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Propagation of Errors
By Hemant More
March 3, 2020
3 Comments
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Science > Physics > Units and
Measurements > Propagation of Errors
In this article, we shall study the propagation of errors in different
mathematical operations.
Propagation of Errors in Addition:
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Suppose a result x is obtained by addition of two quantities say a and
b
i.e. x = a + b
Let Δ a and Δ b are absolute errors in the measurement of a and b and
Δ x be the corresponding absolute error in x.
∴ x ± Δ x = ( a ± Δ a) + ( b ± Δ b)
∴ x ± Δ x = ( a + b ) ± ( Δ a + Δ b)
∴ x ± Δ x = x ± ( Δ a + Δ b)
∴ ± Δ x = ± ( Δ a + Δ b)
∴Δx =Δa +Δb
Thus maximum absolute error in x = maximum absolute error in a +
maximum absolute error in b
Thus, when a result involves the sum of two observed quantities, the
absolute error in the result is equal to the sum of the absolute error in
the observed quantities.
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Propagation
of Errors inPDF
Subtraction:
Suppose a result x is obtained by subtraction of two quantities say a
and b
i.e. x = a – b
Let Δ a and Δ b are absolute errors in the measurement of a and b and
Δ x be the corresponding absolute error in x.
∴ x ± Δ x = ( a ± Δ a) – ( b ± Δ b)
∴x ±Δx =(a–b)± Δa –+Δb
∴ x ± Δ x = x ± ( Δ a + Δ b)
∴ ± Δ x = ± ( Δ a + Δ b)
∴Δx =Δa +Δb
Thus the maximum absolute error in x = maximum absolute error in
a + maximum absolute error in b.
Thus, when a result involves the difference of two observed
quantities, the absolute error in the result is equal to the sum of the
absolute error in the observed quantities.
Propagation of Errors in Product:
Suppose a result x is obtained by the product of two quantities say a
and b
i.e. x = a × b ……….. (1)
Let Δ a and Δ b are absolute errors in the measurement of a and b and
Δ x be the corresponding absolute error in x.
∴ x ± Δ x = ( a ± Δ a) x ( b ± Δ b)
∴ x ± Δ x = ab ± a Δ b ± b Δ a ± Δ aΔ b
∴ x ± Δ x = x ± a Δ b ± b Δ a ± Δ aΔ b
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∴±Δ
x = ± a Δ b ±PDF
b Δ Tools
a ± Δ aΔ
b …… (2)
Dividing equation (2) by (1) we have
The quantities Δa/a, Δb/b and Δx/x are called relative errors in the
values of a, b and x respectively. The product of relative errors in a
and b i.e. Δa × Δb is very small hence is neglected.
Hence maximum relative error in x = maximum relative error in a +
maximum relative error in b
Thus maximum % error in x = maximum % error in a + maximum %
error in b
Thus, when a result involves the product of two observed quantities,
the relative error in the result is equal to the sum of the relative error
in the observed quantities.
Propagation of Errors in Quotient:
Suppose a result x is obtained by the quotient of two quantities say a
and b.
i.e. x = a / b ……….. (1)
Let Δ a and Δ b are absolute errors in the measurement of a and b and
Δ x be the corresponding absolute error in x.
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The values of higher power of Δ b/b are very small and hence can be
neglected.
Now the quantity (Δ aΔ b / ab)is very small. hence can be neglected.
The quantities Δa/a, Δb/b and Δx/x are called relative errors in the
values of a, b and x respectively.
Hence maximum relative error in x = maximum relative error in a +
maximum relative error in b
Thus maximum % error in x = maximum % error in a + maximum %
error in b
Thus, when a result involves the quotient of two observed quantities,
the relative error in the result is equal to the sum of the relative error
in the observed quantities.
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Propagation of Errors in Product of Powers of
Observed Quantities:
Let us consider the simple case . Suppose a result x is obtained by
following relation
x = an ……….. (1)
Let Δ a be an absolute error in the measurement of a and Δ x be the
corresponding absolute error in x.
The values of higher power of Δa/a are very small and hence can be
neglected.
The quantities Δa/a and Δx/x are called relative errors in the values
of a and x respectively.
Hence the maximum relative error in x = n x maximum relative error
in a. i.e. maximum relative error in x is n times the relative error in a.
Consider a general relation
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The quantities Δa/a, Δb/b, Δc/c, and Δx/x are called relative errors
in the values of a, b, c and x respectively.
Thus maximum % error in x is
Examples Explaining Propagation of Error:
Example – 01:
The lengths of the two rods are recorded as 25.2 ± 0.1 cm and
16.8 ± 0.1 cm. Find the sum of the lengths of the two rods with
the limit of errors.
Solution:
We know that in addition the errors get added up
The Sum of Lengths = (25.2 ± 0.1) + (16.8 ± 0.1) = (25.2 + 16.8) ±
(0.1 + 0.1) = 42.0 ± 0.2 cm
Example – 02:
The initial temperature of liquid is recorded as 25.4 ± 0.1 °C and
on heating its final temperature is recorded as 52.7 ± 0.1 °C.
Find the increase in temperature.
Solution:
We know that in subtraction the errors get added up
The increase in temperature = (52.7 ± 0.1) – (25.4 ± 0.1) = (52.7 –
25.4) ± (0.1 + 0.1) = 27.3 ± 0.2 °C.
Example – 03:
During the study, the flow of a liquid through a narrow tube by
experiment following readings were recorded. The values of p,
r, V and l are 76 cm of Hg, 0.28 cm, 1.2 cm3 s-1 and 18.2 cm
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respectively. If these quantities are measured to the accuracies
of 0.5 cm of Hg, 0.01 cm, o.1 cm3 s-1 and 0.1 cm respectively, find
the percentage error in the calculation of η if formula used is
Solution:
Example – 04:
The percentage errors of measurements in a, b, c and d are 1%,
3%, 4% and 2% respectively. These quantities are used to
calculate value of P. Find the percentage error in the calculation
of P, If the formula used is
Solution:
Previous Topic: Errors and Their Types
For More Topics in Physics Click Here
Science > Physics > Units and
Measurements > Propagation of Errors
Error analysis, Errors, Propagation of error in addition, Propagation of error in
division, Propagation of error in equation having index, Propagation of error in
multiplication, Propagation of error in product, Propagation of error in quotient,
Propagation of error in subtraction
← Errors and Their Types
Introduction to Electromagnetic →
Induction
3 replies on “Propagation of Errors”
Eslam
October 25, 2020 at 4:30 pm
Great work, thanks a lot
REPLY
Nandana
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January 1, 2021 at 9:52 pm
Thank you so much, Hemant!
REPLY
Brnson Kuria
January 26, 2021 at 4:22 am
I think the person who did this should also have included University
physics,,however,kudos
REPLY
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