Statistics — JEE Main PYQs

The mean and variance of $n$ observations are 8 and 16, respectively. If the sum of the first $(n-1)$ observations is 48 and the sum of squares of the first $(n-1)$ observations is 496, then the value of $n$ is :

Suppose that the mean and median of the non-negative numbers $21, 8, 17, a, 51, 103, b, 13, 67$, $(a > b)$, are 40 and 21, respectively. If the mean deviation about the median is 26, then $2a$ is equal to:

For 10 observations $x_1, x_2, \ldots, x_{10}$, if $\displaystyle\sum_{i=1}^{10}\left(x_i + 2\right)^2 = 180$ and $\displaystyle\sum_{i=1}^{10}\left(x_i - 1\right)^2 = 90$, then their standard deviation is:

The mean deviation about the mean for the data

$x_i$	5	7	9	10	12	15
$f_i$	8	6	2	2	2	6

is equal to:

A variable $X$ takes values $0, 0, 2, 6, 12, 20, \ldots, n(n-1)$ with frequencies $^{n}C_0, {}^{n}C_1, {}^{n}C_2, {}^{n}C_3, {}^{n}C_4, {}^{n}C_5, \ldots, {}^{n}C_n$, respectively. If the mean of this data is 60, then its median is :

A data consists of 20 observations $x_1, x_2, \ldots, x_{20}$. If $\sum_{i=1}^{20}\left(x_i+5\right)^2=2500$ and $\sum_{i=1}^{20}\left(x_i-5\right)^2=100$, then the ratio of mean to standard deviation of this data is:

Let the mean and the variance of seven observations $2, 4, \alpha, 8, \beta, 12, 14$, $\alpha < \beta$, be 8 and 16 respectively. Then the quadratic equation whose roots are $3\alpha + 2$ and $2\beta + 1$ is :

A set of four observations has mean 1 and variance 13. Another set of six observations has mean 2 and variance 1. Then, the variance of all these 10 observations is equal to :

If the mean and the variance of 6, 4, a, 8, b, 12, 10, 13 are 9 and 9.25 respectively, then $a+b+ab$ is equal to :

Let the Mean and Variance of five observations $x_1 = 1,\ x_2 = 3,\ x_3 = a,\ x_4 = 7$ and $x_5 = \mathrm{b}$, $a > \mathrm{b}$, be 5 and 10 respectively. Then the Variance of the observations $n + x_n,\ n = 1, 2, \ldots, 5$ is

Let the mean and the standard deviation of the observation $2,3,3,4,5,7,a,b$ be $4$ and $\sqrt{2}$ respectively. Then the mean deviation about the mode of these observations is :

The mean and standard deviation of 100 observations are 40 and 5.1, respectively. By mistake one observation is taken as 50 instead of 40. If the correct mean and the correct standard deviation are $\mu$ and $\sigma$ respectively, then $10(\mu + \sigma)$ is equal to

The marks obtained by all the students of class 12 are presented in a frequency distribution with classes of equal width. Let the median class interval be 12-18 with median class frequency 12, and the median of these grouped data be 14. If the number of students whose marks are less than 12 is 18, then the total number of students is

The variance of the numbers $8, 21, 34, 47, \ldots, 320$ is _________.

For a statistical data $x_1, x_2, \ldots, x_{10}$ of 10 values, a student obtained the mean as 5.5 and $\sum_{i=1}^{10} x_i^2 = 371$. He later found that he had noted two values in the data incorrectly as 4 and 5, instead of the correct values 6 and 8, respectively. The variance of the corrected data is

Let $\alpha, \beta \in \mathbf{R}$. Let the mean and the variance of 6 observations $-3, 4, 7, -6, \alpha, \beta$ be 2 and 23, respectively. The mean deviation about the mean of these 6 observations is :

The mean and standard deviation of 20 observations are found to be 10 and 2, respectively. On rechecking, it was found that an observation by mistake was taken 8 instead of 12. The correct standard deviation is

Let $a, b, c \in \mathbb{N}$ and $a < b < c$. Let the mean, the mean deviation about the mean and the variance of the 5 observations $9, 25, a, b, c$ be 18, 4 and $\dfrac{136}{5}$, respectively. Then $2a + b - c$ is equal to ________

The frequency distribution of the age of students in a class of 40 students is given below.

Age	15	16	17	18	19	20
No of Students	5	8	5	12	$x$	$y$

If the mean deviation about the median is 1.25, then $4x+5y$ is equal to :

If the variance of the frequency distribution

$x$	$c$	$2c$	$3c$	$4c$	$5c$	$6c$
$f$	$2$	$1$	$1$	$1$	$1$	$1$

is 160, then the value of $c \in \mathbb{N}$ is

Let the median and the mean deviation about the median of 7 observation 170, 125, 230, 190, 210, a, b be 170 and $\dfrac{205}{7}$ respectively. Then the mean deviation about the mean of these 7 observations is :

Consider 10 observations $x_1, x_2, \ldots, x_{10}$ such that $\sum_{i=1}^{10}(x_i-\alpha)=2$ and $\sum_{i=1}^{10}(x_i-\beta)^2=40$, where $\alpha, \beta$ are positive integers. Let the mean and the variance of the observations be $\frac{6}{5}$ and $\frac{84}{25}$ respectively. Then $\frac{\beta}{\alpha}$ is equal to :

Let $a_1, a_2, \ldots, a_{10}$ be 10 observations such that $\sum\limits_{k=1}^{10} a_k = 50$ and $\sum\limits_{\forall\, k < j} a_k \cdot a_j = 1100$. Then the standard deviation of $a_1, a_2, \ldots, a_{10}$ is equal to :

If the mean and variance of the data 65, 68, 58, 44, 48, 45, 60, $\alpha$, $\beta$, 60 where $\alpha > \beta$, are 56 and 66.2 respectively, then $\alpha^2 + \beta^2$ is equal to ________.

Let M denote the median of the following frequency distribution

Class	0 – 4	4 – 8	8 – 12	12 – 16	16 – 20
Frequency	3	9	10	8	6

Then $20\mathrm{M}$ is equal to :

The variance $\sigma^2$ of the data

$x_i$	0	1	5	6	10	12	17
$f_i$	3	2	3	2	6	3	3

is ______.

Let the mean and the variance of 6 observations $a, b, 68,44,48,60$ be 55 and 194, respectively. If $a>b$, then $a+3 b$ is