Whenever we measure a physical quantity, there are chances that the measured value will be different from the true value. The difference between the measured value and the true value of the physical quantity is called Error in Measurement.

Error = True Value – Measured Value

The errors in measurement are caused due to various reasons and, on the basis of the cause, the errors are classified as:

1. Systematic Error

2. Random Error

3. Least Count Error

 

1. Systematic errors

The errors which occur due to known causes are called systematic errors. Systematic errors always tend to be in one direction, i.e. either positive or negative. Since the systematic errors are due to known causes, they can be minimized by taking proper precautions.

Some of the sources of systematic errors are:

 

(a) Instrumental errors

The errors which are caused due to a fault in the measuring instrument are called instrumental errors. The fault in the measuring instrument may be due to manufacturing defect or due to imperfect design of the instrument or due to wear and tear in the instrument.

Zero error and constant error are the instrumental errors which are caused due to imperfect design of the instrument. These errors can be reduced by using such instruments that have no manufacturing defect or by applying necessary correction.

Backlash error is an instrumental error which is caused due to wear and tear in the instrument. This error can be reduced by using a new instrument that has no manufacturing defect.

Sometimes, it is observed that, even if the measuring instruments are perfect, errors occur. Do you know why ? It is due to change in temperature, pressure, humidity etc. These errors can be reduced by applying necessary correction to the measured values.

 

(b) Imperfection in experimental procedure

When you were in the middle school, your physics teacher or even your geography teacher might have taught you how to use a rain gauge to measure the quantity of rainfall, i.e. the instrument is to be kept in the open space such that only the rain water should fall directly into it and so on. Right ?

If we keep the rain gauge under a tree during rain, then can we expect correct result ? The answer is NO because we didn’t follow the correct procedure. Thus, imperfection in experimental procedure is also a source of systematic error.

 

(c) Personal error

The errors which occur due to carelessness of the user of the instrument are called errors. Personal errors may also be caused due to the lack of knowledge of the user about how to use the instrument properly. 

When errors occur due to carelessness of the user of the instrument, such errors are also called gross errors. 

For example – in an experiment, the actual reading is 21.53 m but the experimenter records it as 21.35 m. This causes error in the measurement and it happens because of the oversights.

 

2. Random errors

The errors which occur due to unknown causes are called random errors. Random errors occur irregularly, i.e. sometimes they may be positive and sometimes they may be negative. The magnitudes of the error are also irregular, i.e. sometimes they are too small and sometime they are too big.

Since the causes of the random errors are not known, it is not possible to remove such errors completely. The only way to minimize such errors is to repeat the experiment a large number of times and take a large number of observations. The mean of all the observations will be very close to the true value of the measured quantity.



3. Least count error

Suppose the true length of a wire is 2.68 cm. If we measure the length of the wire with the help of a meter rule, we are likely to record our reading as 2.6 cm or 2.7 cm because the least count of the meter rule is 0.1 cm.

If we record the reading as 2.6 cm, then the error will be 2.68 cm – 2.6 cm = 0.08 cm

If we record the reading as 2.7 cm, then the error will be 2.68 cm – 2.7 cm = –0.02 cm

 

EXPRESSION OF ERRORS

Errors are expressed in the flowing ways:

1. Absolute error

If we know the true value of a physical quantity, then there is no need to measure it. Right ?

However, if there is need to know whether a given instrument is free from any manufacturing defect, then there is a need to measure a physical quantity, even if its true value is known.

But, if the true value of a physical quantity is not known, then how to determine it ? Because we assume that there is always error in measurement.

If there is no way to know the true value of a physical quantity, then the arithmetic mean of all the observations is considered to be the true value.

What is absolute error, then ?

The difference between a measured value and the true value is called absolute error. Absolute error is denoted by the symbol 

So, if a1, a2, a3, a4, ………an are the values obtained in several measurements, then the arithmetic mean of those values can be calculated as:




Thus, the errors in individual measurements are calculated as:







The errors obtained above may be positive in certain cases and negative in the others. But the absolute error will always be positive.

Mean Absolute Error

The arithmetic mean of all the absolute errors is called Mean Absolute Error and it is denoted by the symbol 




Thus, if we measure a physical quantity only once, then the value that we get may be more or less than the true value.

Denoting the measured value by the symbol a, we can write:



2. Relative or Fractional Error

The ratio of the mean absolute error to the mean value of the physical quantity is called relative error or fractional error.




3. Percentage Error

When the relative error is expressed in percent, then it is called percentage error and it is denoted by the symbol .