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

 

 

Understanding the Median in Grouped Data

The median is the middle value when all data values are arranged in order.

 

For grouped data:
• individual values are unknown
• only the number of values in each class is known

 

Because of this, the median is estimated by locating the median class, which is the class interval that contains the middle value.

 

Total frequency determines where the middle lies

 

 

Step 1: Find the Total Frequency

Add together all the frequencies in the table.

 

This gives the total number of data values.

 

The median position is halfway through this total.

 

 

Step 2: Find the Median Position

To locate the median:
• find half of the total frequency

 

This gives the position of the median within the ordered data.

 

If the total frequency is even, the median lies halfway between the two central positions, but for grouped data this is still handled by finding the median class.

 

 

Step 3: Use Cumulative Frequency

Cumulative frequency shows the total number of values up to each class interval.

 

To find the median class:
• calculate cumulative frequencies
• identify the first class where the cumulative frequency is greater than or equal to the median position

 

This class is the median class.

 

The median lies somewhere within this class interval

 

 

Step 4: Estimate the Median Value

Once the median class is identified, the median is estimated by assuming the data is evenly spread across that class.

 

The estimate is found by:
• starting at the lower boundary of the median class
• moving part of the way through the class, depending on how far the median position lies into it

 

This gives an approximate value for the median.

 

Because this relies on an assumption of even spread, the result is only an estimate.

 

 

Why the Median Is an Estimate

The median is estimated because:
• exact data values within each class are unknown
• values are assumed to be evenly distributed within the class

 

Narrower class intervals usually give a more accurate estimate.

 

Always state that the median is an estimate

 

 

Common Errors to Avoid

Common mistakes include:
• forgetting to use cumulative frequency
• choosing the class with the highest frequency instead of the median class
• using class midpoints instead of boundaries
• stating the median as an exact value

 

Working methodically helps avoid these errors.

 

 

Key Points to Remember

The median of grouped data cannot be found exactly.
Find the total frequency first.
Use cumulative frequency to locate the median class.
Estimate the median from within that class.
The final answer is an estimate, not an exact value.

 

Estimating the median from grouped frequency data allows the centre of large data sets to be described sensibly while recognising the limitations of grouped information.