Question

If each observation of the data is increased by 5, then their mean ______.

Options

• remains the same

• becomes 5 times the original mean

• is decreased by 5

• is increased by 5

Answer 1

If each observation of the data is increased by 5, then their mean is increased by 5.

Explanation:

Let x₁, x₂, ..., x_n be the n observations.

Then, old mean `barx_("old") = (sum_(i = 1)^n x_i)/n`  ...(i)

Now, adding 5 in each observation, the new mean becomes

`barx_("new") = ((x_1 + 5) + (x_2 + 5) + ... + (x_n + 5))/n`

⇒ `barx_("new") = ((x_1 + x_2 + ... + x_n) + 5n)/n`

⇒ `barx_("new") = (sum_(i = 1)^n x_i)/n + 5 = barx_("old") + 5`  ...[From equation (i)]

⇒ `barx_("new") = barx_("old") + 5`

Hence, the new mean is increased by 5.