ERRORS

• ERROR: 
• The difference between true value and measured value is known as error in measurement.

• Accuracy: The accuracy of a measurement is a measure of how close the measured value is to the true value of the quantity.

• Precision: precision tells us to what limit the quantity is measured.

Types of Errors
Errors may be divided into following three types
(I) systematic errors and
(II) Random errors.
(III) Gross errors

(I) Systematic errors:- The systematic errors are those errors that tend to be in one direction, either positive or negative. Some of the sources of systematic errors are :
(a) Instrumental errors that arise from the errors due to imperfect design or zero
error in the instrument, etc. For example, in a vernier callipers the zero mark of vernier scale may not coincide with the zero mark of the main scale, or simply an ordinary metre scale
may be worn off at one end.
(b) Imperfection in experimental technique or procedure 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.
(c) 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. For example, if people, by habit, always hold your head a bit too far to the right while reading the position of a needle on the scale, you will introduce an error due to parallax.


(II) Random errors or Chance Errors
The random errors are those 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 (temperature, voltage supply etc).
For example, when the same person repeats the same observation, he may get different
readings every time.
(III) Gross Error:
The errors due to carelessness of the observer are known as Gross Errors. e.g. Recording
the observations wrongly, using the wrong values of observations in calculations.
Least Count: The smallest value that can be measured by the measuring instrument is
called its least count. All the readings or measured values are good only up to this value.

Least count error The least count error is the error associated with the resolution of the
instrument. For example, a vernier callipers has the least count as 0.01 cm; a
spherometer may have a least count of 0.001 cm.
Least count error belongs to the category of random errors but within a limited size; it
occurs with both systematic and random errors.
Using instruments of higher precision etc., we can reduce the least count error.
Main Point: Repeating the observations several times and taking the arithmetic mean of
all the observations, the mean value would be very close to the true value of the
measured quantity.
Absolute Error, Relative Error and Percentage Error
Absolute Error: The magnitude of the difference between the true value of the quantity
and the individual measurement value is called the absolute error of the measurement.
This is denoted by | Δa |.
If true value is not known then we considered arithmetic mean as the true value. Then
the errors in the individual measurement values are
Ea1 = amean – a1,
Fa2 = amean – a2,
.... .... ....
.... .... ....
Fa n = amean – an
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
Δamean.
Thus,
Δamean = (|Δa1|+|Δa2 |+|Δa3|+...+ |Δan|)/n
Relative Errors: The relative error is the ratio of the mean absolute error Δamean to the
mean value amean of the quantity measured.
Relative error = Δamean /amean
Percentage Error: When the relative error is expressed in per cent, it is called the percentage error (δa).
Thus, Percentage error δa = (Δamean/amean) × 100%