Accuracy and precision of measuring instruments
All measurements are made with the help of instruments. The
accuracy to which a measurement is made depends on several factors.
For example, if length is measured using a metre scale which has
graduations at 1 mm interval then all readings are good only upto this
value. The error in the use of any instrument is normally taken to be half
of the smallest division on the scale of the instrument. Such an error is
called instrumental error. In the case of a metre scale, this error is
about 0.5 mm.
Physical quantities obtained from experimental observation always
have some uncertainity. Measurements can never be made with absolute
precision. Precision of a number is often indicated by following it with
± symbol and a second number indicating the maximum error likely.
For example, if the length of a steel rod = 56.47 ± 3 mm then the
true length is unlikely to be less than 56.44 mm or greater than
56.50 mm. If the error in the measured value is expressed in fraction, it
is called fractional error and if expressed in percentage it is called
percentage error. For example, a resistor labelled “470 Ω, 10%” probably
has a true resistance differing not more than 10% from 470 Ω. So the
true value lies between 423 Ω and 517 Ω.
Significant figures
The digits which tell us the number of units we are reasonably sure of having counted in making a measurement are called significant figures. Or in other words, the number of meaningful digits in a number is called the number of significant figures. A choice of change of different units does not change the number of significant digits or figures in a measurement.
For example, 2.868 cm has four significant figures. But in different
units, the same can be written as 0.02868 m or 28.68 mm or 28680
µm. All these numbers have the same four significant figures.
From the above example, we have the following rules.
i) All the non−zero digits in a number are significant.
ii) All the zeroes between two non−zeroes digits are significant,
irrespective of the decimal point.
iii) If the number is less than 1, the zeroes on the right of decimal
point but to the left of the first non−zero digit are not significant. (In
0.02868 the underlined zeroes are not significant).
iv) The zeroes at the end without a decimal point are not
significant. (In 23080 µm, the trailing zero is not significant).
v) The trailing zeroes in a number with a decimal point are
significant. (The number 0.07100 has four significant digits).
Examples
i) 30700 has three significant figures.
ii) 132.73 has five significant figures.
iii) 0.00345 has three and
iv) 40.00 has four significant figures.
Rounding off
Calculators are widely used now−a−days to do the calculations.
The result given by a calculator has too many figures. In no case the
result should have more significant figures than the figures involved in
the data used for calculation. The result of calculation with number
containing more than one uncertain digit, should be rounded off. The
technique of rounding off is followed in applied areas of science.
A number 1.876 rounded off to three significant digits is 1.88
while the number 1.872 would be 1.87. The rule is that if the insignificant
digit (underlined) is more than 5, the preceeding digit is raised by 1,
and is left unchanged if the former is less than 5.
If the number is 2.845, the insignificant digit is 5. In this case,
the convention is that if the preceeding digit is even, the insignificant
digit is simply dropped and, if it is odd, the preceeding digit is raised
by 1. Following this, 2.845 is rounded off to 2.84 where as 2.815 is
rounded off to 2.82.
Examples
1. Add 17.35 kg, 25.8 kg and 9.423 kg.
Of the three measurements given, 25.8 kg is the least accurately
known.
∴ 17.35 + 25.8 + 9.423 = 52.573 kg
Correct to three significant figures, 52.573 kg is written as
52.6 kg
2. Multiply 3.8 and 0.125 with due regard to significant figures.
3.8 × 0.125 = 0.475
The least number of significant figure in the given quantities is 2.
Therefore the result should have only two significant figures.
∴ 3.8 × 0.125 = 0.475 = 0.48
Errors in Measurement
The uncertainity in the measurement of a physical quantity is
called error. It is the difference between the true value and the measured
value of the physical quantity. Errors may be classified into many
categories.
Constant errors
It is the same error repeated every time in a series of observations.
Constant error is due to faulty calibration of the scale in the measuring
instrument. In order to minimise constant error, measurements are
made by different possible methods and the mean value so obtained is
regarded as the true value.
Systematic errors
These are errors which occur due to a certain pattern or system.
These errors can be minimised by identifying the source of error.
Instrumental errors, personal errors due to individual traits and errors
due to external sources are some of the systematic errors.
Gross errors
Gross errors arise due to one or more than one of the following
reasons.
(1) Improper setting of the instrument.
(2) Wrong recordings of the observation.
(3) Not taking into account sources of error and precautions.
(4) Usage of wrong values in the calculation.
Gross errros can be minimised only if the observer is very careful
in his observations and sincere in his approach.
Random errors
It is very common that repeated measurements of a quantity give
values which are slightly different from each other. These errors have
no set pattern and occur in a random manner. Hence they are called
random errors. They can be minimised by repeating the measurements
many times and taking the arithmetic mean of all the values as the
correct reading.
The most common way of expressing an error is percentage error.
If the accuracy in measuring a quantity x is ∆x, then the percentage
error in x is given by x
x
∆
× 100 %.
All measurements are made with the help of instruments. The
accuracy to which a measurement is made depends on several factors.
For example, if length is measured using a metre scale which has
graduations at 1 mm interval then all readings are good only upto this
value. The error in the use of any instrument is normally taken to be half
of the smallest division on the scale of the instrument. Such an error is
called instrumental error. In the case of a metre scale, this error is
about 0.5 mm.
Physical quantities obtained from experimental observation always
have some uncertainity. Measurements can never be made with absolute
precision. Precision of a number is often indicated by following it with
± symbol and a second number indicating the maximum error likely.
For example, if the length of a steel rod = 56.47 ± 3 mm then the
true length is unlikely to be less than 56.44 mm or greater than
56.50 mm. If the error in the measured value is expressed in fraction, it
is called fractional error and if expressed in percentage it is called
percentage error. For example, a resistor labelled “470 Ω, 10%” probably
has a true resistance differing not more than 10% from 470 Ω. So the
true value lies between 423 Ω and 517 Ω.
Significant figures
The digits which tell us the number of units we are reasonably sure of having counted in making a measurement are called significant figures. Or in other words, the number of meaningful digits in a number is called the number of significant figures. A choice of change of different units does not change the number of significant digits or figures in a measurement.
For example, 2.868 cm has four significant figures. But in different
units, the same can be written as 0.02868 m or 28.68 mm or 28680
µm. All these numbers have the same four significant figures.
From the above example, we have the following rules.
i) All the non−zero digits in a number are significant.
ii) All the zeroes between two non−zeroes digits are significant,
irrespective of the decimal point.
iii) If the number is less than 1, the zeroes on the right of decimal
point but to the left of the first non−zero digit are not significant. (In
0.02868 the underlined zeroes are not significant).
iv) The zeroes at the end without a decimal point are not
significant. (In 23080 µm, the trailing zero is not significant).
v) The trailing zeroes in a number with a decimal point are
significant. (The number 0.07100 has four significant digits).
Examples
i) 30700 has three significant figures.
ii) 132.73 has five significant figures.
iii) 0.00345 has three and
iv) 40.00 has four significant figures.
Rounding off
Calculators are widely used now−a−days to do the calculations.
The result given by a calculator has too many figures. In no case the
result should have more significant figures than the figures involved in
the data used for calculation. The result of calculation with number
containing more than one uncertain digit, should be rounded off. The
technique of rounding off is followed in applied areas of science.
A number 1.876 rounded off to three significant digits is 1.88
while the number 1.872 would be 1.87. The rule is that if the insignificant
digit (underlined) is more than 5, the preceeding digit is raised by 1,
and is left unchanged if the former is less than 5.
If the number is 2.845, the insignificant digit is 5. In this case,
the convention is that if the preceeding digit is even, the insignificant
digit is simply dropped and, if it is odd, the preceeding digit is raised
by 1. Following this, 2.845 is rounded off to 2.84 where as 2.815 is
rounded off to 2.82.
Examples
1. Add 17.35 kg, 25.8 kg and 9.423 kg.
Of the three measurements given, 25.8 kg is the least accurately
known.
∴ 17.35 + 25.8 + 9.423 = 52.573 kg
Correct to three significant figures, 52.573 kg is written as
52.6 kg
2. Multiply 3.8 and 0.125 with due regard to significant figures.
3.8 × 0.125 = 0.475
The least number of significant figure in the given quantities is 2.
Therefore the result should have only two significant figures.
∴ 3.8 × 0.125 = 0.475 = 0.48
Errors in Measurement
The uncertainity in the measurement of a physical quantity is
called error. It is the difference between the true value and the measured
value of the physical quantity. Errors may be classified into many
categories.
Constant errors
It is the same error repeated every time in a series of observations.
Constant error is due to faulty calibration of the scale in the measuring
instrument. In order to minimise constant error, measurements are
made by different possible methods and the mean value so obtained is
regarded as the true value.
Systematic errors
These are errors which occur due to a certain pattern or system.
These errors can be minimised by identifying the source of error.
Instrumental errors, personal errors due to individual traits and errors
due to external sources are some of the systematic errors.
Gross errors
Gross errors arise due to one or more than one of the following
reasons.
(1) Improper setting of the instrument.
(2) Wrong recordings of the observation.
(3) Not taking into account sources of error and precautions.
(4) Usage of wrong values in the calculation.
Gross errros can be minimised only if the observer is very careful
in his observations and sincere in his approach.
Random errors
It is very common that repeated measurements of a quantity give
values which are slightly different from each other. These errors have
no set pattern and occur in a random manner. Hence they are called
random errors. They can be minimised by repeating the measurements
many times and taking the arithmetic mean of all the values as the
correct reading.
The most common way of expressing an error is percentage error.
If the accuracy in measuring a quantity x is ∆x, then the percentage
error in x is given by x
x
∆
× 100 %.
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