Error Analysis
i) Absolute Error
The magnitude of difference between the true value and the measured value of a quantity is called absolute error. If a_1, a_2_, a_3_, ……….a_n are the measured values of any quantity ‘a’ in an experiment performed n times, then the arithmetic mean of these values is called the true value (_a_m) of the quantity.
\(\begin{aligned} \text{am} & = \frac{a_{\text{}1} + a_{\text{}2} + a_{\text{}3} + \ldots + a_{\text{}n}}{n} \end{aligned}\) or
\(\begin{aligned} \text{am} & = \frac{1}{n} \sum_{i=1}^{n} a_{\text{}i} \end{aligned}\)………….. or
The absolute error in measured values is given by
|∆_a_1| = |_a_m – _a_1|
|∆_a_2| = |_a_m – _a_2|
………………
………………
|∆_a_n| = |_a_m – _a_n|
ii) Mean Absolute error
The arithmetic mean of absolute errors in all the measurements is called the mean absolute error.
∆ ∆ ∆ ∆ ∆
a a a a a
nm n=
+ + + +1 2 3 ……………
or = = ∑1
1_n ai_
i
n
∆
If _a_m is the true value and ∆_a_m is the mean absolute error then the magnitude of the quantity may lie between _a_m + ∆_a_m and _a_m - ∆_a_m
iii) Relative error The ratio of the mean absolute error to the mean value is called relative error. This is also called as fractional error. Thus
Relative error =
=
Mean absolute error Mean value
a a
m
m
∆
Relative error expresses how large the absolute error is compared to the total size of the object measured. For example, a driver’s speedometer shows that his car is travelling at 60 km h−1 when it is actually moving at 62 km h−1. Then absolute error of speedometer is 62-60 km h−1 = 2 km h−1 Relative error of the measurement is 2 km h−1 / 62 km h−1 = 0.032.
iv) Percentage error The relative error expressed as a percentage is called percentage error.
Percentage error = ∆_a a_
m
m
×100%
A percentage error very close to zero means one is close to the targeted value, which is good and acceptable. It is always necessary to understand whether error is due to impression of equipment used or a mistake in the experimentation.
EXAMPLE 1.4
In a series of successive measurements in an experiment, the readings of the period of oscillation of a simple pendulum were found to be 2.63s, 2.56 s, 2.42s, 2.71s and
| Mean absolute erro | r | |
|---|---|---|
| Mean value∆am | ||
| am |
| aa++ aa ++…………..12 3 n | or |
|---|---|
| n1n | |
| ∑ani =1 |
2.80s. Calculate (i) the mean value of the period of oscillation (ii) the absolute error in each measurement (iii) the mean absolute error (iv) the relative error (v) the percentage error. Express the result in proper form.
Solution
_t s_1 2 63= . , _t s_2 2 56= . , _t s_3 2 42= . , _t s_4 2 71= . , _t s_5 2 80= .
(i) T t t t t t
m = + + + +
= + + + +
1 2 3 4 5
5 2 63 2 56 2 42 2 71 2 80
5 . . . . .
Tm = 13 12
5 2 624. .= s
Tm= 2 62. s (Rounded off to 2nd decimal place)
(ii) Absolute error |ΔT| = |Tm - t|
∆ ∆ ∆
T s T s T
1
2
3
2 62 2 63 0 01 2 62 2 56 0 06 2 62 2 42 0 2
= - = + = - = + = - = +
. . .
. . .
. . . 0 2 62 2 71 0 09 2 62 2 80 0 18
4
5
s T s T s
∆ ∆
= - = + = - = +
. . .
. . .
(iii) Mean absolute error = Σ ∆_T_
n i
∆_Tm_ = + + + +0 01 0 06 0 20 0 09 0 18 5
. . . . .
∆_T s sm_ = = =0 54 5
0 108 0 11. . . (Rounded
off to 2nd decimal place)
(iv) Relative error:
ST = ∆_T T_
m
m
= =0 11 2 62
0 0. .
. 419
ST = 0.04
(v) Percentage error in T = 0.04 × 100% = 4%
(vi) Time period of simple pendulum = T = (2.62 ± 0.11)s