Question

The mean of the values 1, 2, 3, …….n with respective frequencies x, 2x, 3x …… nx is:

Answer 1

if $x_i$ denotes the observation and $f_i$ denotes respective frequency of observation. then,
mean, $\bar{x}=\frac{\Sigma f_ix_i}{\Sigma f_i}$

so, $\Sigma f_ix_i$ = 1 × x + 2 × 2x + 3 × 3x + ..... n × nx
= 1²x + 2²x + 3²x + 4²x +...... + n²x
= x(1² + 2² + 3² + 4² + .... n²)
= xn(n + 1)(2n + 1)/6

and $\Sigma f_i$ = x + 2x + 3x + 4x + ... + nx
= xn(n + 1)/2

now, mean ={ nx(n + 1)(2n + 1)/6}/{nx(n + 1)/2}
= (2n + 1)/3

Answer 2

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