Errors in Measurement: Do you know that every calculated quantity which is based on measured values may have an error? And when it is used in formulas, these measurement errors can quickly grow in size. For example, let us measure the weights of \(100\) marathon athletes. In the first case, let the scale you were using is one pound off, then this will result in all athlete’s body weight calculations being off by a pound. In the second case, if your scale was accurate but some athletes might be more dehydrated than others, and some might get heavier due to wetting of clothes. In both cases, it will lead to measurement errors. In this article, we will discuss how there will be errors in measurement while doing calculations? What are the types of errors? We will also discuss the combination of errors due to addition, subtraction, multiplication, division, and exponents.

Measurement Error

The difference between a measured quantity and its true value is known as measurement error. It is also known as observational error. Those errors which occur naturally and to be expected with any experiment are random errors, and the error that is caused by a miscalibrated instrument that affects all measurements are systematic errors.

Classification of Errors

We can broadly classify measurement errors as systematic errors, random errors, and least count errors. These are given following:

1. Systematic errors

Those errors that tend to be in one direction, either positive or negative, are called systematic errors. The below figure shows the systematic error:

Types of systematic errors are given below:

1. Instrumental errors: These errors arise due to imperfect design or calibration of the measuring instrument. For example, the temperature graduations of a thermometer may be inadequately calibrated.
2. Imperfection in experimental technique or procedure: A thermometer placed under the armpit to determine the temperature of a human body will always give a temperature lower than the actual value of the body temperature. Some other external conditions may systematically affect the measurement during the experiment, such as changes in temperature, humidity, wind velocity, etc
3. Personal errors: These errors are caused due to an individual’s bias, such as lack of proper setting of the apparatus, or individual’s carelessness in taking observations, etc. For example, you will introduce an error due to parallax while reading the position of a needle on the scale when you hold your head a bit too far to the right.

Systematic errors can be minimized by selecting better instruments, improving experimental techniques, and removing personal bias.

2. Random errors

The random errors are the errors, which occur irregularly and hence are random concerning sign and size. It is not possible to remove random errors completely because the causes of random errors are not known. The figure shows the random error given below:

For example, when you are reading data of sensitive beambalance, then its reading may be changed by the vibrations caused by a person moving in the laboratory. By repeating the observation a largenumber of times and taking the arithmetic mean of all the observations, random errors can be minimized. Thus,
\(a_{\text {mean}}=\frac{a_{1}+a_{2}+\ldots+a_{n}}{n}\)

3. Least count error

The least count is the smallest value that can be measured by the measuring instrument. The least count error belongs to the category of random errors but within a limited size; it occurs with both systematic and random errors. We can reduce the least count error by using higher precision instruments and also by improving experimental techniques. Some types of least count error are given following :

(a) Absolute Error

Absolute error is equal to the difference between the true value and the measured value of a quantity. It may be positive or negative. Usually, the mean value \(a_{m}\) is taken as the true value. So, if

\(a_{m}=\frac{a_{1}+a_{2}+\ldots+a_{n}}{n}\)

Now, according to the definition we have,

\(\Delta a_{1}=a_{m}-a_{1}\)

\(\Delta a_{2}=a_{m}-a_{2}\)

… … …

\(\Delta a_{n}=a_{m}-a_{n}\)

(b) Mean Absolute Error

The arithmetic mean of the magnitudes of absolute errors in all the measurements is called the mean absolute error. Thus,

\(\Delta a_{\text {mean }}=\frac{\left|\Delta a_{1}\right|+\left|\Delta a_{2}\right|+\ldots+\left|\Delta a_{n}\right|}{n}\)

The final result of a measurement can be written as,

\(a=a_{m} \pm \Delta a_{\text {mean }}\)

This implies that the value of \(a\) is likely to lie between

\(a_{m}+\Delta a_{\text {mean }}\) and \(a_{m}-\Delta a_{\text {mean }}\)

(c) Relative or Fractional Error

The relative error is the ratio of mean absolute error to the mean value of the quantity measured. It is also known as fractional error. Thus we can write,

\(\text {Relative error} =\frac{\Delta a_{\text {man }}}{a_{m}}\)

Relative error expressed in percentage is called the percentage error, i.e.

\(\text {Percentage error} =\frac{\Delta a_{m k m}}{a_{m}} \times 100\)

Combination of Errors

If we do an experiment involving several measurements, then the errors in all the measurements may combine, and we must know how they combine. For example, the various combination errors are given following :

(a) Errors of a sum or a difference

Let us suppose two physical quantities \(A\) and \(B\) that have measured values such as \(A \pm \Delta A, B \pm \Delta B\) respectively, where \(\Delta A\) and \(\Delta B\) are their absolute errors. Now we want to find the error \(\Delta Z\) in the sum.

\(Z=A+B\)

We have, by addition, \(Z \pm \Delta Z\)

\(=(A \pm \Delta A)+(B \pm \Delta B)\).

The maximum possible error in \(Z\)

\(\Delta Z=\Delta A+\Delta B\)

Now, for the difference we have,

\(Z=A-B\)

We have,

\(Z \pm \Delta Z=(A \pm \Delta A)-(B \pm \Delta B)\)

\(=(A-B) \pm \Delta A \pm \Delta B\)

Or,

\(\pm \Delta Z=\pm \Delta A \pm \Delta B\)

The maximum value of the error \(\Delta Z\) is again \(\Delta A+\Delta B\).

Hence, we can conclude that when two quantities are added or subtracted, then the absolute error in the final result is equal to the sum of the absolute errors in the individual quantities.

(b) An error of a product or a quotient

Let us suppose \(Z=A B\) and the measured values of \(A\) and \(B\) are given as \(A \pm \Delta A\) and \(B \pm \Delta B\). Then we can write,

\(Z \pm \Delta Z=(A \pm \Delta A)(B \pm \Delta B\)

\(=A B \pm B \Delta A \pm A \Delta B \pm \Delta A \Delta B\)

Now, On dividing the \(L H S\) by \(Z\) and \(R H S\) by \(A B\) we have,

\(1 \pm\left(\frac{\Delta Z}{Z}\right)=1 \pm\left(\frac{\Delta A}{A}\right) \pm\left(\frac{\Delta B}{B}\right) \pm\left(\frac{\Delta A}{A}\right)\left(\frac{\Delta B}{B}\right)\)

And, we shall ignore their product since \(\Delta A\) and \(\Delta B\) are small, Hence the maximum relative error

\(\frac{\Delta Z}{Z}=\left(\frac{\Delta A}{A}\right)+\left(\frac{\Delta B}{B}\right)\)

You can easily verify that this is true for the division also.

Hence, we can conclude that when two quantities are multiplied or divided, then the relative error in the result will be equal to the sum of the relative errors in the multipliers.

(c) Error in case of a measured quantity raised to a power

Suppose \(Z=A^{2}\), then

\(\frac{\Delta Z}{Z}=\left(\frac{\Delta A}{A}\right)+\left(\frac{\Delta A}{A}\right)=2\left(\frac{\Delta A}{A}\right)\)

Hence, we can say that the relative error in \(A^{2}\) is two times the error in \(A\). In general if,

\(Z=A^{p} \frac{B^{q}}{C^{r}}\)

Then

\(\frac{\Delta Z}{Z}=p\left(\frac{\Delta A}{A}\right)+q\left(\frac{\Delta B}{B}\right)+r\left(\frac{\Delta C}{C}\right)\).

Hence we can say that the relative error in a physical quantity raised to the power \(k\) is the \(k\) times the relative error in the individual quantity.

Errors in Measurement – Sample Problems

Q.1. Find the percentage error in the determination of the time period of a pendulum \(T=2 \pi \sqrt{\frac{l}{g}}\) where \(l\) and \(g\) are measured with \(\pm 1 \%\) and \(\pm 2 \%\).

Ans: \(T=2 \pi \sqrt{\frac{l}{g}}\)

or,

\(T=(2 \pi)(l)^{\frac{\pm 1}{2}}(g)^{\frac{-1}{2}}\)

On taking logarithm both sides, we have

\(\ln (T)=\ln \ln (2 \pi)+\frac{1}{2}(\ln \ln l)-\left(\frac{1}{2}\right) \ln \ln (g) \quad \ldots.(i)\)

Now, on differentiating Eq. \((i)\), we have

\(\frac{1}{T} d T=0+\frac{1}{2}\left(\frac{1}{l}\right)(d l)-\frac{1}{2}\left(\frac{1}{g}\right)(d g)\)

Or,

\(\left(\frac{d T}{T}\right)_{\max }= \text {maximum value of} \left(\pm \frac{1}{2} \frac{d l}{l} \mp \frac{1}{2} \frac{d g}{g}\right)\)

\(=\frac{1}{2}\left(\frac{d l}{l}\right)+\frac{1}{2}\left(\frac{d g}{g}\right)\)

This can also be written as

\(\left(\frac{\Delta T}{T} \times 100\right)_{\max }=\frac{1}{2}\left[\frac{\Delta l}{l} \times 100\right]+\frac{1}{2}\left[\frac{\Delta g}{g} \times 100\right]\)

Or percentage error in the time period

\(=\pm\left[\frac{1}{2}(\right. \text {the percentage error in} \,l)+\frac{1}{2}(\text {percentage error in} \left.g)\right]\)

\(=\pm\left[\frac{1}{2} \times 1+\frac{1}{2} \times 2\right]=\pm 1.5 \%\)

Summary

The uncertainty produced in the result of every measurement by any measuring instrument caused a measurement error. These errors can be classified in two ways. The first classification is based on the cause of the error in this group are Systematic errors and Random errors. The second classification is based on the magnitude of errors. Absolute error, mean absolute error, and relative (or fractional) error lie in this group. If we do an experiment involving several measurements, we must know how the errors in all the measurements combine due to sum, difference, product, division, etc.

FAQs on Errors in Measurement

Q.1. What is called a measurement error?
Ans: The difference between a measured quantity and its true value is known as measurement error. The measurement error is a mathematical way to show the uncertainty in the measurement.

Q.2. What are the different sources of error in measurement?
Ans: The most common natural phenomena which may cause measurement errors are temperature variation, humidity, gravity, wind, refraction, etc. For example, an error in the length of a measuring tape is due to a change of temperature.

Q.3. What are the examples of random errors?
Ans: A very simple example is our blood pressure. Even if someone is healthy, it is normal that their blood pressure does not remain the same every time it is measured.

Q.4. What are the examples of systematic errors?
Ans: Systematic errors primarily influence a measurement’s accuracy. Typical causes of the systematic error include observational error, imperfect instrument calibration, and environmental interference.

Q.5. How can we eliminate systematic errors?
Ans: The systematic errors are due to improper calibration and environmental interference. This can be reduced by proper calibration of instruments in experiment environmental conditions.

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