### Theory:

Let us look at an example to find standard deviation of ungrouped data by direct method.
Example:
Find the standard deviation of the data $$5$$, $$8$$, $$10$$, $$11$$ and $$9$$ by direct method.

Explanation:

Let $$n$$ represent the number of values in the given data.

$$n$$ $$=$$ $$5$$

Let $$x_{i}$$ represent the each value of the data.

 $$x_{i}$$ $$x_{i}^{2}$$ $$5$$ $$25$$ $$8$$ $$64$$ $$10$$ $$100$$ $$11$$ $$121$$ $$9$$ $$81$$ $$\sum x_{i} = 43$$ $$\sum x_{i}^{2} = 391$$
The formula to calculate the standard deviation by direct method is given by:

$$\sigma = \sqrt{\frac{\sum x_{i}^{2}}{n}- \left(\frac{\sum x_{i}}{n}\right)^2}$$
Substitute the known values in the above formula.

$$\sigma = \sqrt{\frac{391}{5}- \left(\frac{43}{5}\right)^2}$$

$$=$$ $$\sqrt{78.2 - (8.6)^2}$$

$$=$$ $$\sqrt{78.2 - 73.96}$$

$$=$$ $$\sqrt{4.24}$$

$$=$$ $$2.0591$$

$$\approx$$ $$2.06$$

Therefore, the standard deviation of the given data is $$2.06$$.