Question

In the experiment of a simple pendulum to decide the value of acceleration due to gravity we get the value of the time period $1.328,1.325,1.326,1.330,1.336$ and $1.334$ sec. Find
(A) Average value of time period
(B) Mean absolute error
(C) Relative error
(D) Percentage error

Answer 1

For average value of time period, add all values and divide by the number of values. Mean absolute error is to subtract the mean from each experimental value, find the absolute of these values. Finally, add the absolute values and divide by the number of values. Relative error is given as absolute error divided by the true value. Percentage error is relative error converted to percentage form.
Formula used: $\overline x = \dfrac{1}{n}\sum\limits_{i = 1}^n {{x_i}}$ where $\overline x$ is the average value or mean value and ${x_i}$ is the individual values of different measurement.
$MAE = \dfrac{1}{n}\sum\limits_{i = 1}^n {\left| {{x_i} - x} \right|}$ where $MAE$ means Mean absolute error, $n$ is the number of values, $x$ is the true value, considered as the mean value when no true value is given or known.
$RE = \dfrac{{AE}}{{TV}}$ where $RE$ is the relative error, $AE$ is the absolute value and $TV$ is the true value.
$PE = RE \times 100\%$ where $PE$ is the percentage error.

Complete step by step answer:
- For Average value, we say
$\overline x = \dfrac{{1.328 + 1.325 + 1.326 + 1.330 + 1.336 + 1.334}}{6} = \dfrac{{7.979}}{6}$
$\Rightarrow \overline x = 1.329833... \approx 1.330$
$\therefore \overline x \approx 1.330$
- For Mean Absolute error, first, we subtract the mean from each value to find their respective error and find the absolute value of all of them, as in:
$\left| {{x_1} - \overline x } \right| = \left| {1.328 - 1.330} \right|,\left| {1.325 - 1.330} \right|,\left| {1.326 - 1.330} \right|,\left| {1.330 - 1,330} \right|,\left| {1.336 - 1.330} \right|,\left| {1.334 - 1.330} \right|$ which gives $0.002,0.005,0.004,0.000,0.006,0.004$ respectively.
Now, we add these absolute values together as in:
$\sum {\left| {{x_i} - \overline x } \right|} = 0.002 + 0.005 + 0.004 + 0.000 + 0.006 + 0.004 = 0.021$
Finally, we divide this answer by the number of values
$\dfrac{1}{n}\sum {\left| {{x_i} - \overline x } \right|} = \dfrac{{0.021}}{6} = 0.0035$
$\therefore MAE = 0.0035$
- Relative error is calculated simply by dividing the Absolute error by the mean value as in:
$RE = \dfrac{{0.0035}}{{1.330}} = 0.0026315...$
$\therefore RE = 0.0026$
- For Percentage error, we multiply the relative error by $100\%$ .
$PE = 0.0026 \times 100\%$
$\therefore PE = 0.26\%$.

Note:
A faster method and error proof method would be to use tables to clarify your values as done below.

S/N| ${x_i}$ | ${x_i} - \overline x$ ( From i $\overline x = 1.330$ )| $\left| {{x_i} - \overline x } \right|$
---|---|---|---
1| $1.328$ | $- 0.002$ | $0.002$
2| $1.325$ | $- 0.005,$ | $0.005$
3| $1.326$ | $- 0.004$ | $0.004$
4| $1.330$ | $0.000$ | $0.000$
5| $1.336$ | $0.006$ | $0.006$
6| $1.334$ | $0.004$ | $0.004$
| ${\sum x _i} = 7.979$ | | $\sum {\left| {{x_i} - \overline x } \right|} = 0.021$

- To calculate average value, each entry in column ${x_i}$ is added together to give $\sum {{x_i}} = 7.979$ in row 7 as in:
$1.328 + 1.325 + 1.326 + 1.330 + 1.336 + 1.334 = 7.979 \\\ \\\$ .
Then we complete the solution by dividing the sum by the number of entries as in:
$\overline x = \dfrac{1}{n}\sum {{x_i}} = \dfrac{{7.979}}{6} = 1.330$
- For mean average value, we subtracted the average value from each entry of column ${x_i}$ and placed the corresponding answers in column ${x_i} - \overline x$ . In the next column $\left| {{x_i} - \overline x } \right|$ , we find the absolute of the previous entries, i.e. eliminating all negatives and only using its positive number. Then, similarly as done above, we add all entries of the column, to give $\sum {\left| {{x_i} - \overline x } \right|} = 0.021$ and divide it by the number of entries, as in:
$MAE = \dfrac{1}{n}\sum {\left| {{x_i} - \overline x } \right|} = \dfrac{{0.021}}{6}$ .
Relative error and percentage error are calculated identically as in the step by step solution.