Estimated Median (Grouped Data)

Statement

The estimated median for grouped data allows us to approximate the middle value when data is arranged in class intervals. Since we do not know the exact individual values within each interval, we use a formula based on class boundaries, cumulative frequencies, and class width to estimate the median.

\[\text{Median} = L + \left(\frac{\tfrac{N}{2} - \text{CF}}{f}\right)\times w\]

Where:

• \(L\) = lower boundary of the median class
• \(N\) = total frequency (number of data values)
• \(\text{CF}\) = cumulative frequency before the median class
• \(f\) = frequency of the median class
• \(w\) = class width of the median class

Why it works

• The median is the value that splits the dataset into two equal halves. For raw data, this is straightforward: order the numbers and find the middle one. For grouped data, exact positions are hidden inside intervals, so we estimate.
• We identify the interval where the middle value must lie by checking cumulative frequencies. Then, we proportionally interpolate within that class to estimate where the median sits.

Recipe (Steps to Apply)

1. Add up all frequencies to find \(N\).
2. Calculate \(N/2\) to find the position of the median.
3. Use the cumulative frequency column to locate the median class (the class where the \(N/2\)th value lies).
4. Identify: \(L\), \(\text{CF}\), \(f\), and \(w\) from the table.
5. Substitute these values into the formula.
6. Simplify to find the estimated median.

Spotting it

You use this formula whenever you are asked to estimate the median of data given in a grouped frequency table, rather than as a raw list of numbers.

Common pairings

• Estimated mean (using class midpoints).
• Estimated mode (using modal class formula or frequency comparison).
• Cumulative frequency diagrams (graphical approach to medians).

Mini examples

1. Given: A frequency table of student ages. Find: Median age. Answer: 15.3 years (estimated).
2. Given: Heights grouped into 10 cm intervals. Find: Estimated median height. Answer: 165 cm.

Pitfalls

• Forgetting cumulative frequency: Always calculate it carefully.
• Using class midpoint instead of boundary: The formula requires the lower boundary of the median class, not the midpoint.
• Mixing inclusive vs exclusive boundaries: Be consistent (e.g., 10–20 means lower boundary 10, upper 20; width \(w=10\)).
• Arithmetic slips: Fractions inside the formula must be handled step by step.

Exam Strategy

• Write the formula at the top of your working; examiners reward correct method even if the arithmetic slips.
• Show \(N/2\), the median class, and identified parameters clearly.
• Keep one decimal place in your final answer unless told otherwise.

Worked Micro Example

Suppose you have the grouped data for shoe sizes:

Total \(N=20\). Halfway point \(=10\). The 10th value lies in the 6–8 interval, so that is the median class.

Here, \(L=6\), \(w=3\), \(\text{CF}=4\), \(f=10\).

So, \(\text{Median} = 6 + \frac{10-4}{10}\times 3 = 6 + 1.8 = 7.8\).

Summary

The estimated median formula provides a reliable approximation of the middle of grouped data. By locating the median class through cumulative frequency and interpolating within that class, we overcome the problem of not knowing the raw data. With practice, the calculation becomes straightforward and is a frequent exam question in GCSE statistics and data handling.