Assignment on Errors in Measurement
Md. Khalid Hossain Jewel.
Dept. of AECE.
Islamic University, Kushtia.
Dept. of AECE.
Applied physics, Electronics & communication Engineering.
Islamic University, Kushtia, Bangladesh.
The error in measurement is a mathematical way to show the uncertainty in the measurement. Error is
defined as a difference between the desired and actual performance or behavior of a system or object. Thus the
deviation of the true value from the desired value in instrumentation is called error.
ERROR IN MEASUREMENT
Measurement is the process of comparing an unknown quantity with an accepted standard quantity. It involves
connecting a measurement instrument into the system under consideration and observing the resulting response on
the instrument. The measurement thus obtained is a quantitative measure of the so- called “true value” (since it is
very difficult to define the true value, the term “expected value” is used). Any measurement is affected by many
variables, therefore the results rarely reflect the expected value. For example, connecting a measuring instrument
into the circuit under consideration always disturbs value.
Some factors that affect the measurements are related to the measuring instruments themselves. Other factors that
are related to the person using the instrument .The degree to which a measurement nears the expected value is
expressed in terms of the error of measurement.
Error may be defined as
1. Absolute error
2. Percentage of error or relative error.
Absolute error may be defined as the difference between the expected value of the variable and the measured value
of the variable, or
where e=absolute error
Therefore % Error= (absolute value expected value) ×100
Therefore % error = ( ) ×100
It is more frequently expressed as a accuracy rather than error.
Therefore A = 1-| |, where A is the relative accuracy.
Accuracy is expressed as % accuracy
a= 100%-%error, a=A×100%, where a is the % of accuracy.
TYPES OF ERROR
There are generally two types of error in measurement as
1. Static error
2. Dynamic error.
Static error of a measuring instrument is the numerical difference between the true value of a quality and its value of
quantity and by measurement, i.e. repeated measurement of the same quantity gives different indication.
Dynamic error is the difference between the true value of a quantity changing with time and the value indicated by
Static errors are categorized as
1. Gross errors or human errors
2. Systematic errors
3. Random errors
1. GROSS ERRORS
These errors are mainly due to human mistakes in reading or in using instruments or error in recording observations.
Error may also occur due to incorrect adjustment of instruments and computational mistakes. These errors cannot be
The complete elimination of gross error is not possible, but one can minimize them. Some errors are easily detected
while others may be elusive.
One of the basic gross errors that occur frequently is the improper use of an instrument. The error can be minimized
by taking proper care in reading and recording the measurement parameter.
In general, indicating instruments change ambient conditions to some extent when connected into a complete circuit.
Due to minimizing this error one should be taken at least three separate reading instead of being depended on one
2. SYSTEMATIC ERROR
A systematic error results due to faulty measurement practices. The error of this category is
haracterized by deviation in one direction from the true value. What it means that the error is
introduced, which is either less than or greater than the true value. Systematic error impacts the
accuracy of measurement – not the precision of the measurement.
Systematic errors are biases in measurement which lead to the situation where the mean of many separate
measurements differs significantly from the actual value of the measured attribute. These errors occur due to
shortcomings of the instrument, such as defective or worn parts, or ageing or effects of the environment on the
instrument. Therefore A constant uniform deviation of the operation of an instrument is known as systematic error.
Some sources of systematic error are:
• Errors in the calibration of the measuring instruments.
• Incorrect measuring technique: For example, one might make an incorrect scale Reading because of parallax error.
• Bias of the experimenter. The experimenter might consistently read an instrument incorrectly, or might let
knowledge of the expected value of a result influence the measurements.
There are basically three types of systematic errors:-
a. Instrumental errors.
b. Environmental errors.
c. Observational errors.
a. INSTRUMENTAL ERRORS
Instrumental error refers to the combined accuracy and precision of a measuring instrument, or the difference
between the actual value and the value indicated by the instrument.
These errors are inherent in measuring instruments, because of their mechanical structure. For example, in the
D’Arsonval movement, friction in the bearings of various moving components, irregular spring tensions, stretching of
the spring or reduction in tension due to improper handling or over loading of the instrument.
Instrumental errors can be avoided by:-
1. Selecting a suitable instrument for the particular measurement applications.
2. Appling correction factors after determining the amount of instrumental error.
3. Calibrating the instrument against a standard.
b. ENVIRONMENTAL ERRORS
An environmental error is an error in calculations that are being a part of observations due to environment. Any
experiment performing anywhere in the universe has its surroundings, from which we cannot eliminate our system.
The study of environmental effects has primary advantage of being able us to justify the fact that environment has
impact on experiments and feasible environment will not only rectify our result but also amplify it.
The environmental errors have different causes, which are widening with the passage of time, as the research works
telling us, including; temperature, humidity, magnetic field, constantly vibrating earth surface, wind and improper
In high precision laboratories, where a slightest bug can destroy the whole system, removal or at least minimizing the
environmental errors proved to be very fruitful.
c. OBSERVATIONAL ERRORS
Observational errors are error introduced by the observer. The most common error is the parallax error introduced
in reading a meter scale, and the error of estimation when obtaining a reading from a meter scale.
These errors are caused by the habit of individual observers. For example, an observer may always introduce and
error by consistently holding his head too far to the left while reading a needle and scale reading.
In general, systematic errors can also be subdivided into static and dynamic errors. Static errors are caused by
limitations of the measuring device or the physical laws governing its behavior. Dynamic errors are caused by the
instrument not responding fast enough to follow the changes in a measured variable.
3. RANDOM ERRORS
Random errors are errors that remain after gross and systematic errors have been substantially reduced or at least
accounted for. Random errors are generally an accumulation of a large number of small effects and may be of real
concern only in measurements requiring a high degree of accuracy. Such errors can be analyzed statically.
These errors are due to unknown causes, not determinable in the ordinary processor making measurements. Such
errors are normally small and follow the laws of probability. Random errors can thus be treated mathematically.
For example, suppose a voltage is being monitored by a voltmeter which is read at 15 minutes intervals. Although the
instrument operates under ideal environmental conditions and accurately calibrated before measurements, it still
gives that vary slightly over the period of observation. This variation cannot be corrected by any method of calibration
or any other known method of control.
Random error unlike systematic error is not unidirectional. Some of the measured values are greater
than true alue; some are less than true value. The errors introduced are sometimes positive and
sometimes negative with respect to true value. It is possible to minimize this type of error by repeating
measurements and applying statistical echnique to get closer value to the true value.Another
distinguishing aspect of random error is that it is not biased. It is there because of the limitation of the
instrument in hand and the limitation on the part of human ability. No human being can repeat an
action in exactly the same manner. Hence, it is likely that same person reports different values with the
same instrument, which measures the quantity correctly.
SOURCES OF RANDOM ERRORS:
The sources of error, other than the inability of a piece of hardware to provide true measurements, are as follows:
1. Insufficient knowledge of process parameters and design conditions
2. Poor design.
3. Change in process parameters, irregularities, upsets, etc.
4. Poor maintenance.
5. Errors caused by person operating the instrument or equipment.
6. Certain design limitations.
ESTIMATING RANDOM ERRORS:
There are several ways to make a reasonable estimate of the random error in a Particular measurement. The best
way is to make a series of measurements of a given Quantity (say, x) and calculate the mean x , and the standard
deviation σx from this data.
The mean is defined as
Where xi is the result of the ith measurement and N is the number of measurements.
There are another types of error:-.
Most manufacturers of measuring instruments specify accuracy within a certain % of a full scale reading. For
example, the manufacturer of a certain voltmeter may specify the instrument to be accurate within with full
scale deflection .This specification is called the limiting error. This means that a full scale deflection reading is
guaranteed to be within the limit of 2% of a perfectly accurate reading; however, with a reading less full scale, the
limiting error increases.
PARALLAX ERROR :
Parallax is an apparent displacement or difference in the apparent position of an object viewed along two different
lines of sight, and is measured by the angle or semi-angle of inclination between those two lines.
Therefore A change in apparent position of an object, with respect to the reference marks(s) on an instrument,
caused by imperfect adjustment of the instrument or by a change in the position of the observer or both called
parallax error. It is also called instrumental parallax or error of para
(a)The position of pencil changes with respect to (b) Parallax error is introduced as we may read values
a mark on the background. at an angle.
Fig:(01) Parallax error
To avoid this error separated everywhere by the same distance. The term is used, in particular, in respect of lines
Inherent distortion introduced in the process of quantization of a waveform is known as quantization
noise; quantumization distortion; quantumization noise. In analog to digital conversion, the difference
between the actual analog value and quantized digital value is called quantization error or quantization distortion.
This error is either due to rounding or truncation. The error signal is sometimes considered as an additional random
signal called quantization noise because of its stochastic behavior.
Ways to Improve Accuracy in Measurement:
1. Make the measurement with an instrument that has the highest level of precision. The smaller the
unit, or fraction of a unit, on the measuring device, the more precisely the device can measure. The
precision of a measuring instrument is determined by the smallest unit to which it can measure.
2. Know your tools! Apply correct techniques when using the measuring instrument and reading the
value measured. Avoid the error called "parallax" -- always take readings by looking straight down (or
ahead) at the measuring device. Looking at the measuring device from a left or right angle will give an
3. Repeat the same measure several times to get a good average value.
4. Measure under controlled conditions. If the object you are measuring could change size depending
upon climatic conditions (swell or shrink), be sure to measure it under the same conditions each
time. This may apply to your measuring instruments as well.