Question

A set of measurements of a certain parameter is: $20.00,\; 19.75,\; 18.25,\; 17.01$. Find the relative error for the set of measurements.

• A. $0.12$
• B. $0.06$
• C. $0.09$
• D. $0.17$

Hint:

For numerical problems on errors, always calculate mean first, then mean absolute error, and finally divide by the mean to get relative error.

Answer 1

The Correct Option is B

Concept: For repeated measurements:
Mean value $=\dfrac{\text{sum of observations}}{\text{number of observations}}$
Mean absolute error $=\dfrac{\sum |x_i-\bar{x}|}{n}$
Relative error $=\dfrac{\text{Mean absolute error}}{\bar{x}}$
Step 1: Calculate the mean value \[ \bar{x}=\frac{20.00+19.75+18.25+17.01}{4} =\frac{75.01}{4} =18.75 \]
Step 2: Find absolute errors \[ |20.00-18.75|=1.25 \] \[ |19.75-18.75|=1.00 \] \[ |18.25-18.75|=0.50 \] \[ |17.01-18.75|=1.74 \]
Step 3: Mean absolute error \[ \Delta x=\frac{1.25+1.00+0.50+1.74}{4} =\frac{4.49}{4} \approx1.12 \] Step 4: Relative error \[ \text{Relative error}=\frac{\Delta x}{\bar{x}} =\frac{1.12}{18.75} \approx0.06 \]
Step 5: Hence, the relative error for the given set of measurements is: \[ \boxed{0.06} \]