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

Let ${x_1}, {x_2}, ..., {x_n}$ 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:

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 $I$: Variance of $d_1, d_2, ..., d_n$ is $\sigma^2$. Statement $II$: Mean and mode of $d_1, d_2, ..., d_n$ are $-\bar{x} - a$ and $-M - a$,respectively.

An analysis of monthly wages paid to the workers of two jute mills $A$ and $B$ gives the following data:

Metric	Mill-$A$	Mill-$B$
No. of workers	$500$	$600$
Average daily wage (in rupees)	$186$	$175$
Variance of distribution of wages	$81$	$100$

Then:

The mean and variance of the observations $x_1, x_2, x_3, \ldots, x_{15}$ are respectively $2$ and $4$. If the mean and variance of the observations $y_1, y_2, \ldots, y_{10}$ are respectively $2$ and $5$,then the variance of the combined observations $x_1, x_2, \ldots, x_{15}, y_1, y_2, \ldots, y_{10}$ is

Consider the following statements:
$(1)$ Mode can be computed from a histogram.
$(2)$ Median is not independent of change of scale.
$(3)$ Variance is independent of change of origin and scale.
Which of these is/are correct?