### Theory:

Let us look at an example to find standard deviation of ungrouped data by mean method.
Example:
Find the standard deviation of the data $$5.4$$, $$8.9$$, $$10.1$$, $$11.4$$ and $$9.2$$ by mean method.

Explanation:

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

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

Let $$\overline x$$ represent the mean of the given data.
Mean $$\overline x = \frac{\text{Sum of all the observations}}{\text{Total number of observations}}$$
$$\overline x$$ $$=$$ $$\frac{5.4 + 8.9 + 10.1 + 11.4 + 9.2}{5}$$

$$=$$ $$\frac{45}{5}$$

$$=$$ $$9$$

Let $$x_{i}$$ represent each values in the given data.

 $$x_{i}$$ $$d_{i} = x_{i} - \overline x$$ $$=$$ $$x_{i} - 9$$ $$d_{i}^{2}$$ $$5.4$$ $$-3.6$$ $$12.96$$ $$8.9$$ $$-0.1$$ $$0.01$$ $$10.1$$ $$1.1$$ $$1.21$$ $$11.4$$ $$2.4$$ $$5.76$$ $$9.2$$ $$0.2$$ $$0.04$$ $$\sum d_{i}^{2} = 19.98$$
The formula to calculate the standard deviation by mean method is given by:

$$\sigma$$ $$=$$ $$\sqrt{\frac{\sum d_{i}^{2}}{n}}$$ where $$d_{i} = x_{i} - \overline{x}$$
Substitute the known values in the above formula.

$$\sigma$$ $$=$$ $$\sqrt{\frac{19.98}{5}}$$

$$=$$ $$\sqrt{3.996}$$

$$=$$ $$1.999$$

$$\approx$$ $$2$$

Therefore, the standard deviation for the given data is $$2$$.