Error Analysis, Classification, Types and Definition of Errors | Free online notes, study materials on Physics for NEET, JEE, AIPMT, IIT

No measurement is perfect. Measured value of a physical quantity is always different from the true value. The difference between true and measured value of a physical quantity is termed Error. The errors involved in measurement cannot be removed completely.
Two important terms measurement and error are - Accuracy, Precision and Discrepancy

Accuracy: This indicates how close the measured value is to the true value of the quantity. Accuracy increases with reducing errors. This means as we reduce the errors, the measurement becomes more accurate.

Precision: This indicates as to what resolution or limit a quantity has been measured. It is not necessary that a more precise value will also be more accurate.

Discrepancy: The difference between the two measured values of a physical quantity is known as discrepancy.

Other Topics from this Chapter

Physical Quantities - Fundamental or Basic, Derived, Supplementary quantities

Units - International System of Units, SI Units, Properties and Definitions, Norms for writing Unit Symbols

Notes on Measurements

Significant Figures and its Rules, Rounding Off a Digit

Types and Definition of Errors, Errors in Measurement, Error Analysis

Dimensions, Dimensional Formulae, Dimensional Equation

Dimensional Analysis and its Applications

ERROR ANALYSIS | CLASSIFICATION OF ERRORS | TYPES & DEFINITION OF ERRORS

Classification of errors can be done in two ways. First, based on the cause of errors such as - Systematic Errors and Random Errors. Second classification is based on the magnitude or size of errors such as - Absolute Error, Mean Absolute Error, Relative or Functional Error, Percentage Error.

Systematic Errors

These are the errors that happen because of various causes which are known to us. A systematic error can, therefore, be minimised. Systematic errors may be due to imperfect technique, alteration of the quantity being measured, carelessness or mistakes on the past of observer. These errors occur in one direction only hence, either positive or negative. If the measured value is greater than the true value, the error is said to be positive and if the measured value is lesser than the true value then it will be said as negative. Systematic errors can be of following types -
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Instrumental Error

These errors arise when the measuring instrument or apparatus itself has some defect in it such as:  improper calibration, defect in designing, zero error (in vernier calipers or screw gauge or meter scale). An instrument error is a constant type of error. These errors can be minimised by using more accurate instruments, applying zero correction, as required.

Observational or Personal Error

Such errors are caused mostly due to carelessness or casual behaviour of an observer. For example, if an observer while reading the volume of water in a beaker becomes casual and keeps his eyes below the meniscus, his reading will be wolf because of Parallax. Also termed as Gross Error, may be due (i) negligence towards sources of error due to overlooking of sources of error by the observer, (ii) the observer, without carrying for last count goes on taking wrong observations, (iii) wrong recording of observation, etc.

Error due to Imperfection and Unavoidable Conditions

As the name suggests these errors are due to the external conditions or due to ignoring certain facts. For example, errors due to changes in temperature, pressure, humidity, air resistance etc. may sometimes affect the final result for not taking these factors into consideration. However, these errors can also be reduced by applying proper corrections to the formula used.

How to minimise Systematic Errors

Systematic errors can be minimised by more accurate instruments, and improved experimental techniques. One should take proper precautions and remain unbiased as far as possible while doing experiments. Further, necessary corrections should be done for the instruments having zero errors after talking readings.

Random Errors

The errors arise due to unknown causes. Random errors would occur irregularly and can have any sign - positive or negative. The magnitude or the size of such errors can also vary randomly.  Since the causes of random errors are not known so, it is not possible to remove them completely.

Least Count Error

The smallest value of the measurement that can be directly taken from a measuring instrument is called the least count (LC) or resolution of the instrument. LC error can arise and depends on the precision provided by the measuring instrument. Thus last count error is another type of constant error and can be reduced by using high precision instruments along with improved experimental techniques. (Please find in our separate post - Formula for How to calculate LC of vernier callipers, screw gauge in Errors in Measurement)

Absolute Error

The magnitude of the difference between the true value and the measured value is called absolute error. It is always positive.

Mean Absolute Error

The arithmetic mean of the magnitudes of individual absolute errors is called the mean absolute error. Thus,

Relative or Fractional Error

The relative error or fractional error is the ratio of the mean absolute error Da_mean to the mean value of the quantity measured.

Percentage Error

The relative error expressed in percentage is Percentage Error. It is denoted by da.

Standard Error

The errors which take into account all the factors affecting the accuracy of result, is known as the Standard Error.

Probable Error

The errors calculated by using the principle of probability, are Probable Errors. According to Bessels Formula, 

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Notes on Significant Figures and its Rules - Units, Measurements, Error Analysis and Dimensions

Units, Measurements, Dimensions and Error Analysis

significant figures

The significant figures are a measure of accuracy of a particular measurement of a physical quantity. Thus, significant figures indicate the precision of the measurement which depends on the least count of the measuring instrument.

Significant figures in a measurement are those digits in a physical quantity that are known reliably plus the one digit which is uncertain. For example, the thickness of any material measured = 6.45 cm. Among the digits appearing in the value 6.45, 6 & 4 are reliable or certain, while the digit 5 is uncertain.  

Significant figures are the number of digits up to which we are sure about their accuracy.

Significant figures don't change if we measure a physical quantity in different units.

Significant figures retained after mathematical operations like addition, subtraction, multiplication and division should be equal to the minimum significant figures involved in any physical quantity in the govern operation.  

Other Topics from this Chapter

Physical Quantities - Fundamental or Basic, Derived, Supplementary quantities

Units - International System of Units, SI Units, Properties and Definitions, Norms for writing Unit Symbols

Notes on Measurements

Significant Figures and its Rules, Rounding Off a Digit

Types and Definition of Errors, Errors in Measurement, Error Analysis

Dimensions, Dimensional Formulae, Dimensional Equation

Dimensional Analysis and its Applications

Rules for Counting (Finding) Significant Figures

v       All non-zero digits are significant. For example, X = 2.345 has four significant figures.

v      The zeros appearing between two non-zero digits are counted in significant figures. For example, 1.023 has four significant figures.

v      The zeros occurring to the left of the last non-zero digit are not significant (insignificant). For example, 0.00123 has three significant figures. The two zeros appearing in the left of 1 are not significant.

v       In a number without decimal, zeros to the right of non-zero digit are not significant (insignificant). However if the same value is recorded on the basis of actual measurement the zeros to the right of non-zero digit becomes significant. For example, L = 20000 m has five significant figures but X = 20000 has only one significant figure.

v       In a number containing decimal, the trailing zeros (present to the right) are significant. For example, 1.500 has four significant figures.

v      The exact number of digits appearing in the mathematical formula of different physical quantities has infinite number of significant figures. For example, perimeter of a square is given by (4 x side). Here, 4 is an exact number and has infinite number of significant figures which can be as 4.0, 4.00, 4.000 etc. as per the requirement.

v      The powers of ten are not counted as significant digits. For example, 1.5 X 10⁷ has only two significant figures 1 & 5.

v      Change in the units of measurement does not change the number of significant figures. For example, distance between points is 1234 m, which has four significant figures. The same can be expressed as 1.234 km or 1234 X 10⁵ cm both having four significant figures.

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Rounding Off a Digit

v  If the insignificant digit to be dropped is less than 5, then the preceding digit is left unchanged. For Example, 5.82 is rounded off to 5.8 since 2 < 5.

v  If the insignificant digit to be dropped is more than 5, then the preceding digit is raised by one. For Example, 5.86 can be rounded off to 5.9 since 6 > 5.  

v  If the insignificant digit to be dropped is 5 followed by digits other than zero, then the preceding digit is raised by one. For Example, 5.852 is rounded off to 5.9     

v  If the insignificant digit to be dropped is 5 or 5 followed by zeros, then preceding digit is left unchanged, if it is even. For Example, 5.250 is rounded off to 5.2

v  If the insignificant digit to be dropped is 5 or 5 followed by zeros, then the preceding digit is raised by one, if it is odd. For Example, 5.350 is rounded off to 5.4

v  The exact numbers like 2, 3, 4, p appearing in the mathematical formulae of different physical quantities and are known to have infinite significant figures, are rounded off to a limited number of significant figures as per the requirement.  

Rules for Algebraic Operations (Arithmetic Operations) with Significant Figures

Certain rules need to be followed while doing arithmetic operations such as addition, multiplication, subtraction with significant figures so that the accuracy can be maintained in the final result (as in the original values or inputs).

Addition and Subtraction

If two or more physical quantities are added or subtracted, there should be as many decimal places retained in the final result as are there in the number with the least decimal places. Suppose the numbers to be added or subtracted, the least number of significant digits after decimal is n. Then in the result i.e., after addition or subtraction the number of significant digits after decimal should be n.

Example: 1.1 + 4.54 + 16.084 = 21.724 ⇒ 21.7  

Multiplication and Division

If two measured values are multiplied or divided, there should be as many significant figures retained in the final result, as are there in the original number with the least significant figures. For example say, in the measured values to be multiplied or divided the least number of significant digits be n, then in the product or quotient, the number of significant digits should also be n.

Example: 2.2 x 13.222 = 29.088 ⇒ 29

In the above example the least number of significant digits in the measured values are two so, the result when rounded off to two significant digits become 29.       

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Units, Measurements, Error Analysis and Dimensions - Notes on Measurements

In continuation of section wise notes from the chapter Units, Measurements, Dimensions and Error Analysis containing important study materials and more it goes ...

·    Physical World, Physical Quantities - fundamental or basic, derived, supplementary quantities

·    Units - International System of Units, SI Units, Properties and Definitions, Norms for writing unit symbols

·    International System of Units, SI Units, their Properties, Definitions, How to write unit symbols

·    Notes on Significant Figures, Rules for Countinmg Significant Digits, Rounding Off, Algebraic Operations  

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Measurement

What is Measurement?

Measurement is the comparison of a quantity with a standard of the same physical quantity. (How do we define Physical Quantity?) In other words, measurement of a physical quantity requires is comparison with an arbitrarily chosen, reference standard which is called unit of that quantity.

Measurement of Large Distances -

Large distances, for example distance between stars, planet are measured by indirect methods. Hence, an indirect method of measurement or making rough estimates of quantities using common observations is equally important.
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Parallax Method

Parallax can be defined as the apparent shift in the positron of a body with respect to a specific point on its background with the shift of eye. Let us understand this in this way -

Suppose we want to measure the distance of a very distant planet (P) using Parallax Method, we observe it from two different observatories (L & M) on the earth separated by a distance say, 'b' which is called "basis". Since the planet is too far, both the observatories L & M can be considered at nearly the distance from it. 

LP = MP = D (say). Also, 'b' is too less as compared to 'D'.

Thus LM can be approximately considered as an arc of length 'b' of a circle whose center less at P. The radius of the circle is then, D. So, the angle subtended by LM at P can be used to find D using the relation:

q = b / D 

Angle q is called the Parallax angle or parallactic angle and it should be taken in radian, not in degree.

After calculating the distance of the planet from the earth using Parallax Method, we can also calculate the size of the planet by using the following relation -

d = Df

Where, the angle f is the angular size or angular diameter of the planet.

Measurement of Very Small Distances -

Besides distance / length of normal magnitudes, we can do direct measurements of very small lengths (minute size) of the scale of molecular size using specific instruments like:

Optical Microscope - works in visible light & can resolve particles up to 10^-7 m.

Electron Microscope - uses electron beams focussed by electric and magnetic field, better resolution power than that of optical microscope, can measure as small as 0.6 x 10^–10 m.

Tunnelling Microscope - has better resolution power than electron microscope, can measure the size of a molecule.

Volumetric Method - this method is used to estimate the sizes of large molecules e.g., oleic acid.    

Measurement of Mass - 

Other Topics from this Chapter

Physical Quantities - Fundamental or Basic, Derived, Supplementary quantities

Units - International System of Units, SI Units, Properties and Definitions, Norms for writing Unit Symbols

Notes on Measurements

Significant Figures and its Rules, Rounding Off a Digit

Types and Definition of Errors, Errors in Measurement, Error Analysis

Dimensions, Dimensional Formulae, Dimensional Equation

Dimensional Analysis and its Applications

Mass is a measure of the quantity of matter present in a body. It is one of the basic properties of matter. Variations in conditions such as pressure, temperature, location, etc. do not change the mass of an object. We need to employ different methods to measure masses of different ranges. For example - 

Mass of Small Objects - We can use common balance, spring balance, pedestal balance etc. to find masses of common objects whereas, to estimate very small masses like molecular mass, we have to use analytical tools like Mass Spectrograph  

Mass of Terrestrial Objects - calculated mathematically using Newton’s Law of Gravitation

Also Read: Solved Objective Questions & Test Series on Newton’s Law of Gravitation

Table: Range or Order of Masses

Object	Range of Mass (kg)
Electron	10–30
Proton	10–27
Uranium atom	10–25
Red Blood Cell	10–13
A Dust Particle	10–9
Rain Drop	10–6
Human Body	102
Boeing 747 Aircraft	108
Moon	1023
Earth	1025
Sun	1030
Milky Way Galaxy	1041

Measurement of Time -

Any periodically recurring event can be used as a clock or as a standard of time. The periodic vibrations produced in a cesium atom serve as the most accurate standard of time. A cesium clock is also known as an Atomic Clock is based on these periodic vibrations. Atomic clocks have a high accuracy of ‘1 part in 10¹³’ i.e., they lose or gain no more than 3 ms (3x10^–6 s) in a year. The national standard time interval ‘second’ and its frequency is maintained with a set of four cesium atomic clocks.        

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