

*r*= √((∂*Q*/∂_{q1})^{2}*r*1^{2}) + (∂*Q*/∂_{q2})^{2}r2^{2} +…………. + (∂*Q*/∂_{qn})^{2}*rn*^{2 },

provided that ∂*Q*/∂_{q1}, ∂*Q*/∂_{q2}, … ∂*Q*/∂_{qn} exist

We could also compute the relative error with regard to probable error, found by dividing the whole equation by Q, as follows:

*r/Q*= √((∂*Q*/∂_{q1})^{2}(*r*1^{2}/*Q*^{2})) + (∂*Q*/∂_{q2})^{2}r2^{2}/*Q*^{2}) +…………. + (∂*Q*/∂_{qn})^{2}(*rn*^{2}/*Q*^{2})

Below, we find the probable error in the calculation of the resistance of a two-stage RC oscillator circuit.

The equation used for the calculation of the frequency of a two-stage phase shift RC oscillator will be noted as follows:

*f* = 1/2πRC√(2N

Where,

Ƒ is the output frequency in hertz,

R is the resistance in ohms,

C is the capacitance in farads,

N is the number of RC stages (N = 2)

Let R = 1 kΩ + 0.9 Ω, measured by a digital multimeter, where 0.9 Ω represents the probable error in the measurement of the resistors.

Let C = 1.0 nF + 0.01 pF, measured by a capacitor meter, where 0.01 pF represents the possible error in the measurement of the capacitors.

Therefore,

*f* *= *1*/(*2∗π∗2∗10^{3}∗10^{−9}) = 10^{6}/4∗π = 79.477 kHz

The probable error, *r*, using the above equation is given by:

*r*/*f* = √((∂*f*/∂_{R})^{2}(*r*1^{2}/*f*^{2}) + (∂*f*/∂*C)*^{2 }(r_{2}^{2}/*f*^{2})) ; ∂*f*/∂*R = *−1*/(*2∗π∗*C*∗√(2∗*N*)∗*R*^{2}) ; ∂*f*/∂*C = *−1*/(*2∗π∗*R*∗√(2∗*N*)∗*C*^{2})

*r*1 = 0.9; *r2 *= 0.01∗10^{−12}

Therefore, we find below:

*r*/*f* = √((0.9/10^{3})^{2 }+ (10^{−14}/10^{−9}*)*^{2}) = 9.001∗10^{−4}

r= 10^{6}/4∗π ∗ 9.001 ∗ 10^{−4} = 71.63

The frequency and the probable error are therefore 79.477 ± .07163 kHz.

**Editor’s Note:** The author thanks Tektronix Inc. for use of instruments and the intellectual environment. She also wishes to thank Ramesh PE of Tektronix for presenting her the problem.

*Savitha Muthanna works in the R&D department of Keysight Technologies.*

*This article was originally published on EDN.*

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