Propagation of Errors

Final result of an experiment is calculated from a number of observations taken from different instruments, connected through a formula.

Maximum permissible error in different cases is calculated as follows

Result involving sum of two observed quantities

X is the sum of 2 observed quantities a and b.

X = a + b

Maximum absolute error in X = Maximum absolute error in a + Maximum absolute error in b

Result involving difference of two observed quantities

Suppose X = a - b

Let Da and Db be absolute errors in measurements of quantities a and b, values of a and b and DX be maximum error in X.

 Maximum absolute error in X = Maximum absolute error in a + Maximum absolute error in b

From equations (1) and (2) it is evident that, when result involves sum or difference of 2 observed quantities, absolute error is the sum of absolute errors in the observed quantities.

Result involving the product of two observed quantities

Suppose X = ab

Let Da and Db be absolute errors in measurements of quantities a and b, values of a and b and DX be the maximum possible error in X.

Dividing both sides by X = ab, we get

are relative errors of fractional errors in values of a, b and x. Neglecting  as its product is very small.

The above result is obtained by logarithmic differentiation.

Take log on both sides,

Log X = log a + log b

Differentiating, we get ,

Thus, maximum relative error in X = maximum relative error in a x maximum relative error in b

Maximum absolute error in X = Maximum absolute error in a + Maximum absolute error in b

Result involving quotient of 2 observed quantities

Let Da and Db be absolute errors in measurement of quantities a and b and DX be maximum possible error in X.

Maximum possible relative error in X,

Maximum relative error in X = maximum relative error in a + maximum relative error in b

Maximum percentage error in X,

i.e., Maximum percentage error in X = maximum percentage error in a + maximum percentage in b. From equations 3, 4, 5 and 6, it is seen that when the result involves the multiplication or quotient of 2 observed quantities, the maximum possible relative error in the result is equal to the sum of the relative errors in the observed quantities.

Result involving product of powers of observed quantities

Relative error in an is n times the relative error a

It can be proved that maximum relative error in X,

Also, maximum percentage error in X,

Maximum percentage error in X = l times maximum percentage error in a + m times maximum percentage error in b + n times maximum percentage error in c.