ERRORS NOTES 1.pdf

1. ERRORS AND MEASUREMENTS
Thenumericalvaluesobtainedonmeasuringphysicalquantitiesdependuponthe measuring
instruments,methodsof measurement.
LEAST COUNT (LC) : The smallest value that can be measured by the measuring
instrument.
ACCURACY: It refers to how closely a measured valueagreeswiththetruevalue.
PRECISION: It refers to what limit or resolution the givenphysicalquantifycanbemeasured.
ERROR: Uncertaintyinmeasurementofaphysicalquantityiscalledtheerrorin measurement
(or) thedifferencebetweenthemeasuredvalueandtruevaluephysicalquantity.
Error = True value - Measured value
Truevaluemeansstandardvalue freeofmistakes
Types of Errors:
Errorsarebroadlyclassifiedinto3types:
I) Systematic errors
II) Random errors
III) Gross errors
I) SYSTEMATIC ERRORS: The errors which occur according to a certain pattern (or)
rule. They always occur in one direction. i.e either + ve always (or) – ve always.
Constant Error: Systematic error with a constant magnitude is called constant error.
* Constant error is due to faulty calibration of the scale of a measuring instrument.
Example : Zero error of a screw gauge.
Systematic errors are classified as
a) Environmental Error
b) Imperfection in experimental technique (or) procedure.(Instrumental errors)
c) Personal errors (or) observational errors
a)EnvironmentalError:
The error arises due to external conditions like changes in environment, changes in temperature,
pressure,humidityetc.
Example : Due to rise in temperature a scale gets expandedandthis resultsin errorin
measuringlength.

2. b) Imperfection in Experimental technique or Procedure: (Instrumental errors)
Theerrorduetoexperimental arrangement,procedurefollowedandexperimentaltechniqueis
called instrumental errors.
Example : In calorimetric experiments,the loss of heat due to radiation, the effect on weighing due
tobuoyancyofaircannotbeavoided.
c) Personal errors or observational errors:
These errors are entirely due to personal peculiarities like individual bias, lack of proper settings of
theapparatus,carelessness intakingobservations.
Example:Parallaxerror
II) RANDOM ERRORS: It is a common experience that the repeated measurements of a
quantity give values which are slightly different from each other. These errors
have no set pattern.
Example : The errors due to line voltage changes and back lash error (Back lash errors
are due to screw and nut).
III) GROSS ERRORS: These errors are due to one or more than one of the following
reasons.
i) Improper setting of the instrument
ii) Recording observations wrongly
iii) Not to take in to account the sources of error and precautions
iv) Using some wrong value in calculations
* Nocorrectioncanbeappliedtothesegrosserrors.
* Whentheerrorsareminimized,theaccuracyincreases.
* Thesystematicerrors canbeestimatedandobservations can be corrected.
** Random errors are compensating type. A physical quantity is measured number of times and these
values lie on either side of mean value-with random errors. These errors are estimated by statistical
methodsandaccuracyis achieved.
** Personalerrorslikeparallaxerrorcanbeavoided by taking proper care.
** The instrumental errors are avoided by calibrating the instrument with a standard value and by
applyingpropercorrections.

3. TRUE VALUE AND EXPRESSING ERRORS
TRUE VALUE: If a1, a2, a3 ………….an are readings then true value is given by
1 2 3 1
....
n
i
n i
avg mean
a
a a a a
a a a a
n n

  
      

ABSOLUTEERROR:
The magnitude of the difference between the truevalueofthemeasuredphysicalquantityand
thevalueofindividual measurementiscalledabsoluteerror.
Absolute error = | True value - measured value |
1 1
2 2
mean
mean
n mean n
a a a
a a a
a a a
  
  
  
Theabsoluteerrorisalways positive.
MEAN ABSOLUTE ERROR:
The arithmetic mean of all the absolute errors is considered as the mean absolute error .
The meanabsoluteerroris always positive.
RELATIVE ERROR:
Therelativeerrorofameasuredphysical quantity is the ratio of the mean absolute error to
themeanvalueofthequantitymeasured.
Relative error mean
mean
a
a


Itisapurenumberhavingnounits.
Percentage error 100 %
mean
mean
a
a
a

 

  
 
 
















1
1
2
1 1
i
n
mean a
n
n
a
a
a
a

4. The rules for rounding off numbers
1) The preceding digit is raised by 1 if the insignificant digit to be dropped is more than 5.
Ex: 23.467 ----------- 23.47
542.78923 ----------- 542.79
2) The preceding digit is left unchanged if the insignificant digit to be dropped is less than 5.
Ex: 23.463 ----------- 23.46
542.78323 ----------- 542.78
3) If the insignificant digit to be dropped is equal to 5, then
a) if the preceding digit is even, the insignificant digit is simply dropped
Ex: 23.465 ----------- 23.46
542.78583 ----------- 542.78
b) if it is odd, the preceding digit is raised by 1.
Ex: 23.435 ----------- 23.44
542.71583 ----------- 542.72
Prob 1: We measure the period of oscillation of a simple pendulum. In successive
measurements, the readings turn out to be 2.63 s, 2.56 s, 2.42 s, 2.71 s and 2.80 s.
Calculate the absolute errors, relative error and percentage error.

5. Prob 2: The refractive index 
 of glass is found to have the values 1.49, 1.50, 1.52, 1.54
and 1.48. Calculate the absolute errors, relative error and percentage error.