1.2 Errors in measurement  (Page 2/2)

Procedural error

A faulty measuring process may include inappropriate physical environment, procedural mistakes and lack of understanding of the process of measurement. For example, if we are studying magnetic effect of current, then it would be erroneous to conduct the experiment in a place where strong currents are flowing nearby. Similarly, while taking temperature of human body, it is important to know which of the human parts is more representative of body temperature.

This error type can be minimized by periodic assessment of measurement process and improvising the system in consultation with subject expert or simply conducting an audit of the measuring process in the light of new facts and advancements.

Personal bias

A personal bias is introduced by human habits, which are not conducive for accurate measurement. Consider for example, the reading habit of a person. He or she may have the habit of reading scales from an inappropriate distance and from an oblique direction. The measurement, therefore, includes error on account of parallax.

We can appreciate the importance of parallax by just holding a finger (pencil) in the hand, which is stretched horizontally. We keep the finger in front of our eyes against some reference marking in the back ground. Now, we look at the finger by closing one eye at a time and note the relative displacement of the finger with respect to the mark in the static background. We can do this experiment any time as shown in the figure above. The parallax results due to the angle at which we look at the object.

It is important that we read position of a pointer or a needle on a scale normally to avoid error on account of parallax.

Random errors

Random error unlike systematic error is not unidirectional. Some of the measured values are greater than true value; 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 technique 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.

Least count error

Least count error results due to the inadequacy of resolution of the instrument. We can understand this in the context of least count of a measuring device. The least count of a device is equal to the smallest division on the scale. Consider the meter scale that we use. What is its least count? Its smallest division is in millimeter (mm). Hence, its least count is 1 mm i.e. ${10}^{-3}$ m i.e. 0.001 m. Clearly, this meter scale can be used to measure length from ${10}^{-3}$ m to 1 m. It is worth to know that least count of a vernier scale is ${10}^{-4}$ m and that of screw gauge and spherometer ${10}^{-5}$ m.

Returning to the meter scale, we have the dilemma of limiting ourselves to the exact measurement up to the precision of marking or should be limited to a step before. For example, let us read the measurement of a piece of a given rod. One end of the rod exactly matches with the zero of scale. Other end lies at the smallest markings at 0.477 m (= 47.7 cm = 477 mm). We may argue that measurement should be limited to the marking which can be definitely relied. If so, then we would report the length as 0.47 m, because we may not be definite about millimeter reading.

This is, however, unacceptable as we are sure that length consists of some additional length – only thing that we may err as the reading might be 0.476 m or 0.478 m instead of 0.477 m. There is a definite chance of error due to limitation in reading such small divisions. We would, however, be more precise and accurate by reporting measurement as 0.477 ± some agreed level of anticipated error. Generally, the accepted level of error in reading the smallest division is considered half the least count. Hence, the reading would be :

$$x=0.477\pm \frac{0.001}{2}m$$

$$\Rightarrow x=0.477\pm 0.0005m$$

If we report the measurement in centimeter,

$$\Rightarrow x=47.7\pm 0.05cm$$

If we report the measurement in millimeter,

$$\Rightarrow x=477\pm 0.5mm$$

Mean value of measurements

It has been pointed out that random error, including that of least count error, can be minimized by repeating measurements. It is so because errors are not unidirectional. If we take average of the measurements from the repeated measurements, it is likely that we minimize error by canceling out errors in opposite directions.

Here, we are implicitly assuming that measurement is free of “systematic errors”. The averaging of the repeated measurements, therefore, gives the best estimate of “true” value. As such, average or mean value ( $a_m$ ) of the measurements (excluding "off beat" measurements) is the notional “true” value of the quantity being measured. As a matter of fact, it is reported as true value, being our best estimate.

$$a_m=\frac{a_1+a_2+\dots \dots \dots ..+a_n}{n}$$

$$a_m=\underset{0}{\overset{n}{\Sigma }}\frac{a_i}{n}$$