### Theory:

Let us look at an example to find standard deviation of a grouped data by step deviation method.
Example:
The number of buckets required to fill given volume of water are given below:

 Volume (in litres) $$2 - 4$$ $$4 - 6$$ $$6 - 8$$ $$8 - 10$$ $$10 - 12$$ Number of buckets $$3$$ $$7$$ $$11$$ $$15$$ $$18$$

Find its standard deviation  using step deviation method

Explanation:

Let the assumed mean be $$A = 7$$ and the class width $$c = 2$$.

 Volume Number of buckets($$f_{i}$$) Midpoint($$x_{i}$$) Deviation$$d_{i} = \frac{x_{i} - A}{c}$$ $$d_{i}^{2}$$ $$f_{i}d_{i}$$ $$f_{i}d_{i}^{2}$$ $$2 - 4$$ $$3$$ $$3$$ $$-2$$ $$4$$ $$-6$$ $$12$$ $$4 - 6$$ $$7$$ $$5$$ $$-1$$ $$1$$ $$-7$$ $$7$$ $$6 - 8$$ $$11$$ $$7$$ $$0$$ $$0$$ $$0$$ $$0$$ $$8 - 10$$ $$15$$ $$9$$ $$1$$ $$1$$ $$15$$ $$15$$ $$10 - 12$$ $$18$$ $$11$$ $$2$$ $$4$$ $$36$$ $$72$$ $$\sum_{i = 1}^{5} f_{i} = 54$$ $$\sum_{i = 1}^{5} f_{i}d_{i} = 38$$ $$\sum_{i = 1}^{5} f_{i}d_{i}^{2} = 106$$
The formula to calculate the standard deviation by step deviation method is given by:

$$\sigma = c \times \sqrt{\frac{\sum f_{i} d_{i}^{2}}{N}- \left(\frac{\sum f_{i} d_{i}}{N}\right) ^2}$$ where $$N = \sum_{i = 1}^{n} f_{i}$$ and $$d_{i} = \frac{x_{i} - A}{c}$$.
Substituting the known values in the above formula, we have:

$$\sigma = 2 \times \sqrt{\frac{106}{54}- \left(\frac{38}{54}\right) ^2}$$

$$= 2 \times \sqrt{1.96 - \left(0.7 \right) ^2}$$

$$= 2 \times \sqrt{1.96 - 0.49}$$

$$= 2 \times \sqrt{1.47}$$

$$= 2 \times 1.212$$

$$=$$ $$2.424$$

$$\approx$$ $$2.42$$

Therefore, the standard deviation of the given data is $$2.42$$.