Statistics — JEE Main PYQs

Let $9={\mathrm{x}}_1<{\mathrm{x}}_2<\dots <{\mathrm{x}}_7$ be in an A.P. with common difference d. If the standard deviation of ${\mathrm{x}}_1,{\mathrm{x}}_2\dots ,{\mathrm{x}}_7$ is 4 and the mean is $\overline{\mathrm{x}}$, then …

Let $\mathrm{X}=\{11,12,13,\dots ,40,41\}$ and $\mathrm{Y}=\{61,62,63,\dots ,90,91\}$ be the two sets of observations. If $\bar{x}$ and $\bar{y}$ are their respective means and ${\sigma }^2$ is the variance of all the observations in $\mathrm{X}\cup \mathrm{Y}$, …

Let the six numbers $a_1,a_2,a_3,a_4,a_5,a_6$ be in A.P. and $a_1+a_3=10$. If the mean of these six numbers is $\frac{19}{2}$ and their variance is ${\sigma }^2$, then $8{\sigma }^2$ is equal to

Let the mean and standard deviation of marks of class A of 100 students be respectively 40 and $\alpha (>0)$, and the mean and standard deviation of marks of class B of n students be respectively 55 and $30-\alpha$. If the mean and variance of the marks of the combined class o …

Let S be the set of all values of ${\mathrm{a}}_1$ for which the mean deviation about the mean of 100 consecutive positive integers ${\mathrm{a}}_1,{\mathrm{a}}_2,{\mathrm{a}}_3,\dots \dots ,{\mathrm{a}}_{100}$ is 25 . Then S is

If the variance of the frequency distribution

is 3, then $\alpha$ is equal to

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

The mean and variance of the marks obtained by the students in a test are 10 and 4 respectively. Later, the marks of one of the students is increased from 8 to 12. If the new mean of the marks is 10.2. then their new variance is equal to :

Let ${\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{x}}_3,\dots .,{\mathrm{x}}_{20}$ be in geometric progression with ${\mathrm{x}}_1=3$ and the common ration $\frac{1}{2}$. A new data is constructed replacing each ${\mathrm{x}}_{\mathrm{i}}$ by …

The mean and variance of 10 observations were calculated as 15 and 15 respectively by a student who took by mistake 25 instead of 15 for one observation. Then, the correct standard deviation is ____.

The mean and standard deviation of 40 observations are 30 and 5 respectively. It was noticed that two of these observations 12 and 10 were wrongly recorded. If $\sigma$ is the standard deviation of the data after omitting the two wrong observations from the data, then …

If the mean deviation about median for the number 3, 5, 7, 2k, 12, 16, 21, 24 arranged in the ascending order, is 6 then the median is

Suppose a class has 7 students. The average marks of these students in the mathematics examination is 62, and their variance is 20. A student fails in the examination if he/she gets less than 50 marks, then in worst case, the number of students can fail is

The mean and variance of the data 4, 5, 6, 6, 7, 8, $x$, $y$ where $x<y$ are 6, and $\frac{9}{4}$ respectively. Then $x^4+y^2$ 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 ____ .

The number of values of $a\in \mathbb{N}$ such that the variance of $3,7,12,a,43-a$ is a natural number is:

The mean and standard deviation of 50 observations are 15 and 2 respectively. It was found that one incorrect observation was taken such that the sum of correct and incorrect observations is 70. If the correct mean is 16, then the correct variance is equal to :

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 observation are $\frac{7}{2}$ and $a$ respectively, then $\left(4a+x_5\right)$ is equal to:

The mean of the numbers $a,b,8,5,10$ is 6 and their variance is 6.8. If M is the mean deviation of the numbers about the mean, then 25M is equal to:

Let the mean and the variance of 20 observations $x_1,x_2,\dots x_{20}$ be 15 and 9 , respectively. For $\alpha \in R$, if the mean of ${\left(x_1+\alpha \right)}^2,{\left(x_2+\alpha \right)}^2,\dots ,{\left(x_{20}+\alpha \right)}^2$ is 178, then the square of the …

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

The mean and standard deviation of 20 observations were calculated as 10 and 2.5 respectively. It was found that by mistake one data value was taken as 25 instead of 35. If $\alpha$ and $\sqrt{\beta }$ are the mean and standard deviation respectively for correct data, then …

Let the mean and variance of four numbers 3, 7, $x$ and $y$ $(x>y)$ be 5 and 10 respectively. Then the mean of four numbers $3+2x$, $7+2y$, $x+y$ and $x-y$ is ____.

The mean of 6 distinct observations is 6.5 and their variance is 10.25 . If 4 out of 6 observations are $2,4,5$ and 7 , then the remaining two observations are :

If the variance of 10 natural numbers $1,1,1\dots \dots 1,\mathrm{k}$ less than 10, then the maximum possible value of $k$ is ____

If the mean and variance of six observations $7,10,11,15,a,b$ are 10 and $\frac{20}{3}$, respectively, then the value of $|a-b|$ is equal to

An online exam is attempted by 50 candidates out of which 20 are boys. The average marks obtained by boys is 12 with a variance 2. The variance of marks obtained by 30 girls is also 2. The average marks of all 50 candidates is 15. If $\mu$ is the average marks of girls and …

Consider the following frequency distribution :

\begin{tabular}{lccccc}
Class : & $0-6$ & $6-12$ & $12-18$ & $18-24$ & $24-30$ \\
Frequency : & a & b & 12 & 9 & 5
\end{tabular}

If mean $=\frac{309}{22}$ and median $=14$, then the value $(a-b{)}^2$ is equal to

Let n be an odd natural number such that the variance of $1,2,3,4,\dots ,\mathrm{n}$ is 14 . Then n is equal to

Let $X_1,X_2,\dots X_{18}$ be eighteen observations such that ${\sum }_{i=1}^{18}\left(X_i-\alpha \right)=36$ 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, the …

The mean and variance of 7 observations are 8 and 16 respectively. If two observations are 6 and 8, then the variance of the remaining 5 observations is:

If the mean and variance of the following data :
$6,10,7,13,a,12,b,12$ are 9 and $\frac{37}{4}$ respectively, then $(a-b{)}^2$ is equal to :

Consider the following frequency distribution :

\begin{tabular}{|l|l|l|l|l|l|}
\hline Class: & $10-20$ & $20-30$ & $30-40$ & $40-50$ & $50-60$ \\
\hline Frequency : & $\alpha$ & 110 & 54 & 30 & $\beta$ \\
\hline
\end{tabular}

If the sum of all frequencies is 584 a …

The first of the two samples in a group has 100 items with mean 15 and standard deviation 3 . If the whole group has 250 items with mean 15.6 and standard deviation $\sqrt{13.44}$, then the standard deviation of the second sample is :

Let the mean and variance of the frequency distribution

x: ${\mathrm{x}}_1=2$, ${\mathrm{x}}_2=6$, ${\mathrm{x}}_3=8$, ${\mathrm{x}}_4=9$

f: 4, 4, $\alpha$, $\beta$

be 6 and 6.8 respectively. If $x_3$ is changed from 8 to 7, then the mean for the new data will be :

Consider three observations $\mathrm{a},\mathrm{b}$ and c such that $\mathrm{b}=\mathrm{a}+\mathrm{c}$. If the standard deviation of $a+2,b+2,c+2$ is $d$, then which of the following is true?

Consider the statistics of two sets of observations as follows:

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

Let in a series of $2n$ observations, half of them are equal to a and remaining half are equal to -a. Also by adding a constant $b$ in each of these observations, the mean and standard deviation of new set become 5 and 20, respectively. Then the value of $a^2+b^2$ is eq …

Consider a set of $3n$ numbers having variance 4. In this set, the mean of first $2n$ numbers is 6 and the mean of the remaining n number is 3. A new set is constructed by adding 1 into each of first $2n$ numbers, and subtracting 1 from each of the remaining $n$ numbers. If th …

The mean age of 25 teachers in a school is 40 years. A teacher retires at the age of 60 years and a new teacher is appointed in his place. If the mean age of the teachers in this school now is 39 years, then the age (in years) of the newly appointed teacher is:

If the variance of the first $n$ natural numbers is 10 and the variance of the first $m$ even natural numbers is 16 , then $m+n$ is equal to ____

If the mean and variance of eight numbers $3,7,9,12,13,20,x$ and $y$ be 10 and 25 respectively, then xy is equal to ____

The mean and the standard deviation (s.d.) of 10 observations are 20 and 2 respectively. Each of these 10 observations is multiplied by $p$ and then reduced by $q$, where $\mathrm{p}\ne 0$ and $\mathrm{q}\ne 0$. If the new mean and new s.d. become half of their original value …

The mean and variance of 20 observations are found to be 10 and 4, respectively. On rechecking, it was found that an observation 9 was incorrect and the correct observation was 11. Then the correct variance is:

Let the observations $x_i(1\le i\le 10)$ satisfy the equation ${\sum }_{i=1}^{10}\left(x_i-5\right)=10$ and ${\sum }_{\mathrm{i}=1}^{10}{\left({\mathrm{x}}_{\mathrm{i}}-5\right)}^2=40$. If $\mu$ and $\lambda$ are the mean and the variance of the observations, …

If the mean and standard deviation of the data $3,5,7,a,b$ are 5 and 2 respectively, then $a$ and $b$ are the roots of the equation:

For the frequency distribution:

Variate (x): ${\mathrm{x}}_1{\mathrm{x}}_2{\mathrm{x}}_3\dots .{\mathrm{x}}_{15}$
Frequency (f): $f_1{f}_2{f}_3\dots .f_{15}$
Where $0<x_1<x_2<x_3<\dots <x_{15}=10$ and …

Let $x_i(1\le i\le 10)$ be ten observations of a random variable $x$. If ${\sum }_{i=1}^{10}\left(x_i-p\right)=3$ and ${\sum }_{i=1}^{10}{\left(x_i-p\right)}^2=9$ where $0\ne p\in R$, then the standard deviation of these observations is:

Consider the data on $x$ taking the values $0,2,4,8,\dots ,2^n$ with frequencies ${}^n{C}_0{,}^n{C}_1{,}^n{C}_2,\dots {,}^n{C}_n$ respectively. If the mean of this data is $\frac{728}{2^n}$, then $n$ is equal to ______.

The mean and variance of 7 observations are 8 and 16, respectively. If five observations are $2,4,10,12,14$, then the absolute difference of the remaining two observations is: