how to find a data set of 5 values from a given variance?

Question

the question is to create a data set of 5 values with variance=40 and mean=median, so how do we do that ?

I tried simplifying the variance formula:

40=(x^2)+(mean)^2/(5-1)

so if i assume mean as 10 then : 160 + 100=x^2

and as mean is assumed as 10: total/5=10, so total=50

if the values are a,b,c,d,e , as the median=mean, c=10

so i assumed a data set to be: 1,4,10,15,20 as it gives a total of 50 but it doesn't satisfy the equation created from variance:
**(a^2)+(b^2)+(c^2)+(d^2) + (e^2)=260(160+100)

(a^2)+(b^2)+(100)+(d^2)+ (e^2)=260(160+100)

( a^2)+(b^2)+(d^2)+(e^2)=260-100(160)**

now iam confused how to create the data set as the square of the individual data set doesn't meet the requirements for variance equation.as the data set should be in order d^2 and e^2 values should be greater than 100,but that will add more than 160.

can you please help me with this, iam trying to figure this out for long.

it will be a great help if i can get answers, thank you in advance

Answer 1

in a very easy way, chose a symmetric set around zero with $E(X^2)=40$

Example:

$$\{x_1;x_2;x_3;x_4;x_5\}=\{-8;-6;0;6;8\}$$

In this case you have

$$\mathbb{E}[X]=\text{median}[X]=0$$

$$\mathbb{V}[X]=\mathbb{E}[X^2]=\frac{2\times 64+2\times 36}{5}=40$$

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Of course if for "variance" you mean the Sample variance, say the one divided by $n-1$ the reasoning remains valid but you have to take into consideration the definition of variance you have to apply: in this second case, a nice sample example is the following:

$$\{x_1;x_2;x_3;x_4;x_5\}=\{-8;-4;0;4;8\}$$

Answer 2

Deal with the spread first. If I have a symmetric set of $5$ values around zero, and $E(X^2)=40$, and sum of all squares to be $40\cdot 5=200$, then it would be safe to picture $2$ values (from either positive or negative side of the distribution) where their sum of squares is $100$. Casually pick $6$ and $8$ to be our candidates, then we have:

$${-8, -6, 0, 6, 8}$$

Finally, shift each value with reference to your mean, then we have:

$${2, 4, 10, 16, 18}$$