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 -*

- What is Error
- Formulae to calculate Error
- Solved Problem of Error
- What is RMS Error
- Solved Problem of RMS Error
- What is Uncertainty
- How to measure uncertainty
- Solved Problem
- Practice Problems of Error and Uncertainty Analysis

**The 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.**

*What is Error -* Error = Measured
Value-True Value

- Absolute Error Î”X = XM-XT
- Relative Error = Î”X/X
- Percentage Error = (Î”X/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

Î”Z = Î” X+ Î” Y

**2. Error in Subtraction of two quantities**

Z = X-Y

Î”Z = Î” X+ Î” Y

**3. Error in Product of two quantities**

Z = X.Y

lnZ = lnX + lnY

Î”Z/Z = Î”X/X + Î”Y/Y

**4. Error in Division of two quantities**

Z = X/Y

lnZ = lnX - lnY

Î”Z/Z = Î”X/X + Î”Y/Y

**5. Error in Power of a quantity**

Z=X^n

Î”Z/Z = n(Î”X/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 Viscosityof 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

*The margin of error of measurement.*

__Uncertainty__-True Value ± Uncertainty

For example
X = 10 ± 0.5

How to calculate Uncertainty

__Solved Problem-3__

In a measurement of the viscous drag force experienced by spherical particles in a liquid, the force is found to be proportional towhere V is the measured volume of each particle. If V is measured to be, with an uncertainty of, the resulting relative percentage uncertainty in the measured force is

(1) 2.08

(2) 0.09

(3) 6

(4) 3

**Solution -**Given:

__Practice Problem-1__

The decay constantsof the heavy pseudo-scalar mesons, in the heavy quark limit, are related to their massesby the relation where a is an empirical parameter to be determined. The valuesand correspond to uncorrelated measurements of a meson. The error on the estimate of a is

__Practice Problem-2-__

The experimentally measured values of the variables x and y areand respectively. What is the error in the calculated value offrom the measurements?

(1) 0.12

(2) 0.05

(3) 0.03

(4) 0.07

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