Measurement is the foundation of all experimental science and technology. The result of every measurement by any measuring instrument contains some uncertainty. This uncertainty is called error. Every calculated quantity which is based on measured values, also has an error. We shall distinguish between two terms: accuracy and precision. The accuracy of a measurement is a measure of how close the measured value is to the true value of the quantity. Precision tells us to what resolution or limit the quantity is measured.
The accuracy in measurement may depend on
several factors, including the limit or the resolution
of the measuring instrument. For example, suppose
the true value of a certain length is near 3.678 cm.
In one experiment, using a measuring instrument
of resolution 0.1 cm, the measured value is found to
be 3.5 cm, while in another experiment using a
measuring device of greater resolution, say 0.01 cm,
the length is determined to be 3.38 cm. The first
measurement has more accuracy (because it is closer to the true value) but less precision (its
resolution is only 0.1 cm), while the
second measurement is less accurate but
more precise. Thus every measurement is
approximate due to errors in measurement. In
general, the errors in measurement can be
broadly classified as
(a) systematic errors and
(b) random errors.
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 calibration of the measuring instrument, zero error in the instrument, etc. For example, the temperature graduations of a thermometer may be inadequately calibrated (it may read 104 °C at the boiling point of water at STP whereas it should read 100 °C); 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. Other external conditions (such as changes in temperature, humidity, wind velocity, etc.) during the experiment may systematically affect the measurement.
(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 you, 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.
Systematic errors can be minimised by improving experimental techniques, selecting better instruments and removing personal bias as far as possible. For a given set-up, these errors may be estimated to a certain extent and the necessary corrections may be applied to the readings.
Random 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 (e.g. unpredictable fluctuations in temperature, voltage supply, mechanical vibrations of experimental set-ups, etc), personal (unbiased) errors by the observer taking readings, etc. For example, when the same person repeats the same observation, it is very likely that he may get different readings everytime.
Least count error
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.
The least count error is the error associated with the resolution of the instrument. For example, a vernier callipers has the least count as 0.01cm; 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. If we use a metre scale for measurement of length, it may have graduations at 1 mm division scale spacing or interval.
Using instruments of higher precision, improving experimental techniques, etc., we can reduce the least count error. 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.
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