Errors in Measurement | Absolute Error, Mean Absolute Error, Relative Error (Fractional Error), Percentage Error

Measurement is a fundamental aspect of physics and engineering. However, despite the best efforts, no measurement is perfect. There is always a difference between the measured value and the true value, which is known as the error of measurement. This topic is crucial for exams like JEE, NEET, and CBSE Class 11, as it forms the basis of error analysis in experiments.

Types of Errors in Measurement

1. Absolute Error

Absolute error refers to the magnitude of the difference between the true value and the measured value of a physical quantity.

• Suppose a physical quantity is measured n times, and the measured values are: \begin{array}{l} a_1, a_2, a_3, …, a_n \end{array}
• The arithmetic mean of these values is: \begin{array}{l} a_m = \frac{a_1 + a_2 + … + a_n}{n} \end{array}
• This mean value is taken as the true value of the quantity if it is unknown otherwise.
• The absolute error in each measurement is given by: \begin{array}{l} \Delta a_i = | a_i – a_m | \end{array} where Δa_i is the absolute error for each individual measurement.

2. Mean Absolute Error

Mean absolute error is the average of all absolute errors and is given by:

\begin{array}{l} \bar{a} = \frac{\sum | a_i – a_m |}{n} \end{array}

This helps in determining the precision of the measurement. The final result of the measurement is written as:

\begin{array}{l} a = a_m \pm \bar{a} \end{array}

3. Relative Error (Fractional Error)

Relative error is the ratio of the mean absolute error to the mean value of the quantity measured.

\begin{array}{l} \text{Relative Error} = \frac{\bar{a}}{a_m} \end{array}

This gives an idea of how significant the error is compared to the measured value.

4. Percentage Error

Percentage error is the relative error expressed in percentage:

Percentage Error=\begin{array}{l} \text{Percentage Error} = \left( \frac{\bar{a}}{a_m} \right) \times 100 \% \end{array}

This helps in assessing the reliability of the measurement.

Conceptual Questions with Answers

Q1: What is an error in measurement?

A: Error in measurement is the difference between the true value and the measured value of a physical quantity.

Q2: How is absolute error different from relative error?

A: Absolute error is the direct difference between measured and true value, whereas relative error is the ratio of absolute error to the true value.

Q3: Why is percentage error useful?

A: Percentage error helps compare the reliability of different measurements by giving a standardized percentage value.

Multiple Choice Questions (MCQs)

MCQ 1: If the measured values of a quantity are 4.5, 4.6, and 4.4, find the mean absolute error.

A) 0.1
B) 0.2
C) 0.05
D) 0.15

Answer: A) 0.1
Explanation:

• Mean value: \begin{array}{l} a_m = \frac{4.5 + 4.6 + 4.4}{3} = 4.5 \end{array}
• Absolute errors: \begin{array}{l} |4.5 – 4.5|, |4.6 – 4.5|, |4.4 – 4.5| = 0, 0.1, 0.1 \end{array}
• Mean absolute error: \begin{array}{l} (0 + 0.1 + 0.1) / 3 = 0.1 \end{array}

MCQ 2: Which of the following is a dimensionless quantity?

A) Absolute Error
B) Percentage Error
C) Measured Value
D) None of these

Answer: B) Percentage Error
Explanation: Since percentage error is a ratio of errors expressed in percentage, it has no unit.

Do You Know?

• Even the most precise instruments have least count errors.
• Random errors can be minimized but never eliminated.
• The smaller the percentage error, the more accurate the measurement.

Worksheet

Solve the following:

1. A physical quantity is measured as 10.4, 10.6, 10.5, and 10.3. Find the mean absolute error.
2. The measured value of a length is 5.2 cm with an absolute error of 0.2 cm. Calculate the percentage error.
3. If the measured value is 25.0 and the true value is 25.4, find the absolute error.

Test Paper (Total Marks: 10)

Important Points for Quick Revision

✅ Error is the difference between the measured value and the true value.

✅ Absolute error gives the magnitude of deviation from the true value.

✅ Mean absolute error is the average of absolute errors.

✅ Relative error is the ratio of absolute error to the measured value.

✅ Percentage error helps compare different measurements.

✅ Errors cannot be completely removed but can be minimized.

Test Your Knowledge (Quiz)

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