Class 11 Physics Units and Measurements  Errors in a series of Measurements 
Errors in a series of Measurements
Suppose the values obtained in several measurement are a_{1}, a_{2}, a_{3}, …, a_{n}.
Arithmetic mean, a_{mean} = (a_{1}+ a_{2 }+ a_{3}+ … + a_{n})/n=
Δa_{1} = a_{mean}  a_{1}, Δa_{2} = a_{mean}  a_{2}, ……. ,Δa_{n} = a_{mean} – a_{n}
Δa_{mean} = (Δa_{1} + Δa_{2} +Δa_{3} + …. +Δa_{n}) / n =
For single measurement, the value of ‘a’ is always in the range a_{mean}± Δa_{mean}
So, a = a_{mean} ± Δa_{mean} Or a_{mean}  Δa_{mean}< a <a_{mean} + Δa_{mean}
Relative Error = Δa_{mean} / a_{mean}
δa = (Δa_{mean} / a_{mean}) x 100%
Combinations of Errors
If a quantity depends on two or more other quantities, the combination of errors in the two quantities helps to determine and predict the errors in the resultant quantity. There are several procedures for this.
Suppose two quantities A and B have values as A ± ΔA and B ± ΔB. Z is the result and ΔZ is the error due to combination of A and B.
Criteria 
Sum or Difference 
Product 
Raised to Power 
Resultant value Z 
Z = A ± B 
Z = AB 
Z = A^{k} 
Result with error 
Z ± ΔZ = (A ± ΔA) + (B ± ΔB) 
Z ± ΔZ = (A ± ΔA) (B ± ΔB) 
Z ± ΔZ = (A ± ΔA)^{k} 
Resultant error range 
± ΔZ = ± ΔA ± ΔB 
ΔZ/Z = ΔA/A ± ΔB/B 

Maximum error 
ΔZ = ΔA + ΔB 
ΔZ/Z = ΔA/A + ΔB/B 
ΔZ/Z = k(ΔA/A) 
Error 
Sum of absolute errors 
Sum of relative errors 
k times relative error 
.