Question

In a certain experiment, the mean absolute error (Δ b_mean) for the 4 measurements b₁, b₂, b₃, and b₄ will be-

1. \(\frac {b_{1} +b_{2}+b_{3}+b_{4}} {4}\)
2. \(\frac {(| \Delta b_{1}| +| \Delta b_{2}|+| \Delta b_{3}|+| \Delta b_{4}|)} {4}\)
3. \(\frac {( \Delta b_{1} + \Delta b_{2}+\Delta b_{3}+ \Delta b_{4})} {4}\)
4. \(\frac {(| \Delta b_{1}^2| +| \Delta b_{2}^2|+| \Delta b_{3}^2|+| \Delta b_{4}^2|)} {4}\)

Answer 1

Correct Answer - Option 2 : \(\frac {(| \Delta b_{1}| +| \Delta b_{2}|+| \Delta b_{3}|+| \Delta b_{4}|)} {4}\)

CONCEPT:

• Error: It is the Uncertainty in measurement by any measuring instrument.
• Error in measurement can be broadly classified as

1. Systematic errors
2. Random errors.

• Absolute error is the magnitude of difference between individual measurement and the true value of the quantity being measured.
• Mean absolute error is the arithmetic mean of all the absolute errors.

CALCULATION:

Given that: 4 measurements b1, b2, b3, and b4

mean of the measurements:

\(⇒ b_{mean} = \frac { b_{1} + b_2 +b_3 + b_4}{4}\)      (i)

Absolute error is given by:

⇒ Δ a_n = a_n - a_mean

So absolute error:

⇒ Δ b₁ = b₁ - b_mean               (ii) 

⇒ Δ b2 = b2 - bmean               (iii)

⇒ Δ b3 = b3 - bmean               (iv)

⇒ Δ b4 = b4 - bmean                (v) 

Using (ii), (iii), (iv), and (v) the mean absolute error :

\(\Rightarrow \Delta b_{mean} = \frac {(| \Delta b_{1}| +| \Delta b_{2}|+| \Delta b_{3}|+| \Delta b_{4}|)} {4}\)

Hence option 2) is correct.