Uncertainty In Measurement: Definition, Formula and Examples

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.
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.

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This Story also Contains

1. Uncertainty in the Measurements
2. Some Solved Examples
3. Summary

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

Recommended topic video on(Uncertainty In Measurement)

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.