1.4 Uncertainty in Measurement

Uncertainty In Measurement (Precision and Accuracy)

Very large or very small numbers, having many zeros can be expressed by using scientific notation for such numbers i.e., exponential notation in which any number can be represented in the form \(\mathrm{N} \times 10^{\mathrm{n}}\) where \(\mathrm{n}\) is an exponent having +ve or -ve value and \(\mathrm{N}\) can vary between 1 to 10.
Every experimental measurement has some amount of uncertainty associated with it. However, one would always like the results to be precise and accurate. Precision refers to the closeness of various measurements for the same quantity while accuracy is the agreement of a particular value to the true value of the result.

Significant Figures

The uncertainty in experimental or calculated values is indicated by mentioning the number of significant figures. Significant figures are meaningful digits that are known with certainty. The uncertainty is indicated by writing the certain digits and the last uncertain digit.

The rules for determining the number of significant figures are:

1. All non-zero digits are significant. For ex: in \(285 \mathrm{~cm}\), there are 3 significant figures.
2. Zeros preceding to first non-zero digit are not significant. Such zero indicates the position of the decimal point. For ex: \(0.03\) has one significant figure.
3. Zeros between two non-zero digits are significant. For ex: \(2.005\) has four significant figures.
4. Zeros at the end or right are significant provided they are on the right side of the decimal point. For ex: \(0.200 \mathrm{~g}\) has 3 significant figure.
5. If a number ends in zeros that are not to right of a decimal the zeros may or may not be significant. For e.g., 3500 may have two, three or five significant figures.
6. Counting no. of objects have infinite significant figures.
7. In numbers written in scientific notation, all digits are significant.

In numbers written in scientific notation, all digits are significant e.g., \(4.01 \times 10^{2}\) has three significant figures, and \(8.256 \times 10^{-3}\) has four significant figures. However, one would always like the results to be precise and accurate. Precision and accuracy are often referred to while we talk about the measurement.

Precision refers to the closeness of various measurements for the same quantity. However, accuracy is the agreement of a particular value to the true value of the result. 

Dimensional Analysis

Often while calculating, there is a need to convert units from one system to the other. The method used to accomplish this is called the factor label method or unit factor method or dimensional analysis. This is illustrated below with examples.

Example: A piece of metal is 3 inch (represented by in) long. What is its length in cm?

Answer:

We know that \(1 \mathrm{~in}=2.54 \mathrm{~cm}\) From this equivalence, we can write
\(
\frac{1 \mathrm{~in}}{2.54 \mathrm{~cm}}=1=\frac{2.54 \mathrm{~cm}}{1 \mathrm{in}}
\)
\(
\text { Thus, } \frac{1 \mathrm{in}}{2.54 \mathrm{~cm}} \text { equals } 1 \text { and } \frac{2.54 \mathrm{~cm}}{1 \mathrm{~in}}
\)
also equals 1 . Both of these are called unit factors. If some number is multiplied by these unit factors (i.e., 1), it will not be affected otherwise. Say, the 3 in given above is multiplied by the unit factor. So,
\(
3 \mathrm{~in}=3 \mathrm{~in} \times \frac{2.54 \mathrm{~cm}}{1 \mathrm{in}}=3 \times 2.54 \mathrm{~cm}=7.62 \mathrm{~cm}
\)
Now, the unit factor by which multiplication is to be done is that unit factor \(\left(\frac{2.54 \mathrm{~cm}}{1 \mathrm{ln}}\right.\) in the above case) which gives the desired units i.e., the numerator should have that part which is required in the desired result.

Example:
A jug contains \(2 \mathrm{~L}\) of milk. Calculate the volume of the milk in \(\mathrm{m}^{3}\).
Answer:
Since \(1 \mathrm{~L}=1000 \mathrm{~cm}^{3}\)
and \(1 \mathrm{~m}=100 \mathrm{~cm}\), which gives
\(
\frac{1 \mathrm{~m}}{100 \mathrm{~cm}}=1=\frac{100 \mathrm{~cm}}{1 \mathrm{~m}}
\)
To get \(\mathrm{m}^{3}\) from the above unit factors, the first unit factor is taken and it is cubed.
\(
\left(\frac{1 \mathrm{~m}}{100 \mathrm{~cm}}\right)^{3} \Rightarrow \frac{1 \mathrm{~m}^{3}}{10^{6} \mathrm{~cm}^{3}}=(1)^{3}=1
\)
Now \(2 \mathrm{~L}=2 \times 1000 \mathrm{~cm}^{3}\)
The above is multiplied by the unit factor
\(
2 \times 1000 \mathrm{~cm}^{3} \times \frac{1 \mathrm{~m}^{3}}{10^{6} \mathrm{~cm}^{3}}=\frac{2 \mathrm{~m}^{3}}{10^{3}}=2 \times 10^{-3} \mathrm{~m}^{3}
\)

Example: How many seconds are there in 2 days?

Answer: Here, we know 1 day \(=24\) hours \((\mathrm{h})\)
\(
\text { or } \frac{1 \text { day }}{24 \mathrm{~h}}=1=\frac{24 \mathrm{~h}}{1 \text { day }}
\)
then, \(1 \mathrm{~h}=60 \mathrm{~min}\)
\(
\text { or } \frac{1 \mathrm{~h}}{60 \mathrm{min}}=1=\frac{60 \mathrm{~min}}{1 \mathrm{~h}}
\)
so, for converting 2 days to seconds, The unit factors can be multiplied in series in one step only as follows:
\(
\begin{aligned}
& 2 \text { day } \times \frac{24 \mathrm{~h}}{1 \text { day }} \times \frac{60 \mathrm{~min}}{1 \mathrm{~h}} \times \frac{60 \mathrm{~s}}{1 \mathrm{~min}} \\
& =2 \times 24 \times 60 \times 60 \mathrm{~s} \\
& =172800 \mathrm{~s}
\end{aligned}
\)