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Arithmetic Mean

☼ Arithmetic Mean : The arithmetic mean of n numbers $x_{1}, x_{2},x_{3},...,x_{n}$ is denoted by and is calculated as,

$\overline{x}$ = $\frac{x_{1}, x_{2},x_{3},...,x_{n}}{n}$ = $\frac{Sum-of-observation}{Number-of-observation}$

e.g. Find the average of 12, 20, 28.

Soln. $\overline{x}$ = $\frac{12+20+28}{3}$ = 20.

If the distances of given observations are equal ( or they are in A. P. ), then

Average ( or AM ) = $\frac{\textup{First Observation} + \textup{Last observation}}{2}$

Average ( or AM ) = Middle Observation

e.g. (i) Find the average of first 10 multiples of 15.

Soln. First multiple = 15 and 10^th multiple = 150.

( Also first 10 multiples of 15 are in A.P. )

$\therefore \textup{ } \overline{x}$ = $\frac{15+150}{2}$ = $\frac{165}{2}$ = 82.5

(ii) Find the average of 2, 5, 8, 11, 14, 17, 20

Soln. Here, the given nos. are in A.P. and in ascending order.

$\therefore \textup{ } \overline{x}$ = Middle Observation

= 11