Explain any three main properties of arithmetic mean
Answers
Property 1:
If x is the arithmetic mean of n observations x1, x2, x3, . . xn; then
(x1 - x) + (x2 - x) + (x3 - x) + ... + (xn - x) = 0.
Now we will proof the Property 1:
We know that
x = (x1 + x2 + x3 + . . . + xn)/n
⇒ (x1 + x2 + x3 + . . . + xn) = nx. ………………….. (A)
Therefore, (x1 - x) + (x2 - x) + (x3 - x) + ... + (xn - x)
= (x1 + x2 + x3 + . . . + xn) - nx
= (nx - nx), [using (A)].
= 0.
Hence, (x1 - x) + (x2 - x) + (x3 - x) + ... + (xn - x) = 0.
Property 2:
The mean of n observations x1, x2, . . ., xn is x. If each observation is increased by p, the mean of the new observations is (x + p).
Now we will proof the Property 2:
x = (x1 + x2 +. . . + xn)/n
⇒ x1 + x2 + . . . + xn) = nx …………. (A)
Mean of (x1 + p), (x2 + p), ..., (xn + p)
= {(x1 + p) + (x2 + p) + ... + (x1 + p)}/n
= {(x1 + x2 + …… + xn) + np}/n
= (nx + np)/n, [using (A)].
= {n(x + p)}/n
= (x + p).
Hence, the mean of the new observations is (x + p).
Property 3:
The mean of n observations x1, x2, . . ., xn is x. If each observation is decreased by p, the mean of the new observations is (x - p).
Now we will proof the Property 3:
x = (x1 + x2 +. . . + xn)/n
⇒ x1 + x2 + . . . + xn) = nx …………. (A)
Mean of (x1 - p), (x2 - p), ...., (xn - p)
= {(x1 - p) + (x2 - p) + ... + (x1 - p)}/n
= {(x1 + x2 + …. + xn) - np}/n
= (nx - np)/n, [using (A)].
= {n(x - p)}/n
= (x - p).
Hence, the mean of the new observations is (x + p).