Estimate probable measurement errors using math

Article By : Savitha Muthanna

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.


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 √ ((v1 2 + v2 2 +⋅⋅⋅ ….vn 2) / n-1)

Where v1, v2, v3 and v1n 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, r0, is given by
r0 = 0.6745 √ ((v1 2 + v2 2 +⋅⋅⋅ ….vn 2) / n(n-1))

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

table of 10 actual measurements made on an oscilloscope

M3 = 8.0029 v3 =0.0051 v3 2 = 0.000026
M4 = 8.0048 v4 =0.0032 v4 2 = 0.00001024
M5 = 8.0043 v5 =0.0037 v5 2 = 0.00001369
M6 = 8.0056 v6 = 0.0024 v6 2 = 0.00000576
M7 = 8.0030 v7 = 0.0050 v7 2 = 0.000025
M8 = 8.0029 v8 = 0.0051 v8 2 =0.00002601
M9 =8.0076 v9 = 0.0004 v9 2 = 1.6e-7
M10 = 8.0083 v10= -0.0003 v10 2 = 9e-8
ΣM = 80.0804 Σv = –0.0004
Σv2 = 0
M = 80.0804/10 = 8.0080

r=0.6745 √(0.00073195/9) = 0.6745 (0.00901819149) = 0.00608277016 (r = 0.6745 √ ((v1 2 + v2 2 +⋅⋅⋅ ….vn 2) / n-1))

r0 = 0.00608277016 / 10 = 0.001924 by (r0 = 0.6745 √ ((v1 2 + v2 2 +⋅⋅⋅ ….vn 2) / n(n-1)))

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 (q1, q2, q3….qn), where q1, q2, q3….qn are directly measured quantities.

Then the probable error, r, of the indirect measurement, given that r1, r2….rn are the relative errors of the individual directly measured quantities, is given as:

r= √((∂Q/∂q1)2r12) + (∂Q/∂q2)2r22 +…………. + (∂Q/∂qn)2rn2 ,

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(r12/Q2)) + (∂Q/∂q2)2r22/Q2) +…………. + (∂Q/∂qn)2(rn2/Q2)

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


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.


f = 1/(2∗π∗2∗103∗10−9) = 106/4∗π = 79.477 kHz

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

r/f = √((∂f/∂R)2(r12/f2) + (∂f/∂C)2 (r22/f2)) ; ∂f/∂R = −1/(2∗π∗C∗√(2∗N)∗R2) ; ∂f/∂C = −1/(2∗π∗R∗√(2∗N)∗C2)

r1 = 0.9; r2 = 0.01∗10−12

Therefore, we find below:

r/f = √((0.9/103)2 + (10−14/10−9)2) = 9.001∗10−4

r= 106/4∗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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