Errors in Measurement

The difference between the true value (or mean of repeated measurements) and the observed value is called an error of measurement.

Types of Errors

1. Absolute Error

Suppose a quantity is measured \( n \) times, giving values \( a_1, a_2, a_3, \dots, a_n \).

\[ \bar{a} = \frac{a_1 + a_2 + a_3 + \dots + a_n}{n} \]

The absolute error for the \( i^{th} \) observation is:

\[ \Delta a_i = |a_i - \bar{a}| \]

2. Mean Absolute Error

The mean of all absolute errors:

\[ \Delta a_{\text{mean}} = \frac{\sum_{i=1}^{n} |\Delta a_i|}{n} \]

3. Relative (or Fractional) Error

The ratio of mean absolute error to the mean value:

\[ \text{Relative Error} = \frac{\Delta a_{\text{mean}}}{\bar{a}} \]

4. Percentage Error

Relative error expressed as a percentage:

\[ \delta a = \left( \frac{\Delta a_{\text{mean}}}{\bar{a}} \right) \times 100\% \]

Combination of Errors

When quantities are combined, errors propagate as follows:

(i) Addition or Subtraction

\[ X = A \pm B \quad \Rightarrow \quad \Delta X = \Delta A + \Delta B \]

(ii) Multiplication or Division

\[ X = A \times B \times C \quad \Rightarrow \quad \frac{\Delta X}{X} = \frac{\Delta A}{A} + \frac{\Delta B}{B} + \frac{\Delta C}{C} \]

(iii) Powers and Roots

\[ X = A^k \, B^n \, C^l \quad \Rightarrow \quad \frac{\Delta X}{X} = k\frac{\Delta A}{A} + n\frac{\Delta B}{B} + l\frac{\Delta C}{C} \]

Example-Length by Repeated Measurements

A rod’s length is measured n = 5 times using a scale (in cm): \( a_1 = 12.4,\; a_2 = 12.6,\; a_3 = 12.5,\; a_4 = 12.7,\; a_5 = 12.4 \). Find the mean value, absolute errors, mean absolute error, relative error, and percentage error.

1) Mean Value

\[ \bar{a} = \frac{a_1 + a_2 + a_3 + a_4 + a_5}{5} = \frac{12.4 + 12.6 + 12.5 + 12.7 + 12.4}{5} = \frac{62.6}{5} = 12.52\ \text{cm} \]

2) Absolute Errors

\[ \Delta a_i = \left| a_i - \bar{a} \right| \] \[ \Delta a_1 = |12.4 - 12.52| = 0.12,\quad \Delta a_2 = |12.6 - 12.52| = 0.08,\quad \Delta a_3 = |12.5 - 12.52| = 0.02, \] \[ \Delta a_4 = |12.7 - 12.52| = 0.18,\quad \Delta a_5 = |12.4 - 12.52| = 0.12 \]

3) Mean Absolute Error

\[ \Delta a_{\text{mean}} = \frac{\sum_{i=1}^{5} |\Delta a_i|}{5} = \frac{0.12 + 0.08 + 0.02 + 0.18 + 0.12}{5} = \frac{0.52}{5} = 0.104\ \text{cm} \]

Example-Relative and Percentage Error

\[ \text{Relative error} = \frac{\Delta a_{\text{mean}}}{\bar{a}} = \frac{0.104}{12.52} \approx 8.31 \times 10^{-3} \] \[ \text{Percentage error} = \left( \frac{0.104}{12.52} \right)\times 100\% \approx 0.831\% \]

5) Report the Result (Appropriate Significant Figures)

Since the mean absolute error is \( \approx 0.10\ \text{cm} \), report the length to the same decimal place:

\[ a = \bar{a} \pm \Delta a_{\text{mean}} = (12.52 \pm 0.10)\ \text{cm} \;\;\approx\;\; (12.5 \pm 0.1)\ \text{cm}. \]

Example-Propagation of Errors — Area of a Rectangle

A rectangle’s **length** and **width** are measured as: \( L = (20.0 \pm 0.1)\ \text{cm} \) and \( W = (10.0 \pm 0.1)\ \text{cm} \). Find the **area** \( A = L \times W \) with its uncertainty.

1) Best Estimate

\[ A = L \times W = 20.0 \times 10.0 = 200.0\ \text{cm}^2 \]

2) Fractional (Relative) Error Propagation

For multiplication/division:

\[ \frac{\Delta A}{A} = \frac{\Delta L}{L} + \frac{\Delta W}{W} = \frac{0.1}{20.0} + \frac{0.1}{10.0} = 0.005 + 0.010 = 0.015 \]

Absolute Uncertainty in Area

\[ \Delta A = A \times \frac{\Delta A}{A} = 200.0 \times 0.015 = 3.0\ \text{cm}^2 \]

4) Final Report (Match Significant Figures)

\[ A = (200.0 \pm 3.0)\ \text{cm}^2 \;\;\approx\;\; (200 \pm 3)\ \text{cm}^2. \]

Note: We matched the uncertainty’s decimal place to the main value’s precision and reported the uncertainty with one significant figure (or two when appropriate).

Try yourself

A time interval is measured 6 times (in s): \( 2.10,\ 2.12,\ 2.08,\ 2.11,\ 2.09,\ 2.12 \). Compute \( \bar{t} \), \( \Delta t_{\text{mean}} \), and percentage error. (Hint: follow Example A.)
A cylinder’s diameter is \( d = (2.50 \pm 0.01)\ \text{cm} \) and height is \( h = (5.00 \pm 0.02)\ \text{cm} \). Find the volume \( V = \pi (d/2)^2 h \) and its uncertainty using fractional error rules. (Treat \( \pi \) as exact.)

Quick Recap

Absolute Error: difference between observed and mean value.
Mean Absolute Error: average of absolute errors.
Relative Error: ratio of mean error to mean value.
Percentage Error: relative error × 100.
Propagation rules: errors add in sums, and add fractionally in products or powers.

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