Variance of first n natural numbers is

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13.Statistics

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The variance of the first $n$ natural numbers is

$\frac{{{n^2} - 1}}{{12}}$

$\frac{{{n^2} - 1}}{6}$

$\frac{{{n^2} + 1}}{6}$

$\frac{{{n^2} + 1}}{{12}}$

Solution

(a) Variance $ = {({\rm{S}}{\rm{.D}}{\rm{.}})^2}$$ = \frac{1}{n}\Sigma {x^2} – {\left( {\frac{{\Sigma x}}{n}} \right)^2}$,$\left( {\because \;\;\bar x = \frac{{\Sigma x}}{n}} \right)$

$ = \frac{{n(n + 1)\;(2n + 1)}}{{6n}} – {\left( {\frac{{n(n + 1)}}{{2n}}} \right)^2} $

$= \frac{{{n^2} – 1}}{{12}}$.

Standard 11

Mathematics

Let $x_1, x_2, x_3, x_4, ………. , x_n$ be $n$ observations and let $\bar x$ be their arithmetic mean and $\sigma ^2$ be their variance.

Statement $-1$ : Variance of observations $2x_1, 2x_2, 2x_3, ……, 2x_n$ is $4\sigma ^2$ .

Statement $-2$ : Arithmetic mean of $2x _1, 2x_2, 2x_3, ……, 2x_n$ is $4\bar x$ .

normal

The diameters of circles (in mm) drawn in a design are given below:

Diameters $33-36$ $37-40$ $41-44$ $45-48$ $49-52$

No. of circles $15$ $17$ $21$ $22$ $25$

Calculate the standard deviation and mean diameter of the circles.

[ Hint : First make the data continuous by making the classes as $32.5-36.5,36.5-40.5,$ $40.5-44.5,44.5-48.5,48.5-52.5 $ and then proceed.]

medium

Let the mean and variance of $8$ numbers $x , y , 10$, $12,6,12,4,8$, be $9$ and $9.25$ respectively. If $x > y$, then $3 x-2 y$ is equal to $………..$.

hard

(JEE MAIN-2023)

The mean and standard deviation of $15$ observations are found to be $8$ and $3$ respectively. On rechecking it was found that, in the observations, $20$ was misread as $5$ . Then, the correct variance is equal to……

medium

(JEE MAIN-2022)

What is the standard deviation of the following series

class

0-10

10-20

20-30

30-40

Freq

1

3

4

2

medium

Diameters	$33-36$	$37-40$	$41-44$	$45-48$	$49-52$
No. of circles	$15$	$17$	$21$	$22$	$25$

The variance of the first $n$ natural numbers is

Solution

Similar Questions

Let the mean and variance of $8$ numbers $x , y , 10$, $12,6,12,4,8$, be $9$ and $9.25$ respectively. If $x > y$, then $3 x-2 y$ is equal to $………..$.

The mean and standard deviation of $15$ observations are found to be $8$ and $3$ respectively. On rechecking it was found that, in the observations, $20$ was misread as $5$ . Then, the correct variance is equal to……

What is the standard deviation of the following series class 0-10 10-20 20-30 30-40 Freq 1 3 4 2

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What is the standard deviation of the following series

class

0-10

10-20

20-30

30-40

Freq

1

3

4

2