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Uncertainty in measurement is the range of possible values within which the true/real value of the measurement exists. There are practical techniques to manage these numbers conveniently and deliver the provided information as truthfully as feasible. The purpose of measurement is to provide information about a quantity of interest but there is a chance that all the measurements have a certain degree of uncertainty regardless of their precision and accuracy. This occurs mainly either because of the limitation of the measuring instrument called systematic error or the skill of the experimenter doing the measurements called random error. If the values obtained in an experiment or from a system are very close to each other along with their average values then it can be considered that the measurement is correct and precise.
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Key Terms: Uncertainty in Measurement, Scientific Notation, Accuracy, Precision, Systemic Error, Random Error
Uncertainty in Measurement: Scientific Notation
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Scientific notation represents the numbers that are very huge or very tiny in a form of multiplication of single-digit numbers and 10 raised to the power of the respective exponent. The exponent is positive if the number is very large and it is negative if the number is very small.
- Atoms and molecules consist of very low masses, but they are present in huge numbers.
- Chemists have to deal with numbers as large as 602,200,000,000,000,000,000,000, which is the number of molecules of 2g of hydrogen, and also with numbers as small as 0.00000000000000000000000166g, which is the mass of a Hydrogen atom.
- Apart from these, there are even other constants like the speed of light, charges on particles, Avogadro’s number, and Planck’s constant, which have numbers above these stated magnitudes and are written as ‘SCI’ display mode in scientific calculators.
Significant Figures Video Lecture
To handle these large or small numbers, we use the following notation: m × 10n, which is, m times ten raised to the power of n. In this expression, n is an exponent having positive and negative values and m is that number that varies from 1.000… and 9.999…
- The scientific notation 573.672 can be written as 5.73672 × 102. In this case, the decimal is needed to be moved left by two places but if it is moved three places to its left then the power of 10 will be 3.
- In the same way, 0.000089 can also be written as 8.9 × 10-5. In this, the decimal is moved five places towards the right, and (−5) is the exponent in the scientific notation.
- All of these help us to attain easier handling, better precision, and accuracy while performing operations on numbers with high magnitudes.
Uncertainty in Measurement
Uncertainty in Measurement: Calculation
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Measurement uncertainty is frequently defined as the standard deviation of a probability distribution over the range of possible values that could be attributed to a measured quantity. It helps in yielding accurate results.
Uncertainty in Multiplication and Division
Applying the same rule as discussed above we can solve the given problem as, for example,
(3.9 × 106) × (2.1 × 105) = (3.9 × 2.1)(106+5)
= (3.9 × 2.1) × (1011)
= 11.31 × 1011
Where the values 6 and 5 are the power of 10 are math processing error values.
Similarly for division,
(3.6 × 10-5) ÷ (2.0 × 10-4)
= (3.6 ÷ 2.0)(10(-5)-4)
= 1.8 × 10-1
Where the values (-5) and (-4) that are the power of 10 are math processing error values.
Read More:
| Relevant Concepts | ||
|---|---|---|
| Significant Figures | Valence Bond Theory | Metals and Non-Metals |
| Corrosion | Kelvin Planck Statement | Uses of Electroplating |
Uncertainty in Addition and Subtraction
While doing this operation we have to place these numbers in such a way that they have the same exponents. If there are two numbers 5.43 × 104 and 3.45 × 103, the powers are made equal and after that, the coefficients are added and subtracted.
When adding,
5.43 × 104 + 0.345 × 105
= (5.43 + (0.345×10)) × 104
= 8.88 × 104
Where the values 4 and 5 that are the power of 10 are math processing error values.
Similarly for subtraction,
5.43 × 104 – 0.345 × 105
= (5.43 – (0.345 × 10)) × 104
= (5.43–3.45) × 104
= 1.98 × 104
Where the values 4 and 5 are the power of 10 are math processing error values.
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<Uncertainty in Measurement: Percentage Formula
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The uncertainty of a measured value can be represented in both ways, either in a percentage notation or as a simple ratio.
It is calculated as:
| Percent uncertainty = (Uncertainty/Actual Value) x 100 |
Read More: Refining of Metals Important Notes
Solved Example
A scale incorrectly measures a value as 6 cm because of some marginal errors. If the real measurement of the value is taken as 10 cm then what will be the percentage of error.
Solution: Given,
Approximate value/wrong value = 6 cm
Exact value = 10 cm
Percentage Error = (Approximate Value - Exact Value)/Exact Value) x 100
Percentage Error = (10 – 6)/10 × 100
= 40 %
Read More: Occurrence of Metals
Uncertainty in Measurement: As an Interval
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The most commonly used view of measuring uncertainty is by using random variables such as mathematical models for uncertain quantities and probability distributions. These two methods are sufficient for the representation of measuring uncertainties. However, a mathematical interval might be a better model of uncertainty measurement than a probability distribution in some situations. This may include situations involving periodic measurements, binned data values, censoring, detection limits, or plus-minus ranges of measurements where one cannot assume that the errors among individual measurements are completely independent or where no particular probability distribution seems justified.
Read More:Things to Remember
- Uncertainty in measurement is the range of possible values within which the true/real value of the measurement exists.
- Scientific notation displays the numbers which are very enormous or very tiny in a form of multiplication of single-digit integers and 10 raised to the power.
- The limitation of the measuring instrument is called systematic error
- Limitation of the skill of the experimenter doing the measurements is called random error.
- The percent uncertainty is given by Percent uncertainty = (Uncertainty/Actual Value) x 100.
- In certain circumstances, a mathematical interval may be a more accurate representation of uncertainty measurement than a probability distribution.
Previous Years’ Questions
- Complete combustion of 1.80g of oxygen… [JEE Main – 2021]
- A tertiary butyl carbocation is more stable than a secondary butyl… [NEET – 2020]
- The IUPAC name of the following compound… [JEE Main – 2019]
- Which of the following compounds will show the maximum enol content… [JEE Main – 2019]
- Which of the following compounds will be suitable for Kjeldahl's method… [JEE Main – 2018]
- Which of the following molecules is least resonance stabilized…[JEE Main – 2017]
- According to the IUPAC nomenclature, the correct increasing order… [JKCET – 2016]
- Which of the following compound has all the four types… [BITSAT – 2016]
- Consider the following compounds… [NEET – 2015]
- In the Kjeldahl's method for estimation of nitrogen… [NEET – 2014]
Sample Questions
Ques. State the main difference between Accuracy and Precision? (1 Mark)
Ans. Accuracy can be defined as the degree of closeness to the true value while Precision can be defined as the degree to which an instrument will repetitively produce the same value while doing an experiment.
Ques. What is the Difference between Systematic Error and Random Error? (1 Mark)
Ans. Random error takes place in an experiment because of the uncertain changes in the environment. Whereas a systematic error can be referred to as a constant error that remains the same for all the measurements.
Ques. What is the Absolute Uncertainty Formula? (1 Mark)
Ans. Absolute uncertainty is the uncertainty that we may get from the measurement. For example: If the height of the table is found to be 230 ± 5 mm, then absolute uncertainty is 5 mm.
Ques What is the difference between confidence interval and measurement uncertainty? (3 Marks)
Ans. Measurement uncertainty is defined as a range (also called the interval), where the true value of the measure lies in with some predefined probability around the measured value. This interval is known as coverage interval and measurement uncertainty is usually half its width. All possible effects that cause uncertainty, i.e. both random and systematic effects have to be taken into account in the Coverage interval.
Confidence interval typically refers to some statistical interval estimate. It is somewhat similar to coverage interval and expresses the level of confidence that the true value of a certain statistical parameter resides within the interval. The main difference is that only the mean value found from a limited number of replicates is considered in the confidence interval, not the true value, and only random effects are accounted for.
Ques. How do you write 0.00001 in scientific notation? (2 Marks)
Ans. The scientific notation for 0.0001 is 1 × 10(-4).
Here,
Coefficient = 1
Base = 10
Exponent = -4
Ques. The exact weight of an object is 2.50 kg. A student named David measured 2.46 kg, 2.49 kg, and 2.52 kg respectively. Solve. (3 Marks)
Ans. David’s measurements are 2.46 kg, 2.49 kg and 2.52 kg.
The average of these values is 2.49 kg. Considering the true value being 2.50 kg,
It is proven that the measurements are accurate, but not precise as 2.46 kg and 2.52 kg are not close values.
Ques. Calculate the number of seconds in 3 days. (5 Marks)
Ans. Known: 1 day = 24 hours.
From this equivalence, write 1 day / 24 hours = 1 = 24 hours / 1 day.
This means that both these are unit factors and are here considered equal to 1.
Similarly, find the equivalence from hours to seconds.
Therefore,
3 days = (3 days × (24 hours / 1 day) × (60 min / 1 hour) × (60 s / 1 min)) seconds
3 days = (3 × 24 × 60 × 60) seconds
3 days = 259200 seconds
3 days = 2.592 × 105 seconds.
Thus, 3 days have 2.592 × 105 seconds.
Ques. What are the 5 rules of scientific notation? (5 Marks)
Ans. The five rules of scientific notation are given below:
- The base must always be 10.
- Exponents that are non-zero integers must be either positive or negative in order to be used.
- The coefficient's absolute value is more than or equal to 1, but it must be less than 10.
- Positive and negative numbers, including whole and decimal values, can be coefficients.
- The remaining significant digits of the number are represented by the mantissa.
Ques. Mention the three parts of Scientific Notation (1 Mark)
Ans. The three main parts of a scientific notation are coefficient, base and exponent.
Ques. Multiply 4.3545 × 1.9. Calculate the answer in terms of significant figures. (3 Marks)
Ans. Given: 4.3545 × 1.9 = 8.27355.
However, when considered in terms of significant figures, in these operations, the result must be reported with no more significant figures as in the measurement with the few significant figures.
Thus, one can have maximum only two significant figures as 1.9 has only two significant figures.
Therefore, 4.3545 × 1.9 = 8.2, where 8.2 has only 2 significant figures.
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