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Units and Measurements - Errors in Measurement

Grade 11CBSEPhysics

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Accuracy: It refers to the closeness of a measurement to the true value of the physical quantity. It depends on the minimization of systematic errors.

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Precision: It refers to the limit or resolution to which a physical quantity is measured. It is determined by the least count of the measuring instrument.

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Systematic Errors: These are errors that tend to be in one direction, either positive or negative. Examples include instrumental errors, imperfection in experimental technique, and personal errors.

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Random Errors: These occur irregularly and are random with respect to sign and size. They can be minimized by taking the arithmetic mean of a large number of observations.

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Least Count Error: The smallest value that can be measured by a measuring instrument is called its least count. The error associated with this resolution is the least count error.

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Absolute Error: The magnitude of the difference between the individual measurement and the true value (usually taken as the arithmetic mean aˉ\bar{a}) of the quantity.

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Relative Error: The ratio of the mean absolute error Δamean\Delta a_{mean} to the mean value ameana_{mean} of the quantity measured.

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Percentage Error: The relative error expressed in percent, denoted by δa%\delta a \%.

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Combination of Errors: When a result involves the sum or difference of two quantities, the absolute error in the final result is the sum of the absolute errors in the individual quantities. When a result involves product or quotient, the relative error in the result is the sum of the relative errors in the multipliers.

📐Formulae

Arithmetic Mean: amean=a1+a2+⋯+ann=1n∑i=1nai\text{Arithmetic Mean: } a_{mean} = \frac{a_1 + a_2 + \dots + a_n}{n} = \frac{1}{n} \sum_{i=1}^{n} a_i

Absolute Error: Δai=amean−ai\text{Absolute Error: } \Delta a_i = a_{mean} - a_i

Mean Absolute Error: Δamean=∣Δa1∣+∣Δa2∣+⋯+∣Δan∣n\text{Mean Absolute Error: } \Delta a_{mean} = \frac{|\Delta a_1| + |\Delta a_2| + \dots + |\Delta a_n|}{n}

Relative Error: Δameanamean\text{Relative Error: } \frac{\Delta a_{mean}}{a_{mean}}

Percentage Error: δa%=Δameanamean×100%\text{Percentage Error: } \delta a \% = \frac{\Delta a_{mean}}{a_{mean}} \times 100 \%

Error in Sum (Z=A+B): ΔZ=ΔA+ΔB\text{Error in Sum (} Z = A + B \text{): } \Delta Z = \Delta A + \Delta B

Error in Difference (Z=A−B): ΔZ=ΔA+ΔB\text{Error in Difference (} Z = A - B \text{): } \Delta Z = \Delta A + \Delta B

Error in Product/Quotient (Z=AB or Z=AB): ΔZZ=ΔAA+ΔBB\text{Error in Product/Quotient (} Z = AB \text{ or } Z = \frac{A}{B} \text{): } \frac{\Delta Z}{Z} = \frac{\Delta A}{A} + \frac{\Delta B}{B}

Error in Power (Z=ApBqC−r): ΔZZ=pΔAA+qΔBB+rΔCC\text{Error in Power (} Z = A^p B^q C^{-r} \text{): } \frac{\Delta Z}{Z} = p \frac{\Delta A}{A} + q \frac{\Delta B}{B} + r \frac{\Delta C}{C}

💡Examples

Problem 1:

The resistance R=VIR = \frac{V}{I} where V=(100±5) VV = (100 \pm 5) \text{ V} and I=(10±0.2) AI = (10 \pm 0.2) \text{ A}. Find the percentage error in RR.

Solution:

The percentage error in VV is ΔVV×100=5100×100=5%\frac{\Delta V}{V} \times 100 = \frac{5}{100} \times 100 = 5 \%. The percentage error in II is ΔII×100=0.210×100=2%\frac{\Delta I}{I} \times 100 = \frac{0.2}{10} \times 100 = 2 \%. Since R=VIR = \frac{V}{I}, the relative error ΔRR=ΔVV+ΔII\frac{\Delta R}{R} = \frac{\Delta V}{V} + \frac{\Delta I}{I}. Therefore, the percentage error in R=5%+2%=7%R = 5 \% + 2 \% = 7 \%.

Explanation:

In a quotient, the relative error of the result is the sum of the relative errors of the individual components.

Problem 2:

A physical quantity PP is related to four observables a,b,ca, b, c and dd as P=a3b2cdP = \frac{a^3 b^2}{\sqrt{c} d}. The percentage errors of measurement in a,b,ca, b, c and dd are 1%,3%,4%1\%, 3\%, 4\% and 2%2\% respectively. What is the percentage error in the quantity PP?

Solution:

The relative error in PP is given by ΔPP=3Δaa+2Δbb+12Δcc+1Δdd\frac{\Delta P}{P} = 3 \frac{\Delta a}{a} + 2 \frac{\Delta b}{b} + \frac{1}{2} \frac{\Delta c}{c} + 1 \frac{\Delta d}{d}. Substituting the percentage values: Percentage error in P=3(1%)+2(3%)+12(4%)+1(2%)=3+6+2+2=13%\text{Percentage error in } P = 3(1\%) + 2(3\%) + \frac{1}{2}(4\%) + 1(2\%) = 3 + 6 + 2 + 2 = 13 \%.

Explanation:

The rule for powers states that the relative error in a physical quantity raised to the power nn is nn times the relative error in the individual quantity.

Errors in Measurement - Revision Notes & Key Formulas | CBSE Class 11 Physics