The mean and standard deviation of six observations are and , respectively. If each observation is multiplied by , find the new mean and new standard deviation of the resulting observations.

We are given that,

The mean of six observations .

The standard deviation of six observations .

Let us consider the six observations as , , ,  , and .

The mean of six observations is given by,

When each observation is multiplied by then,

   

So,

   

, for to       …… (1)

Now, the new mean becomes,

Also, we have Standard deviation,

Substituting the values and squaring on both sides,

                       …… (2)

From equations (1) and (2),

So, the new variance of the resulting observation

Hence, the new standard deviation of the resulting observation