Measures of Central Tendency
1. Mean ($\bar{x}$) of Grouped Data:
- Direct Method: $\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$
- Assumed Mean Method: $\bar{x} = a + \frac{\sum f_i d_i}{\sum f_i}$ (where $d_i = x_i - a$)
2. Mode of Grouped Data:
$$\text{Mode} = l + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right) \times h$$
- $l =$ lower limit of modal class (class with maximum frequency $f_1$).
- $f_0 =$ frequency of class preceding modal class.
- $f_2 =$ frequency of class succeeding modal class.
- $h =$ class size.
3. Median of Grouped Data:
$$\text{Median} = l + \left(\frac{\frac{n}{2} - cf}{f}\right) \times h$$
Empirical Relationship:
$$3\,\text{Median} = \text{Mode} + 2\,\text{Mean}$$
