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Physical World and Measurement - Errors in Measurement

Grade 11ICSEPhysics

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

🔑Concepts

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Accuracy refers to how close a measured value is to the true value of the quantity, whereas Precision refers to the resolution or the limit to which the quantity is measured by a measuring instrument.

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Systematic Errors are those errors that tend to be in one direction, either positive or negative. Examples include instrumental errors (like zero error), imperfection in experimental technique, and personal errors.

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

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Least Count Error is the error associated with the resolution of the instrument. For example, a vernier calliper with a least count of 0.01 cm0.01 \text{ cm} has a precision of 0.01 cm0.01 \text{ cm}.

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Absolute Error is the magnitude of the difference between the individual measurement and the true value of the quantity: ∣Δai∣=∣amean−ai∣|\Delta a_i| = |a_{mean} - a_i|.

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Relative Error is the ratio of the mean absolute error to the mean value of the quantity measured: δa=Δameanamean\delta a = \frac{\Delta a_{mean}}{a_{mean}}.

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Percentage Error is the relative error expressed in percent: Percentage Error=δa×100%\text{Percentage Error} = \delta a \times 100\%.

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Combination of Errors: When two quantities are added or subtracted, the absolute error in the final result is the sum of the absolute errors in the individual quantities. When quantities are multiplied or divided, the relative error in the result is the sum of the relative errors in the multipliers.

📐Formulae

amean=a1+a2+a3+...+anna_{mean} = \frac{a_1 + a_2 + a_3 + ... + a_n}{n}

Δai=amean−ai\Delta a_i = a_{mean} - a_i

Δamean=∑i=1n∣Δai∣n\Delta a_{mean} = \frac{\sum_{i=1}^{n} |\Delta a_i|}{n}

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

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

If Z=A±B, then ΔZ=ΔA+ΔB\text{If } Z = A \pm B, \text{ then } \Delta Z = \Delta A + \Delta B

If Z=AB or Z=AB, then ΔZZ=ΔAA+ΔBB\text{If } Z = AB \text{ or } Z = \frac{A}{B}, \text{ then } \frac{\Delta Z}{Z} = \frac{\Delta A}{A} + \frac{\Delta B}{B}

If Z=ApBqCr, then ΔZZ=pΔAA+qΔBB+rΔCC\text{If } Z = A^p B^q C^r, \text{ then } \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=V/IR = 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 relative error in RR is given by ΔRR=ΔVV+ΔII\frac{\Delta R}{R} = \frac{\Delta V}{V} + \frac{\Delta I}{I}. Substituting the values: ΔRR=5100+0.210=0.05+0.02=0.07\frac{\Delta R}{R} = \frac{5}{100} + \frac{0.2}{10} = 0.05 + 0.02 = 0.07. Percentage error =0.07×100=7%= 0.07 \times 100 = 7\%.

Explanation:

Since resistance is the quotient of voltage and current, the relative errors are additive. We convert the absolute uncertainties into relative uncertainties and sum them to find the total percentage error.

Problem 2:

A physical quantity PP is related to four observables a,b,ca, b, c and dd as follows: 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 ΔPP=3Δaa+2Δbb+12Δcc+Δdd\frac{\Delta P}{P} = 3\frac{\Delta a}{a} + 2\frac{\Delta b}{b} + \frac{1}{2}\frac{\Delta c}{c} + \frac{\Delta d}{d}. Percentage error =3(1%)+2(3%)+12(4%)+1(2%)= 3(1\%) + 2(3\%) + \frac{1}{2}(4\%) + 1(2\%) =3%+6%+2%+2%=13%= 3\% + 6\% + 2\% + 2\% = 13\%.

Explanation:

According to the rule of powers, the relative error of a physical quantity raised to the power nn is nn times the relative error of the individual quantity.

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