IEMDaily - Video Lecture Notes

Formula

Absolute, Relative or Percentage Error for a quantity that is sum of measured quantities

y = y_{1} + y_{2} + y_{3} . . y_{n}

Then absolute error will be given by:

Δ y = Δ y_{1} + Δ y_{2} + Δ y_{3} + . . Δ y_{n}

And its relative error will be given by

\frac{Δ y}{y}

Definition

Define and calculate percentage error

Definition:
The relative error expressed in percentage (i.e. multiplied by 100) is called percentage error.

∴ P e r c e n t a g e e r r o r = \frac{Δ a_{m}}{a_{m}} \times 100 %

Problem:
In an experiment, the values of refractive indices of glass were found to be 1.54, 1.53, 1.44, 1.54,1.56 and 1.45 in successive measurements.
Find the Percentage error.

Solution:

M e a n = \frac{1.54 + 1.53 + 1.44 + 1.54 + 1.56 + 1.45}{6}

= 1.51

| M e a n a b s o l u t e e r r o r | = \sqrt{\frac{1}{6} \sum (\bar{x_{1}} - \bar{x})^{2}} = 0.04

R e l a t i v e e r r o r = \frac{A b s o l u t e e r r o r}{M e a n} = \frac{0.04}{1.51} = 0.026 \approx 0.03

e r r o r =

R e l a t i v e

e r r o r \times 100 = 3

Definition

Percentage Error

When the relative error is expressed in percent, it is called the percentage error (

δ a

∴

Percentage error (

δ a

)

= \frac{δ a_{m e a n}}{a_{m e a n}} \times 100

Definition

Absolute, Relative or Percentage Error

y = y_{1}^{a} \times y_{2}^{b}

Relative error is given by:

\frac{Δ y}{y} = a \frac{Δ y_{1}}{y_{1}} + b \frac{Δ y_{2}}{y_{2}}

Absolute error will be given by measuring

Δ y

value.

Example

Absolute, Relative or Percentage Error for a quantity that is product of measured quantities

y = y_{1} \times y_{2}

Relative error is given by:

\frac{Δ y}{y} = \frac{Δ y_{1}}{y_{1}} + \frac{Δ y_{2}}{y_{2}}

Absolute error will be given by measuring

Δ y

value.

Example

Absolute, Relative or Percentage Error for a quantity which is a measured quantity raised to a power

y = y_{1}^{a}

Relative error is given by:

\frac{Δ y}{y} = a \frac{Δ y_{1}}{y_{1}}

Absolute error will be given by measuring

Δ y

value.

Definition

Types of errors

Instrumental (or constant) Error: These errors are caused due to fault construction of instruments. Such errors can be minimized by taking same measurement with different accurate instruments.
Systematic (Persistent) Error: This is an error due to defective setting of an instrument. Such error can be minimised by detecting its causes.
Personal Error: These errors introduced due to fault of an observer taking readings, referred to human error.
Random (Accidental) Error: Even after minimising above types of errors, error may occur due to different factors like pressure, temperature fluctuation in voltage while the experiment is being performed. Such errors cannot be eliminated but can be minimised.

Definition

Random and Systematic Error

Systematic errors: The systematic errors are those errors that tend to be in one direction, either positive or negative. Systematic errors can be minimized by improving experimental techniques, selecting better instruments and removing personal bias as far as possible. Types of systematic errors are as follows:

Instrumental errors
Imperfection in experimental technique or procedure
Personal Errors

Random errors: These are the errors which occur irregularly and hence are random with respect to sign and size. These can arise due to random and unpredictable fluctuations in experimental conditions (e.g. unpredictable fluctuations in temperature, voltage supply, mechanical vibrations of experimental set-ups, etc), personal (unbiased) errors by the observer taking readings, etc.

Definition

Contributing Factors to Systematic Errors

Sources of systematic errors are as follows:

Instrumental errors that 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. For e.g. to determine the temperature of a human body, a thermometer placed under the armpit will always give a temperature lower than the actual value of the body temperature.
Personal errors that arise due to an individual's bias, lack of proper setting of the apparatus or individual's carelessness in taking observations without observing proper precautions, etc.

Definition

Contributing Factors to Random Errors

Random errors can arise due to random and unpredictable fluctuations in experimental conditions (e.g. unpredictable fluctuations in temperature, voltage supply, mechanical vibrations of experimental set-ups, etc), personal (unbiased) errors by the observer taking readings, etc.