Errors Of Measurements

Edited By Vishal kumar | Updated on Jul 02, 2025 05:35 PM IST

Measurement errors may be part of the ease of execution or lead to inaccurate results with scientific conclusions and industrial processes. Practically it is very important to know the sources and types of measurement errors in real-world situations in order to assure the data quality or make any decision based on measurements.

Errors Of Measurements

In this article, we will cover the concept of Errors Of Measurements. This concept is part of the chapter Physics and Measurement, which is a key chapter in Class 11 physics. It is crucial for board exams and competitive exams like the Joint Entrance Examination (JEE Main), National Eligibility Entrance Test (NEET), and other entrance exams such as SRMJEE, BITSAT, WBJEE, VITEEE, among others.

Over the last decade, one question related to this concept was asked in the JEE Main exam (2013-2023). In the NEET exam, four questions from this topic were featured in the same period.

It is the magnitude of the difference between the true value and the measured value of the quantity.

It may be positive in certain cases and negative in certain other cases

If a1,a2,a3………an are a measured value then
am=a1+a2+……ann

where a_m = true value

then

1)Absolute Error for nth reading =Δan=am−an= true value - measured value
So Δa1=am−a1Δa2=am−a2

2) Mean absolute error

Δa¯=|Δa1|+|Δa2|+….|Δan|n

3) Relative error or Fractional error

The ratio of mean absolute error to the mean value of the quantity measured.

Relative error =Δa¯amΔa¯− meanabsolute error am= mean value

4) Percentage error

Percentage error =Δa¯am×100

Error in sum and Error in difference of two physical quantities

1) Error in sum (x=a+b)
- Error in sum ( x=a+b) :-
- absolute error in x=Δx=±(Δa+Δb)
where
Δa= absolute error in measurement of a
Δb= absolute error in measurement of b
Δx= absolute error in measurement of x
- The percentage error in the value of x=
Δxx×100=(Δa+Δb)a+b×100
2) Error in difference ( x=a−b )
- Absolute error in x=Δx=±(Δa+Δb)
- Percentage error in the value of x=
Δxx×100=(Δa+Δb)a−b×100

Error in product and Error in division of two physical quantities

1) Error in product x=ab
- maximum fractional error= Δxx=±(Δaa+Δbb)
where
Δa= absolute error in measurement of a
Δb= absolute error in measurement of b
Δx= absolute error in measurement of x
- The percentage error in the value of x=
Δxx∗100=±(Δaa∗100+Δbb∗100)
=(% error in value of a+% error in value
of b)
2) Error in division
x=ab
- maximum fractional error in x=Δxx=±(Δaa+Δbb)
- The percentage error in the value of x=
Δxx∗100=±(Δaa∗100+Δbb∗100)

=(% error in value of a+% error in value of b)

Error in Quantity Raised to Some Power

when (x=anbm)
- The maximum fractional error in x is:-
Δxx=±(nΔaa+mΔbb)
- Percentage error in the value of x==
Δxx∗100=±(nΔaa∗100+mΔbb∗100)

Solved Example Based On Errors Of Measurements

Example 1: The value of the absolute error of the first measurement in a measured value a₁,a₂..............a_mis equal to [a_m is the true value]

1) a1+a2…ann
2) |am−a1|
3) a1am
4) a1∗am

Solution:

Errors of measurements -

If a1,a2,a3………an are a measured value
then am=a1+a2+……ann
where am= true value
Absolute Error for nth reading =Δan=am−an= true value - measured value
The absolute error of the first measurement, n=1
Δa1=am−a1

Hence, the answer is option (2).

Example 2: In the measurement of the period of a simple pendulum, the readings turn out to be (1) 2.63 s (2) 2.56 s (3) 2.42 s (4) 2.71 s (5) 2.80 s. Calculate the % error in the measurement.

1) ±4
2) ±3
3) ±5
4) ±2

Solution:

As we have learned
percentage error =ΔTmTm×100%

Maen value of time period-

Tm=2.63+2.56+2.42+2.71+2.805=2.624
Tm≈2.62
Absolute error:
ΔT1=2.62−2.63=−0.01s
ΔT2=2.62−2.56=+0.06s
ΔT3=2.62−2.42=+0.20s
ΔT4=2.62−2.71=−0.09s
ΔT5=2.62−2.80=−0.18s

Mean absolute error,
Mean absolute error, ΔTm=∑|ΔTn|n=0.545=0.11 s so, percentage error =±ΔTmTm×100=±0.112.62×100≈±4%

Hence, the answer is the option (1).

Example 3 The resistance R=V1, where V=(50±2)V and I=(20±0.2)A. The percentage error in R is x%. The value of ' x ' to the nearest integer is

1) 5

2) 10

3) 15

4) 20

Solution:

% error in R=ΔRR×100ΔRR×100=ΔVV×100+ΔI1×100⇒ΔRR×100=250×100+2220×100=4+1=5⇒% error in R=5%

Hence, the answer is the option (1).

Example 4: If a tuning fork of frequency (f0)340 Hz, tolerance ±1∘% is used in the resonance column method [v=2f0(l2−l1)], the first and the second resonances are measured at The max. permissible error in speed of sound is :

1) 1.4∘/o
2) 1.8∘/o
3) 1%
4) 0.8%

Solution:

Error in division x=a/b -
Δxx=±(Δaa+Δbb)
( maximum fractional error in x )
- wherein
Δa= absolute error in measurement of a
Δb= absolute error in measurement of b
Δx= absolute error in measurement of x
[Δvv]max=Δf0f0+Δl1+Δl2l2−l1=1100+0.1+0.174−24=[1100+0.250]×100∘/o=1.4∘%Hence, the answer is the option (1),

Example 5:The unit of percentage error is

1) Same as that of the physical quantity

2) percentage error is unitless

3) the error will have its own unit

4) square that of a physical quantity

Solution:

As we know percentage error is the ratio of two similar kinds of physical quantities.
Percentage error =Δa¯am×100%
So, the Percentage error is unitless.

Hence,the answer is option (2).

Summary

This article explains the concept of measurement errors, their types, and their significance in ensuring data quality. It provides formulas for calculating absolute, mean absolute, relative, and percentage errors, along with solved examples to illustrate these calculations. This topic is crucial for various academic and competitive exams in physics.

Frequently Asked Questions (FAQs)

1. What is meant by 'propagation of errors' in physics calculations?

Propagation of errors refers to how uncertainties in individual measurements combine to affect the uncertainty in a final calculated result. When performing calculations with measured values, each with its own uncertainty, these uncertainties "propagate" through the calculation, influencing the precision of the final result.

2. What is meant by 'significant figures' in measurements?

Significant figures are the digits in a measurement that are known with certainty, plus one estimated digit. They indicate the precision of a measurement. For example, if a length is measured as 10.47 cm, it has four significant figures. The '7' is the estimated digit, indicating the measurement's precision to the nearest 0.01 cm.

3. How do you express a measurement result with its uncertainty?

A measurement result is typically expressed as:

4. How does the concept of significant figures relate to measurement uncertainty?

Significant figures are a way of expressing measurement uncertainty. The number of significant figures in a measurement implies the precision of the measuring instrument. For example, a measurement of 10.47 cm suggests uncertainty in the hundredths place (±0.01 cm). Keeping track of significant figures ensures that calculated results don't imply more precision than the original measurements warrant.

5. What is meant by 'zero error' in a measuring instrument?

Zero error occurs when a measuring instrument does not read exactly zero when it should. For example, if a weighing scale shows 0.1 g when nothing is placed on it, it has a zero error of +0.1 g. Zero errors can be positive or negative and must be accounted for in all measurements made with that instrument.

6. How does the concept of 'significant figures' apply to calculations involving measured quantities?

When performing calculations with measured quantities:

7. How does temperature affect measurement errors?

Temperature can introduce errors in various ways:

8. What is the importance of reporting both systematic and random errors in experimental results?

Reporting both systematic and random errors is crucial because:

9. What is 'instrumental error' and how does it contribute to measurement uncertainty?

Instrumental error refers to the inherent limitations and imperfections in measuring devices. It contributes to measurement uncertainty in several ways:

10. What is meant by 'percentage error' and how is it calculated?

Percentage error is a way of expressing relative error as a percentage. It's calculated as:

11. What is meant by 'errors of measurement' in physics?

Errors of measurement refer to the unavoidable differences between the measured value of a quantity and its true value. These errors occur due to limitations in measuring instruments, human factors, or environmental conditions. Understanding measurement errors is crucial for accurately reporting and interpreting experimental results in physics.

12. Why are errors important to consider in scientific measurements?

Errors are important because they help us understand the reliability and limitations of our measurements. Considering errors allows scientists to:

13. What's the difference between accuracy and precision in measurements?

Accuracy refers to how close a measurement is to the true value, while precision refers to how close repeated measurements are to each other. A measurement can be precise (consistent) without being accurate (close to the true value), or accurate without being precise. Ideally, we aim for both accuracy and precision in scientific measurements.

14. How does systematic error differ from random error?

Systematic errors are consistent, predictable deviations from the true value that affect all measurements in the same way. They are often due to faulty equipment or consistent human error. Random errors, on the other hand, fluctuate unpredictably in both magnitude and direction. They are caused by uncontrollable variations in the measurement process or environment.

15. Can you completely eliminate errors in measurements?

It's impossible to completely eliminate all errors in measurements. Even with the most advanced equipment and careful procedures, there will always be some degree of uncertainty. However, we can work to minimize errors through proper calibration, multiple trials, and improved measurement techniques.

16. What is the concept of 'error bars' in graphical representations of data?

Error bars are graphical representations of the variability of data. They indicate the uncertainty in a reported measurement. On a graph, they appear as vertical lines extending above and below each data point. The length of an error bar typically represents one standard deviation, the standard error of the mean, or a confidence interval. Error bars help in visually assessing the significance of differences between data points.

17. What is meant by 'least squares fitting' in the context of measurement errors?

Least squares fitting is a mathematical method used to find the best-fit line or curve for a set of data points, considering measurement uncertainties. It minimizes the sum of the squares of the residuals (differences between observed values and the fitted values provided by the model). This method is widely used in physics to:

18. How do you determine if a measured value is 'statistically significant'?

Statistical significance is determined by comparing the measured difference to the expected variation due to random errors. The process typically involves:

19. What is the difference between 'accuracy class' and 'precision class' in measuring instruments?

Accuracy class and precision class are specifications for measuring instruments:

20. How do you determine the number of significant figures in a measurement?

To determine the number of significant figures:

21. What is absolute error and how is it calculated?

Absolute error is the difference between the measured value and the true or accepted value of a quantity. It's calculated using the formula:

22. How does relative error differ from absolute error?

Relative error is the ratio of the absolute error to the true value of the measurement, often expressed as a percentage. It's calculated as:

23. What is the 'least count' of a measuring instrument?

The least count of a measuring instrument is the smallest division that can be measured accurately with that instrument. It represents the resolution or precision of the instrument. For example, a ruler marked in millimeters has a least count of 1 mm, while a vernier caliper typically has a least count of 0.1 mm or 0.05 mm.

24. How does parallax error occur in measurements?

Parallax error occurs when an observer's line of sight is not perpendicular to the scale of a measuring instrument. This can cause the observer to read a different value than the true measurement. It's common when reading analog scales, like rulers or analog meters. To minimize parallax error, always ensure your line of sight is perpendicular to the scale.

25. How can random errors be reduced in an experiment?

Random errors can be reduced by:

26. What is the difference between 'error' and 'uncertainty' in measurements?

While often used interchangeably, 'error' and 'uncertainty' have distinct meanings in scientific measurements:

27. How do systematic and random errors affect the accuracy and precision of measurements?

Systematic errors affect accuracy but not precision. They consistently shift all measurements away from the true value in the same direction. Random errors affect precision but not necessarily accuracy. They cause measurements to scatter around the true value. An experiment with only systematic error will give precise but inaccurate results, while one with only random error will give imprecise but potentially accurate results (on average).

28. What is the importance of calibration in reducing measurement errors?

Calibration is crucial for reducing systematic errors in measurements. It involves comparing the readings of an instrument to a known standard and adjusting the instrument accordingly. Regular calibration ensures that:

29. How do you choose the appropriate number of decimal places when reporting a measurement?

The number of decimal places in a reported measurement should reflect the precision of the measuring instrument. Generally:

30. What is the difference between 'precision' and 'resolution' in measurements?

While related, precision and resolution are distinct concepts:

31. How does the 'observer effect' contribute to measurement errors in physics?

The observer effect refers to changes that the act of observation makes on the phenomenon being observed. In physics, this can lead to measurement errors. For example:

32. How does the 'signal-to-noise ratio' relate to measurement errors?

The signal-to-noise ratio (SNR) is a measure of how much the desired signal exceeds the background noise. In measurements:

33. What is the concept of 'error propagation' in complex calculations?

Error propagation refers to how uncertainties in individual measurements combine to affect the uncertainty in a final calculated result. In complex calculations:

34. How do you choose between different measuring instruments for an experiment?

Choosing the right measuring instrument involves considering:

35. What is the importance of 'repeatability' in measurements?

Repeatability refers to the closeness of agreement between successive measurements of the same quantity under the same conditions. It's important because:

36. How does the concept of 'significant figures' apply to very large or very small numbers in scientific notation?

For numbers in scientific notation:

37. What is 'standard deviation' and how is it used to express measurement uncertainty?

Standard deviation is a statistical measure of the spread of a set of data points. In measurements:

38. How do you handle discrepant data points in a set of measurements?

Handling discrepant data (outliers) requires careful consideration: