1. Write the data set

The first $50$ even natural numbers are:

$2,4,6,\dots,100$

This is an arithmetic progression with:

• first term $a=2$
• common difference $d=2$
• number of terms $n=50$

2. Use variance formula for an arithmetic progression

These numbers can be written as:

$x_i = 2i \quad (i=1,2,\dots,50)$

If we know the variance of $1,2,3,\dots,n$, then multiplying every term by $2$ multiplies the variance by $2^2=4$.

Now,

$\operatorname{Var}(1,2,3,\dots,n)=\frac{n^2-1}{12}$

For $n=50$,

\operatorname{Var}(1,2,3,\dots,50)=\frac{50^2-1}{12}=rac{2499}{12}

Therefore,

$\operatorname{Var}(2,4,6,\dots,100)=4\cdot \frac{2499}{12} = \frac{2499}{3}=833$

3. Option check

• A: $833$ ✅
• B: $437$ ❌
• C: $\dfrac{437}{4}$ ❌
• D: $\dfrac{833}{4}$ ❌

So the correct answer is:

$\boxed{833}$