error  The  difference  between  observed  value  of  any physical  quantity and  the  `actual'  value is called error of observation.  All  measurements  in science  are generally  inaccurate so that the `actual' or `accurate' value can  not be  found.  We  can  only  specify the value within certain limits. The more accurate the measurements, the closer these  limits. Errors  are of two types, systematic and random. Systematic  errors  occur due to faulty apparatus or inaccurately labeled scale.

Random  error  arises  primarily due to human errors. Taking average  of  a large number  of  reading  reduces the random errors.

Calculation  of  error: Consider  a physical quantity Q, that involves measurement of two other quantities, a and b.

(i)  If  Q  = a+b or Q = a- b, the error in Q, DQ is the sum of errors in a and b,

DQ= Da + Db

(ii) If Q = a´ b or Q = a/b, then

[DQ/Q]= [Da/a] + [Db/b]

(iii) If Q = a ´ bⁿ, then

[DQ/Q]= [Da/a] + n [Db/b]

Taking  a number of readings will reduce the error in the final  answer. Suppose a quantity q_ounits, is measured n times and recorded as q₁, q₂, ....... q_n units.

If Dq_r is the error in measurement in rth reading, then

q_r = q_o + D q_r

The arithmetic mean q of n measurements is

q = (q₁ + q₂ + q₃ +q_n )/n

q = q_o + [Dq₁ +Dq₂ +Dq₃ +…..]/n

q = q_o + e

 The  random  errors have positive as well as negative values, and  as  n is increased e will become smaller. q can be taken as the best value of the measured quantity. The result should be quoted as q ± e.