Page 38 - Class-11-Physics-Part-1_Neat

24 PHYSICS

to sign and size. These can arise due to random as to underestimate the true value of the
and unpredictable fluctuations in experimental quantity.
conditions (e.g. unpredictable fluctuations in The magnitude of the difference
temperature, voltage supply, mechanical between the individual measurement and
vibrations of experimental set-ups, etc), personal the true value of the quantity is called the
(unbiased) errors by the observer taking absolute error of the measurement. This
readings, etc. For example, when the same is denoted by |∆a |. In absence of any other
person repeats the same observation, it is very method of knowing true value, we considered
likely that he may get different readings arithmatic mean as the true value. Then the
everytime.
errors in the individual measurement values
Least count error from the true value, are
The smallest value that can be measured by the ∆a = a – a mean ,
1
1
measuring instrument is called its least count. ∆a = a – a ,
2
mean
2
All the readings or measured values are good only .... .... ....
up to this value. .... .... ....
The least count error is the error ∆a = a – a
associated with the resolution of the instrument. n n mean
For example, a vernier callipers has the least The ∆a calculated above may be positive in
count as 0.01cm; a spherometer may have a certain cases and negative in some other
least count of 0.001 cm. Least count error cases. But absolute error |∆a| will always
belongs to the category of random errors but be positive.
within a limited size; it occurs with both (b) The arithmetic mean of all the absolute errors
systematic and random errors. If we use a metre is taken as the final or mean absolute error
scale for measurement of length, it may have of the value of the physical quantity a. It is
graduations at 1 mm division scale spacing or .
interval. represented by ∆a mean
Using instruments of higher precision, Thus,
improving experimental techniques, etc., we can
reduce the least count error. Repeating the ∆a mean = (|∆a |+|∆a |+|∆a |+...+ |∆a |)/n
2
1
n
3
observations several times and taking the (2.6)
arithmetic mean of all the observations, the
mean value would be very close to the true value n
of the measured quantity. = ∑ |∆a |/n (2.7)
i
= i 1
2.6.1 Absolute Error, Relative Error and If we do a single measurement, the value we
Percentage Error
get may be in the range a ± ∆a
(a) Suppose the values obtained in several mean mean
measurements are a , a , a ...., a . The i.e. a = a ± ∆a
2
3
n
1
arithmetic mean of these values is taken as mean mean
the best possible value of the quantity under or,
the given conditions of measurement as : a mean – ∆a mean ≤ a ≤ a mean + ∆a mean
(2.8)
a = (a +a +a +...+a ) / n (2.4)
mean 1 2 3 n This implies that any measurement of the
or, physical quantity a is likely to lie between
n (a mean + ∆a mean ) and (a mean − ∆a mean ).
mean ∑ (c) Instead of the absolute error, we often use
=
a a / n (2.5)
i
=
i 1 the relative error or the percentage error
This is because, as explained earlier, it is (δa). The relative error is the ratio of the
∆
reasonable to suppose that individual mean absolute error ∆ ∆∆ ∆a mean to the mean
measurements are as likely to overestimate value a mean of the quantity measured.

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