Uncertainty in Measurement

All the measurements have a certain degree of uncertainty regardless of their precision and accuracy. This is because of two factors, the limitation of the measuring instrument (called systematic error) and the skill of the experimenter doing the measurements (called random error). Too often, we come across values that are very close to each other and their average values. In such cases, we can say that the measurement is correct and precise. Specifying the uncertainty formula chemistry is important as it will help us to study the overall effect on output. 

Scientific Notation

• As we are aware of the fact that atoms and molecules have extremely low masses, but they are present in large numbers.

• Chemists deal with figures which are as small as 0.00000000000000000000000166 g (Mass of Hydrogen atoms) and other constants that have very large values, like Avagadros number, Planck’s constant, Speed of light, Charge of particles, etc. Now the question arises how to handle such small and large numbers?

• To handle these large or small numbers, we use the following notation: x x 10\[^{y}\], which is, x times ten raised to the power of y. In this expression, y is an exponent having positive or negative values and x is that number that can vary from 1.000… and 9.999…

• In the same way, scientific notation 823.912 can be written as 8.23912 x 10\[^{2}\]. In this, the decimal is moved to the left side by two places and if it is moved three places to the left then the power of 10 will be 3. In the same manner, 0.00065 can be written as 6.5 x 10\[^{-4}\]. In this, the decimal is moved four places towards the right, so,  −4 is the exponent in the scientific notation.

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Uncertainty Calculation

Uncertainty in Multiplication and Division:

Applying the same rule as discussed above we can solve the given problem as:

(4.3 x 10\[^{7}\]) x (2.7 x 10\[^{3}\]) = (4.3 x 2.7)(10\[^{7+3}\]) 

= (4.3 x 2.7) (10\[^{10}\])

= 11.6 x 10\[^{10}\]

Similarly for division,

\[\frac{4.9 \times 10^{-4}}{3.2 \times 10^{-6}}\] = (4.9 ÷ 3.2)(10\[^{-4-(-6)}\]) = 1.531 x 10\[^{2}\]

Uncertainty in Addition and Subtraction:

While doing addition or subtraction first of all we have to place these numbers in such a way that they have the same exponents. Eg: If there are two numbers 7.32 x 10\[^{3}\] and 9.55 x 10\[^{2}\] 

Now adding both 7.32 x 10\[^{3}\] + 9.55 x 10\[^{2}\] = (7.32 + (0.955 x 10)) x 10\[^{3}\] = 8.275 x 10\[^{3}\]  

Similarly for subtraction, 

7.32 x 10\[^{3}\] - 9.55 x 10\[^{2}\] = (7.32 - (0.955 x 10))10\[^{3}\] = 6.365 x \[^{3}\] 

Percentage Uncertainty Formula:

The uncertainty of a measured value can be represented in a percentage notation or as a simple ratio. It is calculated as:

percent uncertainty = \[\frac{Uncertainity}{\text{Actual value}}\] x 100

Solved Examples

Example: A scale measures wrongly a value as 6 cm due to some marginal errors. If the actual measurement of the value is 10 cm then calculate the percentage 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 %