Units & Measurement

Accuracy, Precision and Errors

The result of every measurement contains some uncertainty. This uncertainty is called error. The accuracy is a measure of how close the measured value is to the true value of the quantity. Precision tells us to what resolution or limit the quantity is measured. There are two types of errors, viz. systematic error and random error.

Systematic Errors: Errors that tend to be in one direction, i.e. either positive or negative. Some of the sources of systematic errors are as follows:

Instrumental errors: Such errors arise from the errors due to imperfect design or calibration of the measuring instrument, zero error in the instrument, etc.
Imperfection in Experimental Technique or Procedure: When you place a thermometer under the armpit, it will always give a temperature lower than the actual value of body temperature. External conditions during the experiment may also affect the measurement.
Personal Errors: Such errors can happen due to personal bias, lack of proper setting of apparatus or carelessness during observations. Parallax error is an example of personal error.

Random Errors: Such errors occur irregularly. These can arise due to random and unpredictable fluctuations in experimental conditions.

Least Count Error: The smallest value that can be measured by a measuring instrument is called the least count of that instrument. Error associated with the resolution of an instrument is called least count error.

Absolute Error: The magnitude of the difference between the individual measurement and the true value of the quantity is called the absolute error of the measurement. This is denoted by |Δa |.

We know that arithmetic mean of several observations is considered as true value. So, errors in individual measurement values from true value are:

Δa₁ = a₁ - a_mean

Δa₂ = a₂ - a_mean

Δa_n = a_n - a_mean

Here; Δa calculated may be positive or negative in different cases. But absolute error |Δa| will always be positive.

Mean Absolute Error: The arithmetic mean of all the absolute errors is taken as the final or mean absolute error of the value of the physical quantity a. It is represented by Δa_mean. Thus,

Δamean = (|Δa₁|+|Δa₂ |+|Δa₃|+...+ |Δa_n|)/n

Relative Error: The ratio of the mean absolute error to the mean value. This can be given by following equation.

(Δa_mean/a_mean)

When relative error is expressed in per cent, it is called percentage error (δa). Thus, Percentage error

δa = (Δa_mean/a_mean) × 100%

Combination of Errors

Error of a Sum or a Difference: When two quantities are added or subtracted, the absolute error in the final result is the sum of the absolute errors in the individual quantities.

Let us assume that two physical quantities A and B have measured values A ± ΔA, B ± ΔB respectively. Here, ΔA and ΔB are their absolute errors. Now, we need to find the error ΔZ in the sum:

Z = A + B

By addition, we have: Z ± ΔZ = (A ± ΔA) + (B ± ΔB)

The maximum possible error in Z

ΔZ = ΔA + ΔB

For the difference Z = A – B, we have

Z ± Δ Z = (A ± ΔA) – (B ± ΔB)

= (A – B) ± ΔA ± ΔB

or, ± ΔZ = ± ΔA ± ΔB

The maximum value of the error ΔZ is again ΔA + ΔB.

Error of a product or a quotient: When two quantities are multiplied or divided, the relative error in the result is the sum of the relative errors in the multipliers.

Let us assume, Z = AB and the measured values of A and B are A ± ΔA and B ± ΔB. Then

Z ± ΔZ = (A ± ΔA) (B ± ΔB)

= AB ± B ΔA ± A ΔB ± ΔA ΔB.

Dividing LHS by Z and RHS by AB we have,

1±(ΔZ/Z) = 1 ± (ΔA/A) ± (ΔB/B) ± (ΔA/A)(ΔB/B).

Since ΔA and ΔB are small, we shall ignore their product. Hence the maximum relative error

ΔZ/ Z = (ΔA/A) + (ΔB/B).