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,the number of observations is $n = 10$.
The mean $\bar{x}$ is calculated as:
$\bar{x} = \frac{\sum_{i=1}^{10} x_i}{10} = \frac{3+6+9+12+15+18+21+24+27+30}{10} = \frac{165}{10} = 16.5$.
The following table shows the calculation for variance:
The sum of squares $\sum (x_i - \bar{x})^2 = 742.5$.
Variance $(\sigma^2) = \frac{1}{n} \sum (x_i - \bar{x})^2 = \frac{742.5}{10} = 74.25$.
Here,the number of observations is $n = 10$.
The mean $\bar{x}$ is calculated as:
$\bar{x} = \frac{\sum_{i=1}^{10} x_i}{10} = \frac{3+6+9+12+15+18+21+24+27+30}{10} = \frac{165}{10} = 16.5$.
The following table shows the calculation for variance:
| $x_i$ | $(x_i - \bar{x})^2$ |
| $3$ | $182.25$ |
| $6$ | $110.25$ |
| $9$ | $56.25$ |
| $12$ | $20.25$ |
| $15$ | $2.25$ |
| $18$ | $2.25$ |
| $21$ | $20.25$ |
| $24$ | $56.25$ |
| $27$ | $110.25$ |
| $30$ | $182.25$ |
The sum of squares $\sum (x_i - \bar{x})^2 = 742.5$.
Variance $(\sigma^2) = \frac{1}{n} \sum (x_i - \bar{x})^2 = \frac{742.5}{10} = 74.25$.
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Similar Questions
Statement $1$: The variance of the first $n$ odd natural numbers is $\frac{n^2 - 1}{3}$.
Statement $2$: The sum of the first $n$ odd natural numbers is $n^2$ and the sum of the squares of the first $n$ odd natural numbers is $\frac{n(4n^2 - 1)}{3}$.
Statement $2$: The sum of the first $n$ odd natural numbers is $n^2$ and the sum of the squares of the first $n$ odd natural numbers is $\frac{n(4n^2 - 1)}{3}$.
DifficultAIEEE 2012
View SolutionFor the frequency distribution:
Variate $(x)$ $x_{1}$ $x_{2}$ $x_{3} \ldots x_{15}$ Frequency $(f)$ $f_{1}$ $f_{2}$ $f_{3} \ldots f_{15}$
where $0 < x_{1} < x_{2} < x_{3} < \ldots < x_{15} = 10$ and $\sum_{i=1}^{15} f_{i} > 0$,the standard deviation cannot be:
| Variate $(x)$ | $x_{1}$ | $x_{2}$ | $x_{3} \ldots x_{15}$ |
| Frequency $(f)$ | $f_{1}$ | $f_{2}$ | $f_{3} \ldots f_{15}$ |
where $0 < x_{1} < x_{2} < x_{3} < \ldots < x_{15} = 10$ and $\sum_{i=1}^{15} f_{i} > 0$,the standard deviation cannot be:
Find the variance and standard deviation for the following data:
$x_i$ $4$ $8$ $11$ $17$ $20$ $24$ $32$ $f_i$ $3$ $5$ $9$ $5$ $4$ $3$ $1$
| $x_i$ | $4$ | $8$ | $11$ | $17$ | $20$ | $24$ | $32$ |
| $f_i$ | $3$ | $5$ | $9$ | $5$ | $4$ | $3$ | $1$ |
Medium
View SolutionThe variance of $20$ observations is $5$. If each observation is multiplied by $2$,find the new variance of the resulting observations.
Difficult
View SolutionIf the mean of $10$ observations is $50$ and the sum of the squares of the deviations of the observations from the mean is $250$,then the coefficient of variation of those observations is
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