Statistics - Algebra - Mathematics Previous Year Questions

The mean deviation from the mean of the discrete data $1,3,4,7,11,18,29,47,78$ is

If $\bar{x}$ is the mean of $n$ observations $x_1, x_2, \ldots ., x_n$ then the mean of the absolute deviations of these observations from $\bar{x}$ is

Assertion (A) The variance of the first $n$ odd natural numbers is $\frac{n^2-1}{3}$.

Reason (R) 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 $\frac{n\left(4 n^2-1\right)}{3}$.

Which of the following alternatives is correct?

If $M$ and $\sigma^2$ represent respectively the mean deviation from the mean and the variance for the data $1,3,5,7$, $11,13,17,19,23$, then $3\left(\sigma^2-M\right)=$

If $X$ is a Poisson variate satisfying the condition $3 P(X=2)=P(X=4)$, then $P(X=6)=$

If the variance of the data $2,3,5,8,12$ is $\sigma^2$ and the mean deviation from the median for this data is $M$, then $\sigma^2-M=$

The mean and standard deviation of 100 observations were calculated as 40 and 5.1 respectively. Later on it was found that one of the observations was taken as 50 in the place of 40 . If the wrong entry is replaced by the correct one, then the sum of the squares of all the observations is

The variance of 50 observations is 7 . Suppose that each observation in this data is multiplied by 6 and then 5 is subtracted from it. Then, the variance of that new data is

If $\alpha, \beta$ are respectively the mean deviation about the mean and variance of the first five prime numbers, then the ordered pair ( $\alpha, \beta$ )

Assertion (A) Variance of $4 x_1, 4 x_2, \ldots, 4 x_n$ is 16 times the variance of $x_1, x_2, x_3, \ldots, x_n$

Reason (R) If $y=a x+b$, then variance of $y$ is a $($ variance of $x)+b$

The correct option among the following is

For the following frequency distribution, the variance is approximately equal to

$$\begin{array}{cccccc} \hline \begin{array}{c} \text { Class } \\ \text { Interval } \end{array} & 0-5 & 5-10 & 10-15 & 15-20 & 20-25 \\ \hline \text { Frequency } & 4 & 1 & 10 & 3 & 2 \\ \hline \end{array}$$

If the mean of the discrete distribution $8,9,6,5, x, 4$, 6, 5 is 6 , then its standard deviation (nearest to two decimal places) is

There are $n$ observations and all of them are negative numbers. The ascending order of these observations is $x_1, x_2, \ldots . x_n$. If the signs of the first term and last term in that order are changed, then the range of the data is

The mean deviation from the mean for the observations $1,3,5,7,11,13,17,19,23$ is

Statement I The range of the ungrouped data does not change even if certain intermediate observations are removed

Statement II The value of the mean deviation of an ungrouped data about the median is always less than or equal to the value of the mean deviation computed about any other measure of central tendency

Statement III For a grouped data, range is approximated as the difference between the lower limit of the largest class and the upper limit of the smallest class

If 10 is the mean deviation of ' $n$ ' observations $x_1, x_2, x_3, \ldots, x_n$, then the mean deviation of the observations $\frac{2 x_1+5}{3}, \frac{2 x_2+5}{3}, \frac{2 x_3+5}{3}, \ldots . \frac{2 x_n+5}{3}$ is