You measure two quantities as $A = 1.0$ m $\pm 0.2$ m, $B = 2.0$ m $\pm 0.2$ m. We should report correct value for $\sqrt{AB}$ as: (a) $1.4$ m $\pm 0.4$ m (b) $1.41$ m $\pm 0.15$ m (c) $1.4$ m $\pm 0.3$ m (d) $1.4$ m $\pm 0.2$ m

You measure two quantities as A = 1.0 m ± 0.2 m, B = 2.0 m ± 0.2 m. We should report correct value for √(AB) a

NCERT Exemplar · Q8

A = 1.0

(a)

1.4

\pm 0.4

(b)

1.41

\pm 0.15

(c)

1.4

\pm 0.3

(d)

1.4

\pm 0.2

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Concept understanding — Measurement Error Estimation

Measurement Error Estimation

Imagine you measure the length of a table five times with a metre scale and get 152.3 cm, 152.4 cm, 152.2 cm, 152.5 cm, 152.3 cm. None of the readings agree exactly — every measurement carries some uncertainty. Measurement Error Estimation is the systematic way of stating how much a measured value can be trusted.

Types of Error

Systematic errors shift every reading in the same direction — a worn instrument, a zero error, or a consistently faulty technique. These can often be removed by calibrating against a known standard.
Random errors scatter unpredictably above and below the true value, caused by small, uncontrollable changes (a slight tremble of the hand, tiny fluctuations in conditions).
Least count error is the smallest possible error for a given instrument — a floor below which no reading, however careful, can be more precise (see Least Count Precision).

Note

Systematic error affects accuracy (closeness to the true value); random error affects precision (how tightly repeated readings cluster together).

Absolute, Mean, Relative and Percentage Error

Suppose you take $n$ readings $a_1, a_2, \dots, a_n$ of the same quantity. The best available estimate of the true value is their mean:

a_{\text{mean}} = \frac{a_1 + a_2 + \cdots + a_n}{n}

The absolute error in each reading is how far it lies from this mean:

\Delta a_i = |a_{\text{mean}} - a_i|

Averaging these gives the mean absolute error — the single number used to report the uncertainty of the whole set:

\Delta a_{\text{mean}} = \frac{|\Delta a_1| + |\Delta a_2| + \cdots + |\Delta a_n|}{n}

The final result is written as $a = a_{\text{mean}} \pm \Delta a_{\text{mean}}$ .

To compare errors across different quantities, use the relative error:

\text{Relative error} = \frac{\Delta a_{\text{mean}}}{a_{\text{mean}}}

and the percentage error, the relative error written as a percentage:

\text{Percentage error} = \frac{\Delta a_{\text{mean}}}{a_{\text{mean}}} \times 100\%

Combining Errors in a Calculation

Most physical quantities are calculated from two or more measured quantities, so their errors combine.

Sum or difference ( $Z = A + B$ or $Z = A - B$ ): absolute errors add —

\Delta Z = \Delta A + \Delta B

Product or quotient ( $Z = AB$ or $Z = A/B$ ): relative errors add —

\frac{\Delta Z}{Z} = \frac{\Delta A}{A} + \frac{\Delta B}{B}

Power ( $Z = A^n$ ): the relative error scales with the power — $\frac{\Delta Z}{Z} = n\,\frac{\Delta A}{A}$ …

Why this formula?

Measurement Error Estimation: Why the Key Formulas Hold

Measurement error estimation is about quantifying how much a measured value might differ from the true value. The core idea is that no measurement is perfect — every reading contains some uncertainty.

1. The Fundamental Idea: True Value vs. Measured Value

Let’s start with the basic relationship:

\text{Measured Value} = \text{True Value} + \text{Error}

The error ( $\varepsilon$ ) is the difference:

\varepsilon = \text{Measured Value} - \text{True Value}

Why this matters: We never know the true value exactly — if we did, there would be no error to estimate. So we must infer the error from repeated measurements.

2. Mean Error (Bias) — Why We Average

If you take $n$ measurements $x_1, x_2, \dots, x_n$ , the mean is:

\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i

Why does the mean estimate the true value?

Assume each measurement has a random error $\varepsilon_i$ with zero mean (no systematic bias). Then:

\bar{x} = \frac{1}{n} \sum_{i=1}^{n} (\text{True} + \varepsilon_i) = \text{True} + \frac{1}{n} \sum_{i=1}^{n} \varepsilon_i

As $n$ increases, the average of random errors $\frac{1}{n} \sum \varepsilon_i$ tends to zero (by the law of large numbers). So:

\boxed{\bar{x} \to \text{True Value}}

Key insight: Averaging cancels out random errors, but not systematic errors (bias).

3. Standard Deviation of the Mean — Why $\sigma / \sqrt{n}$

The standard error of the mean (SEM) is:

\text{SEM} = \frac{\sigma}{\sqrt{n}}

Derivation (why this formula):

Each measurement $x_i$ has variance $\sigma^2$ (spread around the true value).
The variance of the mean $\bar{x}$ is:

\text{Var}(\bar{x}) = \text{Var}\left( \frac{1}{n} \sum x_i \right) = \frac{1}{n^2} \sum \text{Var}(x_i)

Since all $\text{Var}(x_i) = \sigma^2$ and they are independent:

\text{Var}(\bar{x}) = \frac{1}{n^2} \cdot n \sigma^2 = \frac{\sigma^2}{n}

Standard deviation is the square root of variance:

\boxed{\text{SEM} = \sqrt{\frac{\sigma^2}{n}} = \frac{\sigma}{\sqrt{n}}}

Why this makes sense: More measurements ( $n$ larger) reduce uncertainty — but only as $\sqrt{n}$ , not linearly. Doubling $n$ reduces error by only $\approx 30\%$ .

4. Propagation of Errors — Why We Add Variances

When a result $z$ depends on measured quantities $x$ and $y$ (e.g., $z = x + y$ or $z = x \cdot y$ ), errors propagate.

Case 1: Addition/Subtraction

If $z = x + y$ , and errors $\Delta x$ , $\Delta y$ are independent:

(\Delta z)^2 = (\Delta x)^2 + (\Delta y)^2

Why?

Variance of sum = sum of variances (for independent variables):

\sigma_z^2 = \sigma_x^2 + \sigma_y^2

So the uncertainty adds in quadrature (not linearly). This is because errors can partially cancel.

Case 2: Multiplication/Division

If $z = x \cdot y$ , then:

\left( \frac{\Delta z}{z} \right)^2 = \left( \frac{\Delta x}{x} \right)^2 + \left( \frac{\Delta y}{y} \right)^2

Derivation (why relative errors add):

Take natural log: $\ln z = \ln x + \ln y$
Differentiate: $\frac{dz}{z} = \frac{dx}{x} + \frac{dy}{y}$
For small independent errors, variances add:

\left( \frac{\sigma_z}{z} \right)^2 = \left( \frac{\sigma_x}{x} \right)^2 + \left( \frac{\sigma_y}{y} \right)^2

Key insight: Relative uncertainties propagate the same way absolute uncertainties do for sums.

5. The General Formula (Why It's a Taylor Expansion) …

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Using the fractional-error-addition rule (not root-sum-square) for combining errors, $\sqrt{AB}$ works out to $1.4 \pm 0.2\ \text{m}$ — option (D).

Best estimate of $\sqrt{AB}$

\sqrt{AB} = \sqrt{(1.0)(2.0)} = \sqrt{2.0} \approx 1.4142\ \text{m}

Combining the errors

For $Z = \sqrt{AB} = (AB)^{1/2}$ , the NCERT-prescribed rule for combination of errors is: relative errors in a product add, and for a power $p$ , the relative error is multiplied by $|p|$ — here $p = 1/2$ .

\frac{\Delta Z}{Z} = \frac{1}{2}\left(\frac{\Delta A}{A} + \frac{\Delta B}{B}\right)

\frac{\Delta A}{A} = \frac{0.2}{1.0} = 0.2, \qquad \frac{\Delta B}{B} = \frac{0.2}{2.0} = 0.1

\frac{\Delta Z}{Z} = \frac{1}{2}(0.2 + 0.1) = \frac{1}{2}(0.3) = 0.15

Absolute error and rounding

$\Delta Z = 0.15 \times 1.4142 \approx 0.212\ \text{m} \to 0.2\ \text{m (1 significant figure)}$ …

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🧠 The core concept

You have:

$A = 1.0 \pm 0.2$ m
$B = 2.0 \pm 0.2$ m

You want $\sqrt{AB}$ .

Step 1 — Best value

\sqrt{AB} = \sqrt{1.0 \times 2.0} = \sqrt{2.0} \approx 1.414\ \text{m}

Step 2 — Error propagation

For $Z = \sqrt{AB}$ , the relative error formula is:

\frac{\Delta Z}{Z} = \frac{1}{2} \left( \frac{\Delta A}{A} + \frac{\Delta B}{B} \right)

So:

\frac{\Delta Z}{Z} = \frac{1}{2} \left( \frac{0.2}{1.0} + \frac{0.2}{2.0} \right) = \frac{1}{2} (0.2 + 0.1) = 0.15

Thus:

\Delta Z = 0.15 \times 1.414 \approx 0.212 \ \text{m}

Correct report: $1.41 \pm 0.21$ m (or rounded to $1.4 \pm 0.2$ m if using 1 decimal place).

The best match among options is (D) $1.4$ m $\pm 0.2$ m.

✗ Common mistakes & how to avoid them

1. Using absolute error formula for multiplication directly

Mistake: Treating $\sqrt{AB}$ like $A \times B$ and adding absolute errors.
Why wrong: For products/quotients, you must use relative errors, not absolute.
Fix: Always convert to relative error first, then multiply by the value.

2. Forgetting the square root halves the relative error

Mistake: Using $\frac{\Delta Z}{Z} = \frac{\Delta A}{A} + \frac{\Delta B}{B}$ (no factor of $1/2$ ).
Why wrong: For $Z = X^{1/2}$ , relative error in $Z$ is $\frac{1}{2}$ times relative error in $X$ .
Fix: Remember: exponent $n$ multiplies relative error by $|n|$ .

3. Rounding too early

Mistake: Computing $\sqrt{1.0 \times 2.0} \approx 1.4$ then using that to find error.
Why wrong: You lose precision — error calculation needs more digits.
Fix: Keep intermediate results to 3–4 significant figures; round only final answer.

4. Mixing up significant figures in value and error …

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Q.You measure two quantities as $A = 1.0$ m $\pm 0.2$ m, $B = 2.0$ m $\pm 0.2$ m. We should report correct value for $\sqrt{AB}$ as:

Concept understanding — Measurement Error Estimation

Measurement Error Estimation

Types of Error

Absolute, Mean, Relative and Percentage Error

Combining Errors in a Calculation

Why this formula?

Measurement Error Estimation: Why the Key Formulas Hold

1. The Fundamental Idea: True Value vs. Measured Value

2. Mean Error (Bias) — Why We Average

3. Standard Deviation of the Mean — Why $\sigma / \sqrt{n}$

4. Propagation of Errors — Why We Add Variances

Case 1: Addition/Subtraction

Case 2: Multiplication/Division

5. The General Formula (Why It's a Taylor Expansion) …

Best estimate of $\sqrt{AB}$

Combining the errors

Absolute error and rounding

Concept: Propagation of Errors in Product and Square Root

Method: Relative Error Propagation

Steps

🧠 The core concept

✗ Common mistakes & how to avoid them

1. Using absolute error formula for multiplication directly

2. Forgetting the square root halves the relative error

3. Rounding too early

4. Mixing up significant figures in value and error …

Q.You measure two quantities as A=1.0A = 1.0A=1.0 m ±0.2\pm 0.2±0.2 m, B=2.0B = 2.0B=2.0 m ±0.2\pm 0.2±0.2 m. We should report correct value for AB\sqrt{AB}AB​ as:

Best estimate of AB\sqrt{AB}AB​

Combining the errors

Absolute error and rounding

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Q.You measure two quantities as $A = 1.0$ m $\pm 0.2$ m, $B = 2.0$ m $\pm 0.2$ m. We should report correct value for $\sqrt{AB}$ as:

Best estimate of $\sqrt{AB}$

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