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The variate \[x\] and \[u\] are related by \[u = \dfrac{{x - a}}{h}\], then correct relation between \[{\sigma _x}\] and \[{\sigma _u}\], where \[{\sigma _x}\] and \[{\sigma _u}\] are the standard deviation of \[x\] and \[u\] respectively?

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Last updated date: 24th Jul 2024
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Answer
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Hint: Here we are given two variates \[x\] and \[u\], and a relation between both the vartates is given. We are asked to find the relation between the standard deviations of \[x\] and \[u\]. We do this by using the fact that standard deviation is not dependent upon change of origin, but on change of scale. Using this we can find the relation between \[{\sigma _x}\] and\[{\sigma _u}\].

Complete step-by-step solution:
We have two variates \[x\] and \[u\], and the relation between \[x\] and \[u\] is given as,
\[\begin{gathered}
  u = \dfrac{{x - a}}{h} \\
   \Rightarrow u = \dfrac{x}{h} - \dfrac{a}{h} \\
\end{gathered} \]
There is a shift of origin of \[a\] and shift of scale of \[h\] for the new variate \[u\] from \[x\].
Since we know that standard deviation is dependent on change of scale but not on change of origin, we get the relation between \[{\sigma _x}\] and \[{\sigma _u}\]as,
\[
   \Rightarrow {\sigma _u} = \dfrac{{{\sigma _x}}}{h} \\
   \Rightarrow {\sigma _x} = h{\sigma _u} \\
 \]
Thus we have got the relation between \[{\sigma _x}\] and \[{\sigma _u}\] as,
\[{\sigma _x} = h{\sigma _u}\]


Note: Whenever we have a change in the origin of any given data, i.e. we increase or decrease each value of a data, there is no effect in the standard deviation of the data, but when there is shift in scale of the data, i.e. all the data is divided or multiplied by any value, the standard deviation is also changed. Standard deviation is the measure of the value by which all the values of the data differ from the mean of the same data.