### Theory:

Let us look at an example to find standard deviation of a grouped data by assumed mean method.
Example:
Calculate the standard deviation of the following observations using assumed mean method.

 $$x$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$f$$ $$19$$ $$5$$ $$7$$ $$23$$ $$16$$ $$13$$

Explanation:

Let the assumed mean $$A$$ $$=$$ $$3$$.

Let us form a frequency distribution table.

 $$x_{i}$$ $$f_{i}$$ $$d_{i} = x_{i} - A$$ $$=$$ $$x_{i} - 3$$ $$f_{i} d_{i}$$ $$d_{i}^{2}$$ $$f_{i}d_{i}^{2}$$ $$1$$ $$19$$ $$-2$$ $$-38$$ $$4$$ $$76$$ $$2$$ $$5$$ $$-1$$ $$-5$$ $$1$$ $$5$$ $$3$$ $$7$$ $$0$$ $$0$$ $$0$$ $$0$$ $$4$$ $$23$$ $$1$$ $$23$$ $$1$$ $$23$$ $$5$$ $$16$$ $$2$$ $$32$$ $$4$$ $$64$$ $$6$$ $$13$$ $$3$$ $$39$$ $$9$$ $$117$$ $$\sum_{i = 1}^{6} f_{i} = 83$$ $$\sum_{i = 1}^{6} f_{i} d_{i}$$ $$=$$ $$51$$ $$\sum_{i = 1}^{6} f_{i} d_{i}^{2} = 285$$
The  formula to calculate the standard deviation by assumed mean method is given by:

$$\sigma = \sqrt{\frac{\sum f_{i} d_{i}^{2}}{N}- \left(\frac{\sum f_{i} d_{i}}{N}\right)^2}$$ where $$N = \sum_{i = 1}^{n} f_{i}$$.
Substitute the required values in the above formula.

$$\sigma = \sqrt{\frac{285}{83}- \left(\frac{51}{83}\right)^2}$$

$$=$$ $$\sqrt{3.434 - \left(0.614 \right)^2}$$

$$=$$ $$\sqrt{3.434 - 0.378}$$

$$=$$ $$\sqrt{3.056}$$

$$=$$ $$1.748$$

$$\approx$$ $$1.75$$

Therefore, the standard deviation of the given data is $$1.75$$.