Statistics Questions (255)

The mean of $40$ observations is $20$ and their standard deviation is $5$. If the sum of the squares of the observations is $k$, then the value of $\frac{k}{160}$ is

The mean and standard deviation of marks of 200 students were 40 and 15 respectively. Later it was discovered that a score of 40 was wrongly read as 50. The correct standard deviation is

The mean and variance of 7 observations are 8 and 16 respectively. If 5 of the observations are 2, 4, 10, 12, 14, then the remaining two observations are

For a data set of 10 observations with mean $\bar{x} = 50$, $\displaystyle\sum(x_i-\bar{x})^2 = 250$. The coefficient of variation (CV) is

Let \(\bar{X}\) and MD be the mean and the mean deviation about \(\bar{X}\) of \(n\) observations \(x_i,\ i = 1, 2, \ldots, n\). If each of the observations is increased by 5, then the new mean and the mean deviation about the new mean, respectively, are

A student scores the following mark 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:

The mean and variance of 7 observations are 8 and 16 respectively. If 5 of the observations are 2, 4, 10, 12, 14, then the remaining two observations are

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

In ten observations, the mean of all $10$ numbers is $15$, the mean of the first six observations is $16$ and the mean of the last five observations is $12$. The sixth number 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 $a+3b$ is

Given $\bar{x} = 50$, find the coefficient of variation if $\sum(x_i - \bar{x})^2 = 250$.

Out of 100 observations, \(\displaystyle\sum_{i=1}^{100} x_i = 400\) and \(\displaystyle\sum_{i=1}^{100} x_i^2 = 2425\). If two observations 8 and 12 are removed (so \(\sum x_i\) becomes 388 and \(N = 97\) ... wait, \(N = 98\)), find the variance of the remaining observations using \(\sigma^2 = \dfrac{\sum x_i^2}{N} - \left(\dfrac{\sum x_i}{N}\right)^2 = \dfrac{2425}{97} - \left(\dfrac{388}{97}\right)^2\).

Let \(\bar{x}\), \(M\) and \(\sigma^2\) be respectively, the mean, mode and variance of \(n\) observations \(x_1, x_2, ..., x_n\) and \(d_i = -x_i - a\), \(i = 1, 2, ..., n\), where \(a\) is any number.Statement-1: Variance of \(d_1, d_2, ..., d_n\) is \(\sigma^2\).Statement-2: Mean and mode of \(d_1, d_2, ..., d_n\) are \(-\bar{x} - a\) and \(-M - a\), respectively.

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

Runs scored by a batsman in 10 innings are: 38, 70, 48, 34, 42, 55, 63, 46, 54, 44. The mean deviation is

If $\sum x_i = 15(\text{given})$ and $\sum x_i^2 + 2\sum x_i + 8 = 7(\text{given})$, then $\sum x_i + 2\sum x_i = 6u$. Find $u$.

If the mean of the distribution is 2.6, then the value of \(y\) isVariate \(x\)12345Frequency \(f\) of \(x\)45\(y\)12

If $\sum_{i=1}^{n}(x_i - 6) = 5$ and $\sum_{i=1}^{n}(x_i - 6)^2 = 25$, then the standard deviation of observations $3x_1 + 2, 3x_2 + 2, 3x_3 + 2, 3x_4 + 2$ and $3x_5 + 2$ is equal to

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 sum of 100 observations and the sum of their squares are 400 and 2474, respectively. Later on, three observations, 3, 4 and 5, were found to be incorrect. If the incorrect observations are omitted, then the variance of the remaining observations is

Let $x_1, x_2, x_3, \ldots, x_k$ be $k$ observations and $w_i = ax_i + b$ for $i = 1, 2, 3, \ldots, k$, where $a$ and $b$ are constants. If mean of $x_i$ is 52 and their standard deviation is 12 and mean of $w_i$ is 60 and their standard deviation is 15, then the value of $a$ and $b$ should be

Consider the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10. If 1 is added to each number, the variance of the numbers so obtained is

$V_1$ = variance of {13, 16, 19, . . . , 103}. $V_2$ = variance of {3, 6, 9, . . . , 93}. Find $\frac{V_1}{V_2}$.

An automobile driver travels from a plain to a hill station 120 km away at an average speed of 30 km per hour. He then makes the return trip at an average speed of 25 km per hour. He covers another 120 km on the plain at an average speed of 50 km per hour. His average speed (in km/hr) over the entire distance of 360 km will be

Let sets A and B each have 5 elements, with mean 5 and variance 12, and mean 8 and variance 20 respectively. If mean of $(x_i-3)$ and $(y_i+2)$ combined is $\mu$ and variance is $\sigma^2$, then $\mu+\sigma^2$ is

Let \(x_1, x_2, \ldots, x_n\) be \(n\) observations. Let \(w_i = lx_i + k\) for \(i = 1, 2, \ldots, n\), where \(l\) and \(k\) are constants. The mean of \(x_i\)'s is 48 and their standard deviation is 12. Also, the mean of \(w_i\)'s is 55 and standard deviation of \(w_i\)'s is 15. The values of \(l\) and \(k\) should be

For the data $x: 1,3,5,7,9$; frequency $4,24,28,\alpha,8$ with mean 5, then $\dfrac{3\alpha}{m+\sigma^2}$ is equal to _______.

The mean and variance of 7 observations are 8 and 16 respectively. If one observation 14 is omitted and $a$ and $b$ are respectively mean and variance of remaining 6 observations, then $a + 3b - 5$ is equal to ______.

If the mean deviation about the median of the numbers \(a\), \(2a\), ..., \(50a\) is 50, then \(|a|\) equals

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

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?

The mean and the standard deviation (s.d.) of five observations are 9 and 0, respectively. If one of the observations is changed such that the mean of the new set of five observations becomes 10, then their s.d. is

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:

The numbers are 4, 8, 12, . . . , 80. Find the variance.

$n_1 = 50, \bar{z}_1 = 630, \sigma_1 = 90$ and $n_2 = 40, \bar{z}_2 = 540, \sigma_2 = 60$

The mean and variance of 15 numbers are 12 and 14. Another 15 numbers have mean 14 and variance $\sigma^2$. If the combined variance of all 30 is 13, then $\sigma^2$ is equal to

Let one observation equal to the mean is added to $n$ observations. If the variance changes from 78 to 72, then $n$ is equal to

A student obtains 75%, 80% and 85% in three subjects. If the marks of another subject is added, then his average cannot be less than

Consider the following statements regarding the data set \(2x_1, 2x_2, \ldots, 2x_n\) where \(\sigma^2\) is the variance of \(x_1, x_2, \ldots, x_n\):Statement-1: The variance of \(2x_1, 2x_2, \ldots, 2x_n\) is \(4\sigma^2\).Statement-2: The AM of \(2x_1, 2x_2, \ldots, 2x_n\) is \(2\bar{x}\).Which of the following is correct?

If \(\sum_{i=1}^{n}(x_i - a) = n\) and \(\sum_{i=1}^{n}(x_i - a)^2 = na\), where \(n, a > 1\), then the standard deviation of \(n\) observations \(x_1, x_2, \ldots, x_n\) is

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

Standard deviation $\sigma = \sqrt{\frac{1}{n}\sum_{i=1}^{n} x_i^2 - \left(\frac{1}{n}\sum_{i=1}^{n} x_i\right)^2}$

For data $1,2,4,5,x,y$ with mean 5 and variance 10, the mean deviation about the mean is

The variance of first n even natural numbers is:

The following data gives the distribution of height of students:Height (in cm)160150152161156154155Number of students12844337The median of the distribution is

Let \(x_1, x_2, \ldots, x_n\) be \(n\) observations, and let \(\bar{x}\) be their arithmetic mean and \(\sigma^2\) be the variance.Statement 1: Variance of \(2x_1, 2x_2, \ldots, 2x_n\) is \(4\sigma^2\).Statement 2: Arithmetic mean of \(2x_1, 2x_2, \ldots, 2x_n\) is \(4\bar{x}\).

The variance of \(2r\) for \(r = 1, 2, 3, \ldots, 50\) is:

For the distribution $X_i: 0,1,2,3,4,5$; $f_i: k+2,2k,k^2-1,k^2-1,k^2+1,k-3$ with $\sum f_i=62$, then $[\mu^2+\sigma^2]$ is equal to

Let $S$ be the set of all values of $a_1$ for which the mean deviation about the mean of 100 consecutive positive integers $a_1, a_2, a_3, \ldots, a_{100}$ is 25. Then $S$ is

The standard deviation of the data 6, 5, 9, 13, 12, 8, 10 is