Statistics — JEE Main PYQs

If ${\sum }_{\mathrm{i}=1}^{\mathrm{n}}\left({\mathrm{x}}_{\mathrm{i}}-\mathrm{a}\right)=\mathrm{n}$ and ${\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\left({\mathrm{x}}_{\mathrm{i}}-\mathrm{a}\right)}^2=\mathrm{na},(\mathrm{n},\mathrm{a}>1)$ then the standard deviation of n observations …

The mean and variance of 8 observations are 10 and 13.5 , respectively. If 6 of these observations are $5,7,10,12,14,15$, then the absolute difference of the remaining two observations is:

If the variance of the following frequency distribution :

\begin{tabular}{ccccc}
Class & $⋮$ & $10-20$ & $20-30$ & $30-40$ \\
Frequency & $⋮$ & 2 & $x$ & 2
\end{tabular}
is 50 , then x is equal to:

The mean and variance of seven observations are 8 and 16, respectively. If 5 of the observations are $2,4,10,12,14$, then the product of the remaining two observations is:

A student score the following marks in five tests : $45,54,41,57,43$. His score is not known for the sixth test. If the mean score is 48 in the six tests, then the standard deviation of the marks in six tests is:

If the standard deviation of the numbers $-1,0,1,\mathrm{k}$ is $\sqrt{5}$ where $\mathrm{k}>0$, then k is equal to:

The mean of five observations is 5 and their variance is 9.20 . If three of the given five observations are 1,3 and 8 , then a ratio of other two observations is:

If the data $x_1,x_2,\dots .,x_{10}$ is such that the mean of first four of these is 11 , the mean of the remaining six is 16 and the sum of squares of all of these is 2,000 ; then the standard deviation of this data is:

If both the means and the standard deviation of 50 observations $x_1,x_2,\dots \dots \dots ,x_{50}$ are equal to 16, then the mean of $(x_1-4{)}^2,(x_2-4{)}^2,\dots .,(x_{50}-4{)}^2$ is

The outcome of each of 30 items was observed; 10 items gave an outcome $\frac{1}{2}-\mathrm{d}$ each, 10 items gave outcome $\frac{1}{2}$ each and the remaining 10 items gave outcome $\frac{1}{2}+\mathrm{d}$ each. If the variance of this outcome data is $\frac{4}{3}$ then …

If the sum of the deviations of 50 observations from 30 is 50, then the mean of these observations is:

The mean and the variance of five observations are 4 and 5.20 , respectively. If three of the observations are 3,4 and 4 ; then the absolute value of the difference of the other two observations, is :

The mean and the median of the following ten numbers in increasing order $10,22,26$, $29,34,x,42,67,70,y$ are 42 and 35 respectively, then $\frac{y}{x}$ is equal to

A data consists of $n$ observations: ${\mathrm{x}}_1,{\mathrm{x}}_2,\dots \dots ,{\mathrm{x}}_{\mathrm{n}}$. If ${\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\left({\mathrm{x}}_{\mathrm{i}}+1\right)}^2=9\mathrm{n}$ and …

5 students of a class have an average height 150 cm and variance $18{\mathrm{cm}}^2$. A new student, whose height is 156 cm, joined them. The variance (in ${\mathrm{cm}}^2$) of the height of these six students is:

If for some $\mathrm{x}\in \mathrm{R}$, the frequency distribution of the marks obtained by 20 students in a test is

\begin{tabular}{ccccc}
Marks & 2 & 3 & 5 & 7 \\
Frequency & $(x+1{)}^2$ & $2x-5$ & $x^2-3x$ & $x$
\end{tabular}

Then the mean of the marks is:

If ${\sum }_{i=1}^9\left(x_i-5\right)=9$ and ${\sum }_{i=1}^9{\left(x_i-5\right)}^2=45$, then the standard deviation of the 9 items $x_1,x_2,\dots ..,x_9$ is :

If the standard deviation of the numbers 2,3, a and 11 is 3.5, then which of the following is true?

The mean of the data set comprising of 16 observations is 16 . If one of the observation valued 16 is deleted and three new observations valued 3 , 4 and 5 are added to the data, then the mean of the resultant data, is :

The variance of first 50 even natural numbers is

All the students of a class performed poorly in Mathematics. The teacher decided to give grace marks of 10 to each of the students. Which of the following statistical measures will not change even after the grace marks were given ?

Let ${\mathrm{x}}_1,{\mathrm{x}}_2,\dots \dots ,{\mathrm{x}}_{\mathrm{n}}$ be n observations, and let $\overline{\mathrm{x}}$ be their arithematic mean and ${\sigma }^2$ be their variance.

Statement 1: Variance of $2x_1,2x_2,\dots \dots ,2x_n$ is …

If the mean deviation about the median of the numbers $a,2a,\dots ,50a$ is 50, then $|a|$ equals

For two data sets, each of size 5, the variances are given to be 4 and 5 and the corresponding means are given to be 2 and 4, respectively. The variance of the combined data set is

If the mean deviation of number $1,1+d,1+2d,\dots ,1+100d$ from their mean is 255, then the $d$ is equal to

Statement-1: The variance of first $n$ even natural numbers is $\frac{n^2-1}{4}$
Statement-2: The sum of first $n$ natural numbers is $\frac{n(n+1)}{2}$ and the sum of squares of first $n$ natural numbers is $\frac{n(n+1)(2n+1)}{6}$

The mean of the numbers $a,b,8,5,10$ is 6 and the variance is 6.80. Then which one of the following gives possible values of $a$ and $b$?

The average marks of boys in a class is 52 and that of girls is 42. The average marks of boys and girls combined is 50. The percentage of boys in the class is

Suppose a population $A$ has 100 observations $101,102,\dots ,200$, and another population $B$ has 100 observations $151,152,\dots ,250$. If $V_A$ and $V_B$ represent the variances of the two populations, respectively, then $\frac{V_A}{V_B}$ is

If in a frequently distribution, the mean and median are 21 and 22 respectively, then its mode is approximately

Let $x_1,x_2,\dots ,x_n$ 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

Consider the following statements:
(a) Mode can be computed from histogram
(b) Median is not independent of change of scale
(c) Variance is independent of change of origin and scale.

Which of these is/are correct?

In a series of $2n$ observations, half of them equal a and remaining half equal $-a$. If the standard deviation of the observations is 2 , then $|a|$ equals