### Errors in measurements

Errors in measurements – 1) All measurements are carried out with the help of instruments. While making use of various instruments various types of errors creep into our measurements. The chief sources of error two i.e., the observer and the instrument. It is sure that every observer is liable to cause an error in measurements. Side by side every instrument is more or less defective and hence the value of physical quantity provided by it differs from the actual value. Thus the actual error in measurement may be due to a certain complex combination of accuracy and precision of instruments.

2) All measurements are carried out with the help of instruments. While making use of various instruments various types of errors creep into our measurements. The chief sources of error two i.e., the observer and the instrument. It is sure that every observer is liable to cause an error in measurements. Side by side every instrument is more or less defective and hence the value of physical quantity provided by it differs from the actual value.

** Thus the actual error in measurement may be due to a certain complex combination of accuracy and precision of instruments.**

**a) Constant Errors in measurements**– If in spite of taking a series of observations for a physical quantity, the same error are repeated every time, the error said to be a constant one. Such error is caused due to faulty calibration of the scale of measuring instrument. Suppose, the graduations on a meter scale are faulty such that each min division is actually 0.99 mm long. If the length of a table is measured with such a meter scale, the measured value will always be greater than the actual value. If the actual length of table is 99 cm, it will be shown 100 cm by the meter scale.

The error involved will always be 1 cm, how many times the observations may be made. Thus, the error involved cannot be detected, while using the same meter scale. However, if the measurements are made by different meter scale also, then error can be detected. The constant error may appear in different amount in every instrument. Hence, in order to avoid constant error, measurements are made with as many different methods as possible and then assume their mean as the true value.

**b) Systematic Errors in measurements** – Errors occurring according to certain pattern or system are called systematic errors. These errors are due to known reasons and can be eliminated by recognizing the source of error and the rule governing this error.

**Systematic error are classified into following four main categories:**

**i)Personal errors**: The errors arising due to personal peculiarities of the observer are called personal errors. The magnitude of thee errors may be indefinite. In case of an experienced observer, this error is lesser and may behave like constant error. An example of such type of errors is the error in recording the observation for the time period of oscillations of simple pendulum. The observer, in general, starts or stops the watch a little earlier or little later with respect to the instant when the bob passes through the mean position. Time lags involved while starting and stopping the watch may be different or same in magnitude as well as sign.

Thus an indefinite error is involved in every observation. This error can be minimized by measuring the time taken for a large number of oscillations and then taking the mean of a large number of such readings. Larger the number of oscillations for which time is recorded, smaller will be the magnitude of personal error.

Another example of this type of error is parallax error while reading a scale. If an observer do not keep his eyes just above the graduations of the scale while nothing the reading, an indefinite error is liable to occur. This error can be eliminated if the measurements are repeated by different observers.

** ii)Instrumental errors in measurements**: Inherent errors involved in the instruments are called as instrumental errors. A very good example of such type of error is the zero error in a vernier calipers or screw gauge. In case of such errors, even repeated careful observations will fail to detect the presence of these errors. These errors can be detected by measuring a physical quantity with two different instruments of the same type. Because, it is possible to find the reasons of instrumental error, hence by determine the magnitude of error, necessary corrections can be made in the measured value.

**iii) Errors due to external source**:

Systematic errors caused due to change in external conditions like pressure, temperature, wind etc, are called errors due to external sources. For example, change in temperature may lead to expansion of the scale used for measuring the length & thus lead to systematic error. These errors can be easily detected and necessary corrections may be made accordingly. It is also possible to eliminate these errors by keeping control over external conditions in which experiment is performed.

**iv)Error due to internal sources**: These are errors caused due to limitations of experimental arrangement. Through the errors due to external sources can be eliminated by keeping a control over external conditions, but those due to internal sources cannot be eliminated because the reason of this type of error are inherent in the measurement and are unavoidable. For example, loss of energy due to radiation in calorimetry, the effect of buoyant force in weighing, etc., lead to errors of such type which cannot be eliminated altogether but necessary corrections can be made for them.

**c) Random Errors**:

The errors which are neither constant nor occur according to a definite pattern or system are called random errors. It is our common experience that repeated observation of a given physical quantity gives values which are different from each other. The errors involved in different observation have no set pattern. These errors occur irregularly and as such are random with respect to sign and size. These errors may arise due to carelessness of the observer or sudden changes occurring in the experimental set up.

Because the random errors are governed by chance, therefore, it is possible to minimize these errors by repeating the measurements many times and taking the arithmetic mean of all measurements as the correct value of the measured quantity.

Let us assume that x_{1},x_{2}, x_{3},…x_{n} are the values of a physical quantity obtained in different measurements. The best possible value of the physical quantity is given by:

x _{mean}=(x_{1}+x_{2}+ x_{3}….x_{n})/n

This method of elimination of random errors is based on the concept that it is reasonable to assume that individual measurements are as likely to underestimate as to overestimate the value of quantity.

**3. Methods of expressing an errors in measurements**: There are two method of expressing an error:

**a) Absolute errors in measurements: **The magnitude of the difference between the mean value and the measured value of the physical quantity in the observation is called as the absolute error of that ith observation. Thus, if x be the mean value of the measured quantity, and xi be the value obtained in the ith observation, then absolute error in our measurements are given by:

Δx_{1} = x_{mean}-x_{1}.

Δx_{2} = x_{mean}-x_{2}, etc.

If we take the arithmetic mean of all absolute errors, we get the final absolute error Δx_{mean}. When arithmetic mean is taken, only the magnitudes of the absolute errors are taken into account. Thus

*x _{mean}=(|x_{1}|+|x_{2}|+ |x_{3}|….|x_{n}|)/n*

**c) Relative error and percentage errors in measurements**: Relative error is defined as the ratio of the mean absolute error and the value of the quantity being measured. Thus relative error is given by:

δx= Δx_{mean}/x_{mean}

Where x (arithmetical mean) has been taken as the true value.

**The percentage error is expressed as:**

*Percentage error = (Δx/x)100*

True measure of the accuracy of a measurement is the relative percentage error but not the absolute error. When a large physical quantity is to be measured, relative or percentage error is less. In case of measurement of small quantities, special methods are to be used to keep the errors minimum possible.