Measures of central tendency from grouped data

When data is grouped into intervals (classes), we can still calculate measures of central tendency (mean, median, and mode) using approximations. These methods are slightly different from those used with ungrouped data.

Mean from grouped data

To calculate the mean from grouped data, we use the midpoint of each class interval (denoted as \( x_i \)) and the frequency \( f_i \) of each class. The mean is calculated using the formula:

\[ \text{Mean} = \frac{\sum (f_i x_i)}{\sum f_i} \]

Where:

• \( x_i \) is the midpoint of each class.
• \( f_i \) is the frequency of each class.

Median from grouped data

To find the median from grouped data, we first locate the median class (the class where the cumulative frequency is greater than or equal to half of the total frequency).

Then, we use the following formula:

\[ \text{Median} = L + \left( \frac{\frac{n}{2} - F}{f_m} \right) \times h \]

Where:

• \( L \) is the lower boundary of the median class.
• \( n \) is the total number of data points.
• \( F \) is the cumulative frequency before the median class.
• \( f_m \) is the frequency of the median class.
• \( h \) is the class width.

Modal class from grouped data

The modal class is the class with the highest frequency.