Relative Standard Deviation Formula
The relative standard deviation (RSD) is a special type of standard deviation (SD). The relative standard deviation formula helps us understand whether the standard deviation is small or large when compared to the mean for the set of data. For example, if the standard deviation is 0.1 and the mean is 3.5, the RSD for this set of numbers is 100 x 0.1 / 3.5 = 2.86%.
The standard deviation is 2.86% of the mean of 3.5 is very small. In other words, the data points lie close to the mean. On the other hand, if the relative standard deviation percentage was large, say, 56%, this would mean that the data is more spread out. Let us learn the relative standard deviation formula in more detail.
What is the Relative Standard Deviation Formula?
The relative standard deviation gives an idea about the precision of the data in an experiment. The more precise the data, the smaller the is the RSD. The RSD is generally written with the mean and a plus/minus symbol: 3.5 ± 2.86%. The relative standard deviation formula is given as:
Relative standard deviation = 100 × s / x̄
where,
 s = the sample standard deviation
 x̄ = sample mean
Let us see how to use the relative standard deviation formula in the following solved examples section.
Solved Examples Relative Standard Deviation Formula

Example 1: Following is the data of scored marks obtained by 4 students in the math examination: 60, 98, 65, 85. Use the relative standard deviation formula to find RSD.
Solution:
Formula of the mean is given by:
\(\bar{x}=\frac{\sum x}{n}\\ \bar{x}=\frac{50+88+55+75}{4}=67\)
Calculation of standard deviation:
Formula for standard deviation:
\(\mathrm{S}=s=\sqrt{\frac{\sum\left(x\bar{x}^{2}\right)}{n1}}\\ \mathrm{S}=\sqrt{\frac{938}{3}}\\ S=17.66\)
Relative standard deviation \(=\frac{s \times 100}{\bar{x}}\)
\(\frac{17.66 \times 100}{77}\)
\(=22.93 \%\)Answer: The RSD is 22.93%

Example 2: 4 measurements were collected as a sample 51,55,49 and 52. Calculate the relative standard deviation.
Solution:
\(\begin{array}{l}
\text { average, } \bar{x}=\frac{51+55+49+52}{4}=\frac{207}{4}=51.7 \\
\text { standard deviation, } S=\sqrt{\frac{(5151.7)^{2}+(5551.7)^{2}+(4951.7)^{2}+(5251.7)^{2}}{41}} \\
=\sqrt{\frac{(0.7)^{2}+(3.3)^{2}+(2.7)^{2}+(0.3)^{2}}{3}} \\
=\sqrt{\frac{0.49+10.89+7.29+0.09}{3}} \\
=\sqrt{6.25} \\
=2.5
\end{array}\)
Relative standard deviation, \(\mathrm{RSD}=100 \mathrm{~S} / \overline{\mathrm{x}}=\frac{2.5}{51.7} \times 100=4.86 \%\)
This can be represented as \(51.7 \pm 2.5\) or \(51.7 \pm 4.86 \%\)Answer: The standard deviation is \(51.7 \pm 4.86 \%\)
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