Question

In an experiment to measure focal length of a concave mirror, the values of focal length is successive observations are $17.3\;{\text{cm}}$,$17.8\;{\text{cm}}$, $18.3\;{\text{cm}}$, $18.2\;{\text{cm}}$, $17.9\;{\text{cm}}$ and $18.0\;{\text{cm}}$. Calculate mean absolute error and percentage error. Express the result in a proper way.

Answer 1

Hint: Firstly, calculate the mean of the observed focal lengths of the concave mirror. Secondly, find the percentage error in each observation by subtracting it from mean value obtained. Finally, calculate absolute error and percentage error of all the mean values.

Complete step by solution:
We know that the mean of a number terms is given by the formula written below.
\[m = \dfrac{{{\text{sum}}\;{\text{of}}\;{\text{terms}}}}{{{\text{number}}\;{\text{of}}\;{\text{terms}}}}\]
Now, calculate the mean of all the observations of focal length,
 \[{f_{{\text{mean}}}} = \dfrac{{17.3 + 17.8 + 18.3 + 18.2 + 17.9 + 18.0}}{6}\]
Simplifying above expression,
\[{f_{{\text{mean}}}} = 17.9\;{\text{cm}}\]
Now, we know that the absolute errors, $\Delta x$ can be obtained from by subtracting mean value obtained from each of the observations, i.e. $\Delta x = {x_{{\text{actual}}}} - {x_{{\text{mean}}}}$.
Let the errors obtained in each observation are ${e _1},\;{e_2},\;{e_3},\;...{e_n}$, then use the formula $\Delta x = {x_{{\text{actual}}}} - {x_{{\text{mean}}}}$ to find the error.
\[{e_1} = 17.3 - 17.9\]
   \[ = - 0.6\]
\[{e_2} = 17.8 - 17.9\]
    $ = - 0.1$
\[{e_3} = 18.3 - 17.9\]
    $ = 0.4$
\[{e_4} = 18.2 - 17.9\]
     $ = 0.3$
${e_5} = 17.9 - 17.9$
     $ = 0.0$
\[{e_6} = 18 - 17.9\]
     \[ = 0.1\]
We know that the formula for finding error is given by,
\[\Delta {f_{{\text{mean}}}} = \dfrac{{\left| {{e_1}} \right| + \left| {{e_2}} \right| + \left| {{e_3}} \right| + ... + \left| {{e_n}} \right|}}{{{\text{number}}\;{\text{of}}\;{\text{terms}}}}\]
Now, substitute the errors values obtained in the above formula.
\[\Delta {f_{{\text{mean}}}} = \dfrac{{\left| { - 0.6} \right| + \left| { - 0.1} \right| + \left| { - 0.4} \right| + \left| {0.3} \right| + \left| 0 \right| + \left| {0.1} \right|}}{6}\]
           \[ = 0.25\]
Since the absolute error is obtained, we just need to find the percentage error.
As the formula for percentage error is given by,
${\text{percentage}}\;{\text{error}} = \dfrac{{{\text{error}}}}{{{\text{mean}}\;{\text{value}}}} \times 100$
Now, substitute the values obtained into above formula,
${\text{percentage}}\;{\text{error}} = \dfrac{{0.25}}{{17.9}} \times 100$
                              $ = 1.39\% $

Therefore, the absolute error is $0.25$ and the error percentage is $1.39\;\% $.

Note:In these types of questions, find the mean value of all the given terms. Once the mean value is obtained, you can subtract it from each observation to find the absolute value.
The absolute value can be obtained by subtracting the actual value, ${x_a}$ , from measured value ${x_0}$, i.e., $\Delta x = {x_0} - {x_a}$. You can directly add the modulus into the formula of absolute value to get the result.