Let the data set be : Di , for i = 1, to 10, ie., D₁, D₂, ... D₁₀
mean μ of N observations = 10.
So total of the N observations = 10 N
The new data set contains : 5 Di + 1, ie., 5 D₁+1, 5 D₂+2,..., 5 D_N+1
Sum of the new data set = 5 (D₁ + D₂ + D₃ ... D_N ) + (1+1+...N times)
= 5 * 10 N + N
= 51 N
So mean of the new data set = 51 N / N = 51
If each data item is multiplied by x then the mean is also multiplied by x.
If each data item is added by x then the mean is also added by x.
If each item is multiplied by a and b is added to each item, then the mean becomes a times plus b.