Accuracy Precision Measuring Instruments – Engineersfield

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
☛ 30700 has three significant figures.
☛ 132.73 has five significant figures.
☛ 0.00345 has three and
☛ 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
● 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+ 25.8 + 9.423 = 52.573 kg
Correct tree significant figures, 52.573 kg is written as 52.6 kg

● 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.
● Improper setting of the instrument.
● Wrong recordings of the observation.
● Not taking into account sources of error and precautions.
● 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 Δx, then the percentage error in x is given by x x Δ × 100 %.