Consider three set of data having same mean and MD but their ranges are changing. It is interesting to see how SD changes with change in the range of the data.
SET 1: 1, 3,5,7,9,11,13,15,17,19 Range:1-19 Mean=10, MD=5 SD= 6.05
SET 2: 2,3,5,7,7,9,13,15,14,23 Range: 1-23 Mean=10 MD=5 SD=6.28
SET 3: 3,5,5,7,7,8,10,12,13,30 Range: 1-30 Mean =10 MD=5 SD=7.70
It can be observed that all the three sets have same mean and MD. It is to be highlighted that while MD do not change with change in range, SD show changes with every change in ranges. This clearly establishes the supremacy of SD as compared to MD in dealing with variation in the data.
Relative standard deviation is also called percentage relative standard deviation formula, is the deviation measurement that tells us how the different numbers in a particular data set are scattered around the mean. This formula shows the spread of data in percentage.
If the product comes to a higher relative standard deviation, that means the numbers are very widely spread from its mean.
If the product comes lower, then the numbers are closer than its average. It is also knows as the coefficient of variation.
The formula for the same is given as:
\[\large RSD=\frac{s\times 100}{\overline{x}}\]
Where,
RSD = Relative standard deviation
s = Standard deviation
\[\begin{array}{l}\overline{x}\end{array} \]
= Mean of the data.
Solved Examples
Question 1: Following are the marks obtained in by 4 students in mathematics examination: 60, 98, 65, 85. Calculate the relative standard deviation ?
Solution:
Formula of the mean is given by:
\[\begin{array}{l}\overline{x}\end{array} \]
=
\[\begin{array}{l}\frac{\sum x}{n}\end{array} \]
\[\begin{array}{l}\overline{x}\end{array} \]
=
\[\begin{array}{l}\frac{60+ 98+ 65+ 85}{4}=77\end{array} \]
Calculation of standard deviation:
\[\begin{array}{l}x\end{array} \] | \[\begin{array}{l}x-\overline{x}\end{array} \] | \[\begin{array}{l}\left[x-\overline{x}\right]^{2}\end{array} \] |
60 | -17 | 289 |
98 | 21 | 441 |
65 | -12 | 144 |
85 | 8 | 64 |
\[\begin{array}{l}\sum \left[x-\overline{x}\right]^{2}=938\end{array} \] |
Formula for standard deviation:
S =
\[\begin{array}{l}s=\sqrt{\frac{\sum \left[x-\overline{x}^{2}\right]}{n-1}}\end{array} \]
S =
\[\begin{array}{l}\sqrt{\frac{938}{3}}\end{array} \]
S = 17.66
Relative standard deviation =
\[\begin{array}{l}\frac{s\times 100}{\overline{x}}\end{array} \]
=
\[\begin{array}{l}\frac{17.66\times 100}{77}\end{array} \]
= 22.93%
The relative standard deviation is the ratio of the standard deviation to the absolute value of the mean of the given data. The relative standard deviation [R.S.D] is a measure of the dispersion of the data, that is, it tells us how much the data is spread apart. The relative standard deviation tells us how much larger the standard deviation of our data is, as compared to the mean of the data.
The formula for relative standard deviation is given as,
Relative Standard Deviation R.S.D = σ/|µ| where,
µ is the mean of the data,
and, σ is the standard deviation of the data.
Relative Standard Deviation [R.S.D] vs Coefficient of Variation [C.V]:
The relative standard deviation [R.S.D] is similar to the coefficient of variation of the data except for the fact that to calculate R.S.D we have to divide by the absolute value of the mean. Hence the relative standard deviation will always be positive irrespective of whether the mean is positive or negative.
Disadvantages of Relative Standard Deviation:
Note that it is not possible to calculate relative standard deviation if the mean is equal to 0 as we cannot divide by 0. If the mean is very close to zero then the value of R.S.D will be very large and in such cases it is much better to calculate other measures of dispersion to know about the spread of the data
How to calculate relative standard deviation?
- Calculate the mean of the data using the formula x̄ = ∑xi /n where xi denotes the sample values and n denotes the size of the sample.
- Calculate the variance of the data using the formula, Variance=∑[xi– x̄]2/n
- Take the square root of the variance to find the standard deviation.
- Find the relative standard deviation using the formula, R.S.D = σ/|µ|. Notice that we divide by the absolute value of the mean.
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