(a) Suppose the values obtained in several measurements are $a_1$, $a_2$, $a_3$...., $a_n$. The arithmetic mean of these values is taken as the best possible value of the quantity under the given conditions of measurement as :
$a_{mean}=\frac{a_1+a_2+a_3+ ...+a_n}{n}$, or
$a_{mean}=\frac{\sum_{n}^{i=1}a_i}{n}$
This is because, as explained earlier, it is reasonable to suppose that individual measurements are as likely to overestimate as to underestimate the true value of the quantity.
The magnitude of the difference between the individual measurement and the true value of the quantity is called the absolute error of the measurement. This is denoted by |∆a |. In absence of any other method of knowing true value, we considered arithmatic mean as the true value. Then the errors in the individual measurement values from the true value, are
$\Delta a=a_1-a_{mean}$
$\Delta a=a_2-a_{mean}$
....
$\Delta a_n=a_n-a_{mean}$
The ∆a calculated above may be positive in certain cases and negative in some other cases. But absolute error |∆a| will always be positive.
(b) The arithmetic mean of all the absolute errors is taken as the final or mean absolute error of the value of the physical quantity a. It is represented by ∆$a_{mean}$.
Thus,
$a_{mean}=\frac{|\Delta a_1+|\Delta a_2|+|\Delta a_3|+ ...+|\Delta a_n|}{n}$, or
$a_{mean}=\frac{\sum_{n}^{i=1}|\Delta a_i|}{n}$
If we do a single measurement, the value we get may be in the range $a_{mean}$ ± ∆$a_{mean}$
i.e $a=a_{mean}$ ± ∆$a_{mean}$
or, $a_{mean}$ - ∆$a_{mean}$ ≤ a ≤ $a_{mean}$ + ∆$a_{mean}$
This implies that any measurement of the physical quantity a is likely to lie between
($a_{mean}$ + ∆$a_{mean}$) and ($a_{mean}$ - ∆$a_{mean}$)
(c) Instead of the absolute error, we often use the relative error or the percentage error (δa). The relative error is the ratio of the mean absolute error ∆$a_mean$ to the mean value $a_{mean}$ of the quantity measured.
Relative error = $\frac{\Delta a_{mean}}{a_{mean}}$ (2.9)
When the relative error is expressed in per cent, it is called the percentage error (δa).
Thus, Percentage error
δa = $\frac{\Delta a_{mean}}{a_{mean}}$ × 100% (2.10)
Let us now consider an example.
Example 1
Two clocks are being tested against a standard clock located in a national laboratory. At 12:00:00 noon by the standard clock, the readings of the two clocks are :
|
|
Clock 1 |
Clock 2 |
|
Monday |
12:00:05 |
10:15:06 |
|
Tuesday |
12:01:15 |
10:14:59 |
|
Wednesday |
11:59:08 |
10:15:18 |
|
Thursday |
12:01:50 |
10:15:07 |
|
Friday |
11:59:15 |
10:14:53 |
|
Saturday |
12:01:30 |
10:15:24 |
|
Sunday |
12:01:19 |
10:15:11 |
If you are doing an experiment that requires precision time interval measurements, which of the two clocks will you prefer?
Answer
The range of variation over the seven days of observations is 162 s for clock 1, and 31 s for clock 2. The average reading of clock 1 is much closer to the standard time than the average reading of clock 2. The important point is that a clock’s zero error is not as significant for precision work as its variation, because a ‘zero-error’ can always be easily corrected. Hence clock 2 is to be preferred to clock 1.
Example 2
We measure the period of oscillation of a simple pendulum. In successive measurements, the readings turn out to be 2.63 s, 2.56 s, 2.42 s, 2.71s and 2.80 s. Calculate the absolute errors, relative error or percentage error.
Answer
The mean period of oscillation of the pendulum
$T=\frac{(2.63+2.56+2.42+ 2.71+2.80)s}{5}$
$=\frac{13.12 \ s}{5}=2.623 \ s=2.62 \ s$
As the periods are measured to a resolution of 0.01 s, all times are to the second decimal; it is proper to put this mean period also to the second decimal.
The errors in the measurements are
2.63 s – 2.62 s = 0.01 s
2.56 s – 2.62 s = – 0.06 s
2.42 s – 2.62 s = – 0.20 s
2.71 s – 2.62 s = 0.09 s
2.80 s – 2.62 s = 0.18 s
Note that the errors have the same units as the quantity to be measured.
The arithmetic mean of all the absolute errors (for arithmetic mean, we take only the magnitudes) is
$\Delta T_{mean}=\frac{(0.01+0.06+0.20+0.09+0.18)s}{5}$
$=\frac{0.54 \ s}{5}=0.11 \ s$
That means, the period of oscillation of the simple pendulum is (2.62 ± 0.11) s i.e. it lies between (2.62 + 0.11) s and (2.62 – 0.11) s or between 2.73 s and 2.51 s. As the arithmetic mean of all the absolute errors is 0.11 s, there is already an error in the tenth of a second. Hence there is no point in giving the period to a hundredth. A more correct way will be to write
T = 2.6 ± 0.1 s
Note that the last numeral 6 is unreliable, since it may be anything between 5 and 7. We indicate this by saying that the measurement has two significant figures. In this case, the two significant figures are 2, which is reliable and 6, which has an error associated with it. You will learn more about the significant figures in section 2.7.
For this example, the relative error or the percentage error is
δa = $\frac{0.1}{2.6}$ × 100% = 4%
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