Find the mean and variance for the first $10$ multiples of $3$
Find the mean and variance for the first $10$ multiples of $3$
The first $10$ multiples of $3$ are
$3,6,9,12,15,18,21,24,27,30$
Here, number of observations, $n=10$
Mean, $\bar x = \frac{{\sum\limits_{i = 1}^{10} {{x_i}} }}{{10}} = \frac{{165}}{{10}} = 16.5$
The following table is obtained.
| ${x_i}$ | $\left( {{x_i} - \bar x} \right)$ | ${\left( {{x_i} - \bar x} \right)^2}$ |
| $3$ | $-13.5$ | $182.25$ |
| $6$ | $-10.5$ | $110.25$ |
| $9$ | $-7.5$ | $56.25$ |
| $12$ | $-4.5$ | $20.25$ |
| $15$ | $-1.5$ | $2.25$ |
| $18$ | $1.5$ | $2.25$ |
| $21$ | $4.5$ | $20.25$ |
| $24$ | $7.5$ | $56.25$ |
| $27$ | $10.5$ | $110.25$ |
| $30$ | $13.5$ | $182.25$ |
Variance $\left( {{\sigma ^2}} \right) = \frac{1}{n}\sum\limits_{i = 1}^{10} {{{\left( {{x_1} - \bar x} \right)}^2} = } \frac{1}{{10}} \times 742.5 = 74.25$
$742.5$
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