Explanation:

When each of n observations having mean x is divided by a non-zero constant k, the new mean will be obtained by dividing the sum of new observations by the total number of new observations.

Formula:

Let a1, a2, …, an be n observations with mean x. If each observation is divided by a non-zero constant k, then the new observations are:

b1 = a1 / k, b2 = a2 / k, …, bn = an / k

The new mean can be calculated as:

New Mean = (b1 + b2 + … + bn) / n

Substituting values of b1, b2, …, bn, we get:

New Mean = [(a1 / k) + (a2 / k) + … + (an / k)] / n

New Mean = (1 / k) * [(a1 + a2 + … + an) / n]

New Mean = (1 / k) * x

Therefore, the new mean is x/k.

Example:

Let us consider an example where we have 5 observations with mean 10. If each observation is divided by 2, then the new mean will be:

a1 = 8, a2 = 9, a3 = 10, a4 = 11, a5 = 12

n = 5

x = (a1 + a2 + a3 + a4 + a5) / n = 50 / 5 = 10

k = 2

b1 = a1 / k = 8 / 2 = 4, b2 = a2 / k = 9 / 2 = 4.5, b3 = a3 / k = 10 / 2 = 5, b4 = a4 / k = 11 / 2 = 5.5, b5 = a5 / k = 12 / 2 = 6

New Mean = (b1 + b2 + … + bn) / n

New Mean = (4 + 4.5 + 5 + 5.5 + 6) / 5

New Mean = 25 / 5

New Mean = 5

Therefore, the new mean is 5.