Question

Standard deviation of a set of 50

observation is 6.5. If value of each

observation is increased by 5, then the

standard deviation is​

Answer 1

Standard deviation of a set of 50

observation is 6.5. If value of each

observation is increased by 5, then the

standard deviation is 6.5

Answer 2

Find the standard deviation for the given data.

Explanation:

• The variance of a data set is square of its standard deviation, $\sigma=6.5\ \ \ \ \ \ \ \ \ ->\sigma^2= 42.25$    
• The variance of a data set can be expressed in terms of the expectation value as we know that E(x) is the mean($\bar x$) of the data,                  $\sigma^2= E((x-E(x))^2) \\\sigma^2=E(x^2-2xE(x)+E(x)^2)\\\sigma^2=E(x^2)-2E(x)E(x)+E(E(x^2))\\\sigma^2=E(x^2)-E(x)^2\\\sigma^2=E(x^2)-\bar x^2\\-> E(x^2)-\bar x^2=42.25$----(a)
• Also we have that Expectation of a constant is the constant itself.
• Also while increasing each value of the data set by five the mean of the data remains the same.
• Hence, from (a) we get the new variance as,  
• $\sigma'^2=E((x+5)^2)-E(x+5)^2\\\sigma'^2=E(x^2+10x+25)-E(x)^2-2E(x)E(5)-E(5)^2\\\sigma'^2=E(x^2)+10E(x)+E(25)-\bar x^2-10\bar x-25\\\sigma'^2=E(x^2)-\bar x^2 \\\sigma'^2=\sigma ^2\ \ \ \ \ \ \ \ \ \ ->\sigma'=\sigma=6.5$              
• The increase in each value does not affect the standard deviation.
• Hence, the standard deviation will be the same as before, that is $6.5$