(D) The first $n$ even natural numbers are $2, 4, 6, \dots, 2n$.The mean $\bar{x} = \frac{2(1+2+\dots+n)}{n} = \frac{2 \times n(n+1)}{2n} = n+1$.The v

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Statement-$I$ is false and Statement-$II$ is true.

(D) The first $n$ even natural numbers are $2, 4, 6, \dots, 2n$.The mean $\bar{x} = \frac{2(1+2+\dots+n)}{n} = \frac{2 \times n(n+1)}{2n} = n+1$.The variance is given by $\sigma^2 = \frac{1}{n} \sum_{i=1}^n (2i)^2 - (n+1)^2$.$\sigma^2 = \frac{4}{n} \times \frac{n(n+1)(2n+1)}{6} - (n+1)^2 = \frac{2(n+1)(2n+1)}{3} - (n+1)^2$.$\sigma^2 = (n+1) \left[ \frac{4n+2-3n-3}{3} \right] = \frac{(n+1)(n-1)}{3} = \frac{n^2-1}{3}$.Thus,Statement-$I$ is false.For $n=20$,the mean is $20+1 = 21$.The variance is $\frac{20^2-1}{3} = \frac{399}{3} = 133$.The difference is $133 - 21 = 112$.Thus,Statement-$II$ is true.

Class 11 Mathematics Statistics Word problem -Statistics

Easy

Based on the following statements,choose the correct option.
Statement-$I:$ The variance of the first $n$ even natural numbers is $\frac{n^2-1}{4}$.
Statement-$II:$ The difference between the variance of the first $20$ even natural numbers and their arithmetic mean is $112$.

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A
Both Statements are true and $II$ is a correct explanation of $I$.
B
Both Statements are true but $II$ is not a correct explanation of $I$.
C
Statement-$I$ is true and Statement-$II$ is false.
D
Statement-$I$ is false and Statement-$II$ is true.

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Expenditure in $Rs. (x)$	$0-50$	$50-100$	$100-150$	$150-200$	$200-250$
No. of families $(f)$	$24$	$33$	$37$	$b$	$25$

Based on the following statements,choose the correct option.Statement-$I:$ The variance of the first $n$ even natural numbers is $\frac{n^2-1}{4}$.Statement-$II:$ The difference between the variance of the first $20$ even natural numbers and their arithmetic mean is $112$.

Similar Questions

Let ${x_1}, {x_2}, ..., {x_n}$ be $n$ observations such that $\sum x_i^2 = 400$ and $\sum x_i = 80$. Then a possible value of $n$ among the following is:

Let the mean and the variance of $5$ observations $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}$ be $\frac{24}{5}$ and $\frac{194}{25}$ respectively. If the mean and variance of the first $4$ observations are $\frac{7}{2}$ and $a$ respectively,then $(4a + x_{5})$ is equal to

The frequency distribution of daily working expenditure of families in a locality is as follows: Expenditure in $Rs. (x)$ $0-50$ $50-100$ $100-150$ $150-200$ $200-250$ No. of families $(f)$ $24$ $33$ $37$ $b$ $25$ If the mode of the distribution is $Rs. 140$,then the value of $b$ is:

Two distributions $A$ and $B$ have the same mean. If their coefficients of variation are $6$ and $2$ respectively and $\sigma_A$ and $\sigma_B$ are their standard deviations,then:

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 place of $40$. If the wrong entry is replaced by the correct one,then the sum of the squares of all the observations is:

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Based on the following statements,choose the correct option.
Statement-$I:$ The variance of the first $n$ even natural numbers is $\frac{n^2-1}{4}$.
Statement-$II:$ The difference between the variance of the first $20$ even natural numbers and their arithmetic mean is $112$.

The frequency distribution of daily working expenditure of families in a locality is as follows:

Expenditure in $Rs. (x)$ $0-50$ $50-100$ $100-150$ $150-200$ $200-250$

No. of families $(f)$ $24$ $33$ $37$ $b$ $25$

If the mode of the distribution is $Rs. 140$,then the value of $b$ is: