When data is grouped into class intervals, the exact values are not known. This means the mean cannot be found exactly, but an estimate can be calculated using class midpoints.

 

 

Understanding Grouped Data

A grouped frequency distribution shows:
• class intervals
• the frequency in each interval

 

Because individual values are unknown, assumptions must be made about where values lie within each interval.

 

To estimate the mean, each class interval is represented by its midpoint.

 

The midpoint is assumed to represent all values in that class

 

 

Finding Class Midpoints

The midpoint of a class interval is found by adding the lower and upper boundaries and dividing by 2.

 

For example, for a class interval from 10 to 20, the midpoint represents the centre of that interval.

 

Each class interval has exactly one midpoint.

 

 

Steps to Estimate the Mean

To estimate the mean of grouped data:

• find the midpoint of each class interval
• multiply each midpoint by its frequency
• add all the results together
• add all the frequencies together
• divide the total of midpoint multiplied by frequency by the total frequency

 

The calculation follows this structure:

$$
estimated\ mean = \frac{\sum (midpoint \times frequency)}{\sum frequency}
$$

 

This formula must be applied carefully using all classes.

 

 

Why the Mean Is Only an Estimate

The mean is an estimate because:
• exact data values are unknown
• all values in a class are assumed to be at the midpoint

 

If data is unevenly spread within a class interval, the estimate may be less accurate.

 

Smaller class widths usually give a better estimate.

 

 

Using the Estimated Mean

The estimated mean can be used to:
• compare two grouped data sets
• describe the centre of grouped data
• support conclusions in statistical problems

 

However, it should not be treated as an exact value.

 

Always state that the mean is an estimate

 

 

Common Errors to Avoid

Common mistakes include:
• forgetting to calculate midpoints
• multiplying frequency by class boundaries instead of midpoints
• dividing by the number of classes instead of total frequency
• claiming the mean is exact

 

Careful organisation avoids these errors.

 

 

Key Points to Remember

Grouped data does not allow an exact mean.
Class midpoints are used to represent each interval.
Each midpoint is multiplied by its frequency.
The total is divided by the total frequency.
The result is an estimate, not an exact mean.

 

Estimating the mean of grouped data allows large data sets to be summarised sensibly while recognising the limitations of grouped information.