Key Points

For grouped data, the mean is calculated using the formula $\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$. Here, $f_i$ is the frequency of the $i$-th class and $x_i$ is its class mark (mid-point).

The class mark of a class interval is its mid-point, which represents all values within that class. It is calculated as Class Mark $= \frac{\text{Upper class limit} + \text{Lower class limit}}{2}$.

This method simplifies calculations with large numbers. The formula is $\bar{x} = a + \frac{\sum f_i d_i}{\sum f_i}$, where $a$ is the assumed mean and $d_i = x_i - a$ is the deviation of each class mark from the assumed mean.

This is a further simplification of the assumed mean method, used when class sizes are equal. The formula is $\bar{x} = a + h \left( \frac{\sum f_i u_i}{\sum f_i} \right)$, where $h$ is the class size, $a$ is the assumed mean, and $u_i = \frac{x_i - a}{h}$.

The mode for grouped data is the value with the highest frequency, found using the formula: Mode $= l + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h$.

In the mode formula, $l$ is the lower limit of the modal class, $h$ is the class size, $f_1$ is the frequency of the modal class, $f_0$ is the frequency of the class preceding the modal class, and $f_2$ is the frequency of the class succeeding the modal class.

The modal class is the class interval with the maximum frequency in a grouped frequency distribution. The mode of the data lies within this class interval.

The median for grouped data represents the middle value and is calculated using the formula: Median $= l + \left( \frac{\frac{n}{2} - \text{cf}}{f} \right) \times h$.

In the median formula, $l$ is the lower limit of the median class, $n$ is the total number of observations, $h$ is the class size, $f$ is the frequency of the median class, and $\text{cf}$ is the cumulative frequency of the class preceding the median class.

To find the median class, first calculate $\frac{n}{2}$, where $n$ is the total frequency. The median class is the class interval whose cumulative frequency is just greater than or equal to $\frac{n}{2}$.

The cumulative frequency of a class is the sum of the frequencies of that class and all classes preceding it. It is required to find the median class.

For a moderately skewed distribution, the three measures of central tendency are related by the empirical formula: $3 \times \text{Median} = \text{Mode} + 2 \times \text{Mean}$.

Before applying the formulas for median or mode, ensure the class intervals are continuous. For discontinuous classes like $50-52, 53-55$, convert them to continuous classes like $49.5-52.5, 52.5-55.5$.