# ERROR AND UNCERTAINTY ANALYSIS : EXPLANATION, SOLVED EXAMPLES AND PRACTICE PROBLEMS OF ERROR AND UNCERTAINTY

Error is a very important concept of Physics and Mathematics. It is generally said that nothing is Perfect, this statement is also true in the context of scientific calculations or measurements. Some content of Error always exists in any scientific results. So it is not possible to find the exact value of measurements. The study of error can help us to reduce the content of error in calculations. This topic is also an inseparable part of many exams like CSIR-NET/JRF, GATE, JEST, IIT-JAM, NEET, JEE, etc.

What to be discussed -

1. What is Error
2. Formulae to calculate Error
3. Solved Problem of Error
4. What is RMS Error
5. Solved Problem of RMS Error
6. What is Uncertainty
7. How to measure uncertainty
8. Solved Problem
9. Practice Problems of Error and Uncertainty Analysis

What is ErrorThe deviation of the measured value from true value is known as error. When we measured any quantity, it has two things, the first is the true value of that quantity which we cannot measure exactly and the other is the uncertainty in that measurement.
Error = Measured Value-True Value
•  Absolute Error  ΔX = XM-XT
•  Relative Error  = ΔX/X
•  Percentage Error = (ΔX/X) *100
Propagation of Error - Error is propagating through the Arithmetic operations. You should also remember that total Error is always the sum of errors in all quantities, whether it is addition or substruction. While we make mistake or error in measuring the two quantities and when we apply an arithmetic operation even subtraction then the total error will be the sum of each error. The following are the ways to find the total absolute error while we applying arithmetic operations on two or more quantities.

1. Error in Addition of two quantities
Z = X+Y
ΔZ = Δ X+ Δ Y

2. Error in Subtraction of two quantities
Z = X-Y
ΔZ = Δ X+ Δ Y

3. Error in Product of two quantities
Z = X.Y
lnZ = lnX + lnY
ΔZ/Z = ΔX/X + ΔY/Y

4. Error in Division of two quantities
Z = X/Y
lnZ = lnX - lnY
ΔZ/Z = ΔX/X + ΔY/Y

5. Error in Power of a quantity
Z=X^n
ΔZ/Z = n(ΔX/X)

Solved Problem -1

A Student measures the displacement x from the equilibrium of a stretched spring and reports it be with a 1% error. The spring constant k is known to be 10N/m with 0.5% error. The percentage error in the estimate of the potential energy is

(1) 0.8%
(2) 2.5%
(3) 1.5%
(4) 3.0%

Solution - The given Potential energy is

For Relative Error, we have-
Root Mean Square Error  - It is generally defined as the square root of the mean square. Mathematically, RMS Error for the previous problem can be written as -

Solved Problem - 2
The Viscosity of a liquid is given by poiseuille's formula . Assume that l and V can be measured very accurately. but the pressure P has an RMS error of 1% and the radius has an independent RMS error of 3%. The RMS error of the viscosity is closest to
(1) 2%
(2) 4%
(3) 12%
(4) 13%

Solution -

where l and V measured accurately, So we can take these as constants for error calculation

Uncertainty The margin of error of measurement.
Solution - Given:

Practice Problem-1

Practice Problem-2-
(1) 0.12
(2) 0.05
(3) 0.03
(4) 0.07