### Theory:

Let us look at an example to find standard deviation of ungrouped data by assumed mean method.
Example:
Find the standard deviation of the data $$5$$, $$8$$, $$10$$, $$11$$ and $$12$$ which represents the number cookies in $$5$$ bottles by assumed 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 $$\overline x = \frac{\text{Sum of all the observations}}{\text{Total number of observations}}$$
$$\overline x$$ $$=$$ $$\frac{5 + 8 + 10 + 11 + 12}{5}$$

$$=$$ $$\frac{46}{5}$$

$$=$$ $$9.2$$

Here, the mean is not an integer value.

So, let us find the standard deviation by assumed mean method.

Let $$A$$ be the assumed mean, which is the middle most value.

Here, $$A$$ $$=$$ $$10$$

Let $$x_{i}$$ represent the marks scored by each student.

 $$x_{i}$$ $$d_{i} = x_{i} - A$$ $$=$$ $$x_{i} - 10$$ $$d_{i}^{2}$$ $$5$$ $$-5$$ $$25$$ $$8$$ $$-2$$ $$4$$ $$10$$ $$0$$ $$0$$ $$11$$ $$1$$ $$1$$ $$12$$ $$2$$ $$4$$ $$\sum d_{i} = -4$$ $$\sum d_{i}^{2} = 34$$
The  formula to calculate the standard deviation by assumed mean method is given by:

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

$$\sigma = \sqrt{\frac{34}{9}- \left(\frac{-4}{9}\right)^2}$$

$$=$$ $$\sqrt{3.778 - (-0.444)^2}$$

$$=$$ $$\sqrt{3.778 - 0.198}$$

$$=$$ $$\sqrt{3.58}$$

$$=$$ $$1.892$$

$$\approx$$ $$1.89$$

Therefore, the standard deviation of the given data is $$1.89$$.