## Mean, median and mode for grouped data

Let $(x_i,f_i), i=1,2, \cdots , n$ be given frequency distribution.

## Sample mean

The mean of $X$ is denoted by $\overline{x}$ and is given by

### $\overline{x} =\dfrac{1}{N}\sum_{i=1}^{n}f_ix_i$

In case of continuous frequency distribution, $x_i$’s are the mid-values of the respective classes.

## Sample median

The median is given by

### $\text{Median } = l + \bigg(\dfrac{\frac{N}{2} - F_<}{f}\bigg)\times h$

where

• $N$, total number of observations
• $l$, the lower limit of the median class
• $f$, frequency of the median class
• $F_<$, cumulative frequency of the pre median class
• $h$, the class width

## Sample mode

The mode of the distribution is given by

### $\text{Mode } = l + \bigg(\dfrac{f_m - f_1}{2f_m-f_1-f_2}\bigg)\times h$

where

• $l$, the lower limit of the modal class
• $f_m$, frequency of the modal class
• $f_1$, frequency of the class pre-modal class
• $f_2$, frequency of the class post-modal class
• $h$, the class width