1. Given data

We have $10$ observations $a_1,a_2,\dots,a_{10}$ such that $\sum_{k=1}^{10} a_k = 50$ and $\sum_{k<j} a_k a_j = 1100.$

We need the standard deviation.

2. Find $\sum a_k^2$ using the identity

Use $\left(\sum_{k=1}^{10} a_k\right)^2 = \sum_{k=1}^{10} a_k^2 + 2\sum_{k<j} a_k a_j.$

Substitute the given values: $50^2 = \sum_{k=1}^{10} a_k^2 + 2(1100).$

So, $2500 = \sum a_k^2 + 2200,$ which gives $\sum_{k=1}^{10} a_k^2 = 300.$

3. Compute the mean

For $n=10$ observations, the mean is $\bar a = \frac{\sum a_k}{10} = \frac{50}{10} = 5.$

4. Compute the variance

Variance of $n$ observations is $\sigma^2 = \frac{1}{n}\sum a_k^2 - \bar a^2.$

Substitute values: $\sigma^2 = \frac{300}{10} - 5^2 = 30 - 25 = 5.$

5. Compute the standard deviation

$\sigma = \sqrt{5}.$

6. Match with options

The correct option is:

$\boxed{\sqrt{5}}$

So, Option D is correct.