Example of a sample where mean, variance and median are equal

Question

Find examples of two samples: $X = (X_1, X_2, X_3, X_4, X_5)$ and $Y = (Y_1, Y_2, Y_3, Y_4, Y_5, Y_6)$ where sample mean, sample variance and median are all equal to $2$.

After many failures, I managed to find the first sample: $X = (0, 2, 2, 2, 4)$.

$$\bar{X} = \frac{0 + 2+ 2+ 2+4}{5} = 2$$

$$S_X^2 = \frac{1}{4} \left ( (0-2)^2 + 3\cdot(2-2)^2+ (4-2)^2 \right ) = 2$$

However, I struggle to find the second one. Is there any way to "see" this, without using the method of randomly guessing numbers and seeing if they fit? Especially since I also have to find another sample of size five, but this time the statistics have to equal 3.

Answer 1

Y could be (0,1,2,2,3,4). I just wrote out a number line and tried to balance the numbers around 2. Your second problem can't be done. The numbers can't balance around 3 and have the squares sum to 12.

Answer 2

For $n=6$ you know that their sum equals $12$ and the sum of their squares equals $34$. Not let three of them be $2$ and one of them zero. Then solve the quadratic system $$y_1+y_6=6\quad\text{and}\quad y_1^2+y_6^2=32.$$