The standard deviation of a data is 6, when each observation is increased by 1, then the S.D. of the new data is

We know that X and Y are two variables such that  Y = X + b, where b is a constant.

Then, $\mathrm{Var}\left(Y\right)=\mathrm{Var}\left(X\right)\Rightarrow {\sigma }_X={\sigma }_Y$

Thus, the standard deviation of the new data is also 6