Uncertainty In Measurement: Definition, Formula and Examples

Edited By Shivani Poonia | Updated on Jul 02, 2025 07:38 PM IST

The concept of uncertainty in measurement is very important and has been a fundamental part of scientific questions for centuries. There was a mathematician named Sir William Thomson, generally known as Lord Kelvin who made a great contribution to the discovery or the study of uncertainty in measurement in the mid-19th century. Kelvin emphasizes the error analysis in the experimental science.

This Story also Contains

Uncertainty in the Measurements
Significant Figures
Dimensional Analysis
Some Solved Examples
Summary

Uncertainty In Measurement: Definition, Formula and Examples

However, nowadays there are modern methods for current the uncertainty in the statistical methods express and manage it Because as time passes the theories develop according to the advancement in the science. Before Kelvin, there was the scientist Karl Friedrich Gauss who also had great significance to the statistical analysis of data through the Gaussian distribution.

Uncertainty in the Measurements

Scientific Notation

For denoting numbers as large as 602,200,000,000,000,000,000,000 for the molecules of 2 g of hydrogen gas or as small as 0.00000000000000000000000166 g mass of an H atom, we need to use scientific notation i.e., exponential notation in which any number can be represented in the form of N × 10ⁿ where n is an exponent having positive or negative values and N is a number (called digit term) which varies between 1.000... and 9.999.... Thus, we can write 232.609 as 2.32609 ×10² in scientific notation.

Addition and Subtraction

For adding or subtracting two numbers, first, the numbers are written such that they have the same exponent.

After that, the digit terms are added or subtracted.

4.5 X 10⁴ + 2.5 X 10⁵ = 4.5 X 10⁴ + 25.0 X 10⁴ = 29.5 X 10⁴

4.5 X 10⁴ - 2.5 X 10⁵ = 4.5 X 10⁴ - 25.0 X 10⁴ = (-)20.5 X 10⁴ (negative)

Multiplication and Division

For multiplying or dividing two numbers, the same rules are followed as for exponential numbers.

(4.5 X 10⁴) X (2.5 X 10⁵) = (4.5 X 2.5)X (10⁴⁺⁵) = 11.25 X 10⁹ = 1.125 X 10¹⁰

(4.5 X 10⁴ ) / (2.5 X 10⁵) = (4.5 / 2.5) X 10^4-5 = 1.8 X 10^-1

Experimental calculations or Measurements

Precision refers to the closeness of various measurements for the same quantity.

Accuracy is the agreement of a particular value to the true value of the result.

For example, if the true value for a result is 2.00 g and student ‘A’ takes two measurements and reports the results as 1.95 g and 1.93 g. These values are precise as they are close to each other but are not accurate. Another student ‘B’ repeats the experiment and obtains 1.94 g and 2.05 g as the results for two measurements. These observations are neither precise nor accurate. When the third student ‘C’ repeats these measurements and reports 2.01 g and 1.99 g as the result, these values are both precise and accurate.

Significant Figures

Every experimental measurement has some amount of uncertainty associated with it because of the limitations of the measuring instrument and the skill of the person making the measurement. Hence, the result of any experiment is written indicating the significant figures which includes all the digits known with certainty plus one which is uncertain

There are certain rules for determining the number of significant figures. These are stated below:

(1) All non-zero digits are significant.

(2) Zeros preceding to first non-zero digit are not significant. Such zero indicates the position of the decimal point.

(3) Zeros between two non-zero digits are significant.

(4) Zeros at the end or right of a number are significant provided they are on the right side of the decimal point.

(5) Counting numbers of objects, for example, 2 balls or 20 eggs, have infinite significant figures as these are exact numbers and can be represented by writing an infinite number of zeros after placing a decimal i.e., 2 = 2.000000 or 20 = 20.000000

In numbers written in scientific notation, all digits are significant e.g., 4.01×10² has three significant figures, and 8.256 × 10^–3 has four significant figures.

Addition and Subtraction of Significant Figures

The result cannot have more digits to the right of the decimal point than either of the original numbers.

12.11
18.0
1.012

31.122

Here, 18.0 has only one digit after the decimal point and the result should be reported only up to one digit after the decimal point, which is 31.1.

Multiplication and Division of Significant Figures

In these operations, the result must be reported with no more significant figures as in the measurement with a few significant figures.

2.5×1.25 = 3.125
Since 2.5 has two significant figures, the result should not have more than two significant figures, thus, it is 3.1.

Rounding off the numbers

While limiting the result to the required number of significant figures as done in the above mathematical operation.the following points for rounding off the numbers.

1. If the rightmost digit to be removed is more than 5, the preceding number is increased by one. For example, 1.386. If we have to remove 6, we have to round it to 1.39.
2. If the rightmost digit to be removed is less than 5, the preceding number is not changed. For example, 4.334 if 4 is to be removed, then the result is rounded upto 4.33.
3. If the rightmost digit to be removed is 5, then the preceding number is not changed if it is an even number but it is increased by one if it is an odd number. For example, if 6.35 is to be rounded by removing 5, we have to increase 3 to 4 giving 6.4 as the result. However, if 6.25 is to be rounded off it is rounded off to 6.2.

Dimensional Analysis

While calculating, the units should be the same all across the equation. The units for each physical quantity should be the same throughout the equation.

eg. If the temperature is involved, the units should be Kelvin on both sides

Some Solved Examples

Example.1

1.The minimum amount of $O_2(g)$ consumed per gram of reactant is for the reaction :

(Given atomic mass :Fe=56 O=16, $M g=24, P=31, C=12, H=1)$)

1) (correct)$4 \mathrm{Fe}(\mathrm{s})+3 \mathrm{O}_2(\mathrm{~g}) \rightarrow 2 \mathrm{Fe}_2 \mathrm{O}_3(\mathrm{~s})$

2)$P_4(s)+5 O_2(g) \rightarrow P_4 O_{10}(s)$

3) $\mathrm{C}_3 \mathrm{H}_8(\mathrm{~g})+5 \mathrm{O}_2(\mathrm{~g}) \rightarrow 3 \mathrm{CO}_2(g)+4 \mathrm{H}_2 \mathrm{O}(\mathrm{I})$

4)$2 \mathrm{Mg}(\mathrm{s})+\mathrm{O}_2(\mathrm{~g}) \rightarrow 2 \mathrm{MgO}(\mathrm{s})$

Solution

The amount of Fe consumed for 3 moles of O₂ = 4 x 56 = 224g, Thus, the amount of O₂ consumed per gram of Fe = 3/224 g, Similarly, for other elements, we have: per gram P₄ required = 5/124 moles, per gram C₃H₈ required = 5/44 moles, per gram Mg required = 1/48 moles. Thus, the amount of O₂ is the minimum for Fe.

Hence, the answer is the option (1).

Example.2

2.If the true value of a quantity is 5g, which of the following measurements are both precise and accurate?

1)4.90g, 4.89g

2) (correct)4.99g, 5.01g

3)4.84g, 4.86g

4)5.03g, 5.05g

Solution

The true value of a quantity is 5g. A student measures it twice and gets 4.99g and 5.01g.

4.99g & 5.01g are accurate because they are the closest to the true value. They are precise because they are close to each other.

Hence, the answer is the option (2).

Example.3

3. If the true value of a quantity is 3g, which of the following values is most precise?

1) (correct)2.98g, 2.97g

2)3.01g, 2.99g

3)2.94g, 2.97g

4)2.97g, 2.99g

Solution

Since 2.98g & 2.97g are closest to each other for the same quantity, thus they are most precise for the given value.

Hence, the answer is the option (1).

Example.4

4. Expression of 175000 in scientific notation:

1)0.0175

2)0.000175

3) (correct)1.75 x 10⁵

4)0.175

Solution

In scientific notation, 175000 is written as 1.75 x 10⁵

Hence, the answer is the option (3).

Example.5

5. Using the rules for significant figures, the correct answer for the expression

$\frac{0.02858 \times 0.112}{0.5702}$ will be

1)$0.005613$

2) (correct)$0.00561$

3)$0.0056$

4)0.006

Solution

The correct answer is rounded off to the same number of significant figures as present in the number with the least significant figure

In the given case, the number with the least significant figure is 0.112 (3 significant figures).

Thus, the result will also contain only 3 significant figures.

$\frac{0.02858 \times 0.112}{0.5702}$

$=0.00561(3 \mathrm{~S} . \mathrm{F})$

Hence, the answer is the option (2).

Summary

The uncertainty principle is the inherent limitation of chemistry that helps to determine the value of any physical quantity. It can arise from various sources like from instruments, measurements, and environmental factors, and also from the ability of an observer to notice it. There are various types of uncertainty such as systematic uncertainty which occurs consistently or repeatedly and other one is random uncertainty which occurs from unpredictable fluctuations and variations in measurement.

Frequently Asked Questions (FAQs)

1. What is the concept of propagation of uncertainty?

Propagation of uncertainty refers to how uncertainties in individual measurements combine to affect the uncertainty of a final calculated result that depends on those measurements.

2. What is the role of significant digits in expressing uncertainty?

Significant digits convey the precision of a measurement. The number of significant digits used should be consistent with the uncertainty of the measurement.

3. How does uncertainty affect the comparison of experimental results?

When comparing results, overlapping uncertainty ranges indicate that the results may not be significantly different, even if the central values differ.

4. How does the principle of significant figures apply to calculations involving uncertain measurements?

In calculations, the result should not imply greater precision than the least precise measurement used. The number of significant figures in the result should reflect this limitation.

5. How does uncertainty affect the interpretation of trends in data?

Large uncertainties can obscure trends in data, making it difficult to draw reliable conclusions. Trends should be interpreted in light of the associated uncertainties.

6. How does significant figures relate to uncertainty in measurements?

Significant figures indicate the precision of a measurement. The last significant figure usually carries some uncertainty, reflecting the limitations of the measuring device or process.

7. How does rounding affect uncertainty in measurements?

Rounding can introduce additional uncertainty by discarding information. It's important to round only at the end of calculations to minimize cumulative rounding errors.

8. How does the choice of measuring instrument affect uncertainty?

The precision and accuracy of the measuring instrument directly impact the uncertainty of the measurement. More precise instruments generally lead to lower uncertainty.

9. What is the importance of reporting uncertainty along with measurements?

Reporting uncertainty provides context for the measurement, allowing others to assess its reliability and make informed decisions based on the data.

10. What is the concept of "least count" in relation to uncertainty?

The least count is the smallest division on a measuring instrument's scale. It often represents the lower limit of uncertainty for that instrument.

11. What is uncertainty in measurement?

Uncertainty in measurement refers to the doubt or lack of exactness in the result of a measurement. It acknowledges that no measurement is perfect and there's always a range within which the true value likely lies.

12. Why is understanding uncertainty important in scientific measurements?

Understanding uncertainty is crucial because it helps scientists assess the reliability and accuracy of their results. It allows for more honest reporting of data and helps in making informed decisions based on the measurements.

13. What are the main sources of uncertainty in measurements?

The main sources of uncertainty include limitations of measuring instruments, environmental factors, human error, and the inherent variability of the quantity being measured.

14. How is uncertainty different from error in measurements?

Error is the difference between a measured value and the true value, while uncertainty is a quantification of doubt about the measurement result. Errors can sometimes be corrected, but uncertainty is always present.

15. What is the formula for calculating relative uncertainty?

Relative uncertainty = (Absolute uncertainty / Measured value) × 100%

16. What is 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, regardless of their accuracy.

17. What is meant by "absolute uncertainty" in measurements?

Absolute uncertainty is the range within which the true value is expected to lie. It's often expressed as ± a certain value in the same units as the measurement.

18. How does sample size affect uncertainty in measurements?

Generally, increasing the sample size (number of measurements) reduces uncertainty by minimizing the impact of random errors and providing a more representative average.

19. What is the relationship between confidence level and uncertainty?

The confidence level indicates how certain we are that the true value falls within the stated uncertainty range. Higher confidence levels typically result in wider uncertainty ranges.

20. How do systematic and random errors contribute to uncertainty?

Systematic errors cause consistent deviations in measurements and can be corrected if identified. Random errors cause fluctuations around the true value and contribute to uncertainty.

21. How does calibration of instruments affect measurement uncertainty?

Proper calibration reduces systematic errors and can significantly decrease measurement uncertainty by ensuring the instrument provides accurate readings.

22. How does temperature affect measurement uncertainty?

Temperature changes can cause expansion or contraction of materials and instruments, potentially introducing errors and increasing uncertainty in measurements.

23. How does the concept of uncertainty apply to qualitative observations?

While uncertainty is typically associated with quantitative measurements, qualitative observations can also have uncertainty in terms of subjective interpretations or limitations in observational methods.

24. What is meant by "standard uncertainty" in measurements?

Standard uncertainty is the uncertainty of a measurement expressed as one standard deviation, providing a consistent way to express uncertainty across different types of measurements.

25. How does the resolution of a measuring device relate to uncertainty?

The resolution of a device (smallest detectable change) often sets a lower limit on the uncertainty of measurements made with that device.

26. What is the difference between Type A and Type B uncertainties?

Type A uncertainties are evaluated using statistical methods from repeated observations, while Type B uncertainties are evaluated using other means, such as manufacturer specifications or calibration certificates.

27. What is the concept of "uncertainty budget" in measurements?

An uncertainty budget is a comprehensive list of all sources of uncertainty in a measurement, quantifying their individual contributions to the overall uncertainty.

28. What is the relationship between uncertainty and the detection limit of an instrument?

The detection limit is often related to the uncertainty of measurements near zero. Measurements near or below the detection limit have high relative uncertainty.

29. What is the importance of stating the confidence level when reporting uncertainty?

Stating the confidence level (e.g., 95% confidence interval) provides context for the reported uncertainty, indicating how likely it is that the true value falls within the stated range.

30. How does uncertainty in measurements impact scientific theories and laws?

Uncertainty in measurements can limit the precision of scientific theories and laws. As measurement techniques improve and uncertainties decrease, theories may need to be refined or revised.

31. What is the concept of "combined standard uncertainty"?

Combined standard uncertainty is the standard uncertainty of a result that depends on several other quantities, taking into account how the uncertainties of these quantities combine.

32. How does the choice of measurement method affect uncertainty?

Different measurement methods may have different sources and magnitudes of uncertainty. Choosing an appropriate method can minimize uncertainty for a specific measurement task.

33. What is the role of uncertainty in determining significant differences between experimental results?

Uncertainty helps determine whether observed differences between results are statistically significant or could be due to random variations within the uncertainty range.

34. How does uncertainty relate to the concept of precision in repeated measurements?

The spread of repeated measurements (precision) contributes to the uncertainty of the final result. Higher precision (smaller spread) generally leads to lower uncertainty.

35. What is the importance of considering correlation between variables when calculating combined uncertainties?

Correlations between variables can significantly affect the combined uncertainty. Ignoring correlations can lead to over- or underestimation of the total uncertainty.

36. How does the concept of uncertainty apply to digital measurements?

Digital measurements also have uncertainty, often related to the resolution of the device and the analog-to-digital conversion process.

37. What is the relationship between uncertainty and the reproducibility of an experiment?

Reproducibility is affected by uncertainty. Experiments with lower uncertainty are generally more reproducible, as the range of possible outcomes is narrower.

38. How does uncertainty in measurements affect the validation of scientific models?

Uncertainty in measurements sets limits on how precisely a model can be validated. Models must be consistent with experimental results within their uncertainty ranges.

39. What is the concept of "expanded uncertainty" in measurement?

Expanded uncertainty is obtained by multiplying the standard uncertainty by a coverage factor, typically providing a larger confidence interval (e.g., 95% or 99% confidence).

40. How does the principle of uncertainty affect the reporting of very small or very large quantities?

For very small or large quantities, relative uncertainty often provides more meaningful information than absolute uncertainty. Scientific notation may be used to express both the quantity and its uncertainty clearly.