Question

Six successive measurements of length to the accuracy of 0.01cm gave the following values:
5.12cm, 5.11cm, 5.10cm, 5.13cm, 5.13cm and 5.12cm
Find mean length, mean absolute error and percentage error.

Answer 1

Hint: As a first step, one could read the question well and hence note down the given values. One could recall the formulas for finding the mentioned quantities. For finding the mean absolute error one could take the absolute error for each value and then simply take the mean to get the answer.

Complete step-by-step solution:
In the question we are provided with six successive measurements of length with an accuracy of 0.01cm. We are supposed to find the mean length, mean absolute error and percentage error.
Mean length could easily be taken by taking the sum of the set of measurements given and then dividing this value by the total number of values. That is mean of the given set of measurements would be,
$Mean=\dfrac{5.12+5.11+5.10+5.13+5.13+5.12}{6}=5.118\approx 5.12cm$
Mean absolute error basically is the difference between the measured value with the true value.
$\left| \Delta {{a}_{1}} \right|=\left| 5.12-5.12 \right|=0$
$\left| \Delta {{a}_{2}} \right|=\left| 5.12-5.11 \right|=0.01$
$\left| \Delta {{a}_{3}} \right|=\left| 5.12-5.10 \right|=0.02$
$\left| \Delta {{a}_{4}} \right|=\left| 5.12-5.13 \right|=0.01$
$\left| \Delta {{a}_{5}} \right|=\left| 5.12-5.13 \right|=0.01$
$\left| \Delta {{a}_{4}} \right|=\left| 5.12-5.12 \right|=0$
The mean absolute error (MAE) would be the mean of the above set of absolute errors, that is,
$MAE=\dfrac{0+0.01+0.02+0.01+0.01+0}{6}=0.008$
Now that we have found the mean absolute error (MAE), finding the percentage error is pretty easy. You could divide the mean absolute error by the real value and then multiply it by 100. That is,
$P.E=\dfrac{0.008}{5.12}\times 100=0.15%$
Therefore, we found the mean value, mean absolute error and the percentage error to be 5.12cm, 0.008cm and 0.15% respectively.

Note: MAE that is the mean absolute error could be defined as the measure of errors that is measured between paired observations that is used to express the same phenomenon. Arithmetic means no new stuff to you people as we have been learning this from smaller classes in mathematics.