Find the variance of the following data: $6,8,10,12,14,16,18,20,22,24$
From the given data we can form the following Table The mean is calculated by step-deviation method taking $14$ as assumed mean. The number of observations is $n=10$
${x_i}$ | ${d_i} = \frac{{{x_i} - 14}}{2}$ |
Deviations orom mean $\left( {{x_i} - \bar x} \right)$ |
$\left( {{x_i} - \bar x} \right)$ |
$6$ | $-4$ | $-9$ | $81$ |
$8$ | $-3$ | $-7$ | $49$ |
$10$ | $-2$ | $-5$ | $25$ |
$12$ | $-1$ | $-3$ | $9$ |
$14$ | $0$ | $-1$ | $1$ |
$16$ | $1$ | $1$ | $1$ |
$18$ | $2$ | $3$ | $9$ |
$20$ | $3$ | $5$ | $25$ |
$22$ | $4$ | $7$ | $49$ |
$24$ | $5$ | $9$ | $81$ |
$5$ | $330$ |
Therefore $Mean\,\,\bar x = $ assumed mean $ + \frac{{\sum\limits_{i = 1}^n {{d_i}} }}{n} \times h$
$ = 14 + \frac{5}{{10}} \times 2 = 15$
and Veriance $\left( {{\sigma ^2}} \right) = \frac{1}{n}\sum\limits_{i = 1}^{10} {{{\left( {{x_i} - \bar x} \right)}^2} = \frac{1}{{10}} \times 330 = 33} $
Thus Standard deviation $\left( \sigma \right) = \sqrt {33} = 5.74$
If the standard deviation of the numbers $ 2,3,a $ and $11$ is $3.5$ then which of the following is true ?
One set containing five numbers has mean $8$ and variance $18$ and the second set containing $3$ numbers has mean $8$ and variance $24$. Then the variance of the combined set of numbers is
Let $n \geq 3$. A list of numbers $x_1, x, \ldots, x_n$ has mean $\mu$ and standard deviation $\sigma$. A new list of numbers $y_1, y_2, \ldots, y_n$ is made as follows $y_1=\frac{x_1+x_2}{2}, y_2=\frac{x_1+x_2}{2}$ and $y_j=x_j$ for $j=3,4, \ldots, n$.
The mean and the standard deviation of the new list are $\hat{\mu}$ and $\hat{\sigma}$. Then, which of the following is necessarily true?
The mean and standard deviation of $20$ observations are found to be $10$ and $2$ respectively. On rechecking, it was found that an observation $8$ was incorrect. Calculate the correct mean and standard deviation in each of the following cases:
If it is replaced by $12$
Given that $\bar{x}$ is the mean and $\sigma^{2}$ is the variance of $n$ observations $x_{1}, x_{2}, \ldots, x_{n}$ Prove that the mean and variance of the observations $a x_{1}, a x_{2}, a x_{3}, \ldots ., a x_{n}$ are $a \bar{x}$ and $a^{2} \sigma^{2},$ respectively, $(a \neq 0)$