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Statistics & Probability

Statistics: JEE Main & Advanced PYQs

169 previous year questions on Statistics from JEE Main and JEE Advanced papers, each with its exam and year, an answer and a solution.

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Questions 1-50 of 169

  1. Let $\mu$ be the mean and $\sigma$ be the standard deviation of the distribution $x_i$ : $0,1,2,3,4,5$ $f_i$ : $k+2,\ 2k,\ k^{2}-1,\ k^{2}-1,\ k^{2}+1,\ k-3$ where …
  2. Let the mean and the variance of $6$ observations $a,b,68,44,48,60$ be $55$ and $194$, respectively. If $a>b$, then $a+3b$ is
  3. Let $a_1,a_2,\ldots,a_{10}$ be $10$ observations such that $\sum_{k=1}^{10}a_k=50$ and $\sum_{\forall k<j}a_k\cdot a_j=1100$. Then the standard deviation of …
  4. For $10$ observations $x_1,x_2,\ldots,x_{10}$, if $\sum_{i=1}^{10}(x_i+2)^{2}=180$ and $\sum_{i=1}^{10}(x_i-1)^{2}=90$, then their standard deviation is:
  5. Let sets $A$ and $B$ have $5$ elements each. Let the mean of the elements in sets $A$ and $B$ be $5$ and $8$ respectively and the variance of the elements in sets $A$ …
  6. 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 …
  7. 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 $25$ …
  8. Let the Mean and Variance of five observations $x_1=1$, $x_2=3$, $x_3=a$, $x_4=7$ and $x_5=b$, $a>b$, be $5$ and $10$ respectively. Then the Variance of the observations …
  9. 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 …
  10. The mean and standard deviation of $10$ observations are $20$ and $8$ respectively. Later on, it was observed that one observation was recorded as $50$ instead of $40$. …
  11. 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 $20M$ is equal to :
  12. 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. …
  13. 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
  14. 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 …
  15. 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 …
  16. 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 …
  17. 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 :
  18. 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 …
  19. Three rotten apples are accidently mixed with fifteen good apples. Assuming the random variable $x$ to be the number of rotten apples in a draw of two apples, the …
  20. 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 $8$ …
  21. Let the median and the mean deviation about the median of $7$ observation $170,125,230,190,210,a,b$ be $170$ and $\frac{205}{7}$ respectively. Then the mean deviation …
  22. A box contains $10$ pens of which $3$ are defective. A sample of $2$ pens is drawn at random and let $X$ denote the number of defective pens. Then the variance of $X$ is
  23. Let the mean and variance of $12$ observations be $\frac{9}{2}$ and $4$ respectively. Later on, it was observed that two observations were considered as $9$ and $10$ …
  24. 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 …
  25. Two dice $A$ and $B$ are rolled. Let the numbers obtained on $A$ and $B$ be $\alpha$ and $\beta$ respectively. If the variance of $\alpha-\beta$ is $\frac{p}{q}$, where …
  26. Let the mean of $6$ observations $1,2,4,5,x$ and $y$ be $5$ and their variance be $10$. Then their mean deviation about the mean is equal to
  27. The mean and variance of a set of $15$ numbers are $12$ and $14$ respectively. The mean and variance of another set of $15$ numbers are $14$ and $\sigma^{2}$ …
  28. Let $x_1,x_2,\ldots,x_{100}$ be in an arithmetic progression, with $x_1=2$ and their mean equal to $200$. If $y_i=i(x_i-i)$, $1\le i\le100$, then the mean of …
  29. 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 $10$ …
  30. If $\sum_{i=1}^{16}(a_i-6)=8$ and $\sum_{i=1}^{16}(a_i-6)^{2}=40$, then the standard deviation of $a_1,a_2,\ldots,a_{16}$ is
  31. 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 :
  32. 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 …
  33. Consider the statistics of two sets of observations as follows : Observation I : Size $10$, Mean $2$, Variance $2$ Observation II : Size $n$, Mean $3$, Variance $1$ If …
  34. Consider the following frequency distribution : Class : $10-20$, $20-30$, $30-40$, $40-50$, $50-60$ Frequency : $\alpha$, $110$, $54$, $30$, $\beta$ If the sum of all …
  35. 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 $4$ …
  36. Let $x_i$ ($1\le i\le10$) be ten observations of a random variable $X$. If $\sum_{i=1}^{10}(x_i-p)=3$ and $\sum_{i=1}^{10}(x_i-p)^2=9$ where $0\ne p\in\mathbf{R}$, then …
  37. 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
  38. The mean and standard deviation of the marks of $10$ students were found to be $50$ and $12$ respectively. Later, it was observed that two marks $20$ and $25$ were …
  39. 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 …
  40. The mean of $10$ numbers $7\times8,\ 10\times10,\ 13\times12,\ 16\times14,\ \ldots$ is $\_\_\_\_\_\_$ .
  41. If the mean and variance of the frequency distribution $x_i$ : $2,4,6,8,10,12,14,16$ $f_i$ : $4,4,\alpha,15,8,\beta,4,5$ are $9$ and $15.08$ respectively, then the value …
  42. If the variance of the terms in an increasing A.P., $b_1,b_2,b_3,\ldots,b_{11}$ is $90$, then the common difference of this A.P. is ________.
  43. 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 $3x-2y$ is equal to $\_\_\_\_\_\_$ .
  44. If the mean of the frequency distribution Class : $0-10$, $10-20$, $20-30$, $30-40$, $40-50$ Frequency : $2,3,x,5,4$ is $28$, then its variance is $\_\_\_\_\_\_$ .
  45. Let the mean and the variance of $20$ observations $x_1,x_2,\ldots,x_{20}$ be $15$ and $9$, respectively. For $\alpha\in\mathbf{R}$, if the mean of …
  46. Let $X_1,X_2,\ldots,X_{18}$ be eighteen observations such that $\sum_{i=1}^{18}(X_i-\alpha)=36$ and $\sum_{i=1}^{18}(X_i-\beta)^{2}=90$, where $\alpha$ and $\beta$ are …
  47. Let a random variable $X$ take values $0,1,2,3$ with $P(X=0)=P(X=1)=p$, $P(X=2)=P(X=3)$ and $E(X^{2})=2E(X)$. Then the value of $8p-1$ is :
  48. Let $X=\{x\in\mathbf{N}:1\le x\le17\}$ and $Y=\{ax+b:x\in X$ and $a,b\in\mathbf{R},a>0\}$. If mean and variance of elements of $Y$ are $17$ and $216$ respectively then …
  49. 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 …
  50. Consider the following frequency distribution : Class : $0-6$, $6-12$, $12-18$, $18-24$, $24-30$ Frequency : $a$, $b$, $12$, $9$, $5$ If mean $=\frac{309}{22}$ and …

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