Measures of Central Tendency

Let $x_1,x_2,x_3,\dots ..,x_{20}$ be in geometric progression with $x_1=3$ and the common ration $\frac{1}{2}$. A new data is constructed replacing each $x_i$ by ${\left(x_i-i\right)}^2$. If $\overline{x}$ is the mean of new data, then the greatest integer less than or equal to $\overline{x}$ is

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 ${\sigma }^2$ is the variance of marks of $50$ candidates, then $\mu +{\sigma }^2$ is equal to

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 ______.

Consider the following frequency distribution :

If the sum of all frequencies is $584$ and median is $45$, then $|\alpha -\beta |$ is equal to            .

Consider the following frequency distribution:

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

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 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:

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 both the mean and the standard deviation of $50$ observations $x_1,\mathrm{ }{x}_2,\mathrm{ }\dots ,\mathrm{ }{x}_{50}$ are equal to $16$, then the mean of ${\left(x_1-4\right)}^2,\mathrm{ }{\left(x_2-4\right)}^2,\mathrm{ }\dots ,{\left(x_{50}-4\right)}^2$ is

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

Then the mean of the marks 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:

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

The mean of a set of $30$ observation is $75$. If each observations is multiplied by non-zero number $\lambda$ and then each of them is decreased by $25$, their mean remains the same. Then, $\lambda$ is equal to :

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

A factory is operating in two shifts, day and night, with $70$ and $30$ workers, respectively.If per day mean wage of the day shift workers is, $₹54$ and per day mean wage of all the workers is $₹60$, then per day mean wage of the night shift workers (in $₹$) is :

The mean of a data set comprising of $\mathrm{16}$ observations is $\mathrm{16}$. If one of the observation value $\mathrm{16}$ is deleted and three new observations valued $3,\; 4$ and $\mathrm{5}$ are added to the data, then the mean of the resultant data is

In a set of $2n$ distinct observations, each of the observation below the median of all the observations is increased by $5$ and each of the remaining observations is decreased by $3$. Then, the mean of the new set of observations :