# Errors in Measurement

## Errors in Measurement

In many experiments in the physics laboratory, the aim is to determine the value of a physical constant.

To determine a physical constant, we have to measure the various quantities which are connected with that physical constant by a formula.

For example, to determine the density ($\rho$) of a metal block, we have to measure its mass (m) and its volume (V) which are related to $\rho$ by the formula

$\rho = \frac {m}{V}$

The accuracy in the value of $\rho$ obviously depends upon the accuracy in the measurements of m and V.

Measurements of the quantities in the formula involve errors which are of two types:

1. Random Errors, and
2. Systematic Errors

## Random Errors

Random errors may be due to

• Small changes in the conditions of measurement, and
• The incorrect judgement of the observer in making a measurement

Examples:

1. Suppose you are asked to locate a null point in an electrical experiment using a meter bridge. If you determine the null point 3 or 4 times, you will notice that it shows small variations. The small changes in the null point are an example of random errors. This error is caused by small changes in the conditions of the experiment, such as heating effect or current and other relevant causes.
2. Suppose you are determining the weight of a body with the help of a spring balance. You will usually make an error in estimating the coincidence of the pointer with the scale reading or in assessing the correct position of the pointer when it lies between two consecutive graduations of the scale. The error, which is due to incorrect judgement of the observer is also a random error.

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Random errors cannot be traces to any systematic or constant cause of error. They do not obey any well-defined law of action.

Their character can be understood or appreciated from the illustration of firing shots at a target using a rifle. The target is usually a bull’s eye with concentric rings round it.

The result of firing a large number of shots at the target is well known. The target will be marked by a well grouped arrangement of shots. A large number of shots will be nearer a certain point and other shots will be grouped around it on all sides.

These shots which are grouped on both sides of the correct point obey the law of probability which means that large random errors are less probable to occur than small ones. A study of the target will show that the random shots lie with as many to one side of the center as to the other. They will also show that small deviations from the center are more numerous than large deviations and that a large deviation is very rare.

Method of minimizing Random Errors

If we make a large number of measurements of the same quantity then it is very likely that the majority of these measurements will have small errors which might be positive or negative.

The error will be positive or negative depending on whether the observed measurement is above or below the correct value.

Thus random errors can be minimized by taking the arithmetic mean of a large number of measurements of the same quantity. This arithmetic mean will be very close to the correct result. If one or two measurement  differ widely from the rest, they should be rejected while finding the mean.

## Systematic Errors

During the course of some measurement, certain sources of error operate contantly opr systematically making the measurement systematically greater or smaller than correct reading. These errors, whose cause can be traced, are called systematic errors. All instrumental errors belong to this category, such as

• Zero error in vernier calipers and micrometer screw,
• Index error in an optical bench,
• End error in a meter bridge,
• Faulty graduations of a measuring scale, etc.

Elimination of systematic errors:

To eliminate systematic errors, different methods are used in different cases.

1. In some cases, the errors are determined previously and the measurements corrected accordingly. For example, the zero error in an instrument is determined before a measurement us made and each measurement is corrected accordingly.
2. In some cases the error is allowed to occur and finally eliminated with the data obtained from the measurement. The heat loss due to radiation is taken into account and corrected for from the record of the temperature at different times.
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