Let $X _{1}, X _{2}, \ldots, X _{18}$ be eighteen observations such that $\sum_{ i =1}^{18}\left( X _{ i }-\alpha\right)=36 \quad$ and $\sum_{i=1}^{18}\left(X_{i}-\beta\right)^{2}=90,$ where $\alpha$ and $\beta$ are distinct real numbers. If the standard deviation of these observations is $1,$ then the value of $|\alpha-\beta|$ is ...... .

  • [JEE MAIN 2021]
  • A

    $4$

  • B

    $2$

  • C

    $3$

  • D

    $5$

Similar Questions

Find the variance and standard deviation for the following data:

${x_i}$ $4$ $8$ $11$ $17$ $20$ $24$ $32$
${f_i}$ $3$ $5$ $9$ $5$ $4$ $3$ $1$

Consider the statistics of two sets of observations as follows :

  Size Mean Variance
Observation $I$ $10$ $2$ $2$
Observation $II$ $n$ $3$ $1$

If the variance of the combined set of these two observations is $\frac{17}{9},$ then the value of $n$ is equal to ..... .

  • [JEE MAIN 2021]

The mean of $5$ observations is $4.4$ and their variance is $8.24$. If three observations are $1, 2$ and $6$, the other two observations are

If the mean deviation about the mean of the numbers $1,2,3, \ldots ., n$, where $n$ is odd, is $\frac{5(n+1)}{n}$, then $n$ is equal to

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The mean and variance of $5$ observations are $5$ and $8$ respectively. If $3$ observations are $1,3,5$, then the sum of cubes of the remaining two observations is

  • [JEE MAIN 2023]