Grouped data: estimated mean, modal class and median class

Q: What is grouped data and why is it used?

Grouped data is data organized into "bins" or "classes" (class intervals) to make large datasets easier to handle and analyze.

Q: How do you find the midpoint (\(x\)) of a class interval?

The midpoint is found by adding the start and end values of the class interval together and dividing by 2: \(x = \frac{\text{start} + \text{end}}{2}\).

Q: What is the formula for the estimated mean of grouped data?

The formula is: \(\text{Estimated Mean} = \frac{\sum fx}{\sum f}\), where \(f\) is frequency and \(x\) is the midpoint.

Q: How do you identify the modal class in a grouped frequency table?

The modal class is the class interval that has the highest frequency (\(f\)).

Q: What is a common mistake when identifying the modal class?

The most common mistake is writing down the frequency number instead of the actual class interval (e.g., writing '12' instead of '\(10 < x \le 20\)').

Q: How do you determine which class contains the median?

Find the middle position (\(\frac{n}{2}\)) and use a running total of frequencies to identify the class where that position falls.

Q: What is a quick way to check if your estimated mean is likely correct?

Check if the calculated mean falls within the range of the smallest and largest values in your data table.

Mathematics (Specification B) · Pearson Edexcel IGCSE · 4 min read · Updated: Sep 7, 2026

Welcome to Grouped Data!

In your previous studies, you probably learned how to find the average height of five friends. But what if you had to find the average height of 500 students in a school? Writing down every single number would take forever! This is where grouped data comes in. We put the numbers into "bins" or "classes" to make them easier to handle.

In this chapter, we will learn how to estimate the mean and identify the modal class and median class from these groups. Don't worry if it seems like a lot of numbers at first; we will break it down into simple, repeatable steps.

Note: For basic mean, median, and mode of single numbers, see the "Mean, median and mode" chapter.

1. Understanding Grouped Frequency Tables

When data is grouped, it is usually presented in a table with class intervals. For example:

Height (\(h\)) in cm: \(140 < h \le 150\)
Frequency (\(f\)): \(4\)

This means there are 4 students whose heights are somewhere between 140 cm and 150 cm. Because we don't know their exact heights (one could be 141 cm, another 149 cm), we have to use estimates for our calculations.

2. The Estimated Mean

Since we don't know the exact values in a group, we make a sensible guess: we assume every value in that group is exactly in the middle. This is called the midpoint.

Step-by-Step: Calculating the Estimated Mean

Step 1: Find the Midpoint (\(x\)) for each class.
Add the start and end of the class together and divide by 2.
Example: For the class \(10 < w \le 20\), the midpoint \(x = \frac{10 + 20}{2} = 15\).

Step 2: Multiply the Frequency by the Midpoint (\(f \times x\)).
This gives us an estimate of the total "weight" of that group.

Step 3: Find the Total Frequency (\(\sum f\)).
Add up all the frequencies to see how many pieces of data you have in total.

Step 4: Find the Total of the \(fx\) column (\(\sum fx\)).
Add up all your results from Step 2.

Step 5: Divide!
The formula for the estimated mean is:
\(\text{Estimated Mean} = \frac{\sum fx}{\sum f}\)

Quick Review: Why is it called an "estimate"? Because we used midpoints rather than the actual raw data. It’s not a mistake; it’s just the best guess we can make with the information provided!

The "Mode" usually means the most common value. In grouped data, we look for the Modal Class.

This is the simplest thing to find! Just look down the Frequency (\(f\)) column. The class with the highest frequency is the modal class.

Example:
Class A: Frequency \(5\)
Class B: Frequency \(12\)
Class C: Frequency \(7\)
The modal class is Class B because \(12\) is the highest number.

Common Mistake: Students often write down the frequency (12) as the answer. Don't do that! The answer is the class interval itself (e.g., \(10 < x \le 20\)).

4. The Median Class

The median is the middle value. To find which class the middle value sits in, we follow these steps:

Step 1: Find the total frequency (\(n\)).
This is the same as \(\sum f\).

Step 2: Find the middle position.
Calculate \(\frac{n}{2}\). (For example, if there are 40 items, the middle is the 20th item).

Step 3: Track the running total.
Look at the frequencies and keep adding them up until you reach the middle position. The class where you "hit" or "pass" that middle number is the median class.

Did you know? This "running total" is often called cumulative frequency, but for this specific syllabus, you just need to be able to identify the class containing the median without drawing a graph.

5. Summary Table (The "All-in-One" Method)

When you see a grouped data question, it is often helpful to add columns to the table provided in the exam. It usually looks like this:

Class Interval | Frequency (\(f\)) | Midpoint (\(x\)) | \(f \times x\)
\(0 < t \le 10\) | \(3\) | \(5\) | \(15\)
\(10 < t \le 20\) | \(7\) | \(15\) | \(105\)
\(20 < t \le 30\) | \(2\) | \(25\) | \(50\)
Totals: | \(\sum f = 12\) | | \(\sum fx = 170\)

From this table:
1. Estimated Mean: \(170 \div 12 = 14.17\) (to 2 decimal places).
2. Modal Class: \(10 < t \le 20\) (it has the highest frequency of 7).
3. Median Class: Total \(n=12\), so the middle is the 6th value. The first class has 3 values. The second class adds 7 more (total 10). Since 6 is between 4 and 10, the median is in the \(10 < t \le 20\) class.

Top Tips for Success

1. Watch the inequalities: Sometimes classes look like \(0 \le x < 10\) and sometimes like \(1 - 10, 11 - 20\). Always find the exact middle. For \(1 - 10\), the midpoint is \(\frac{1 + 10}{2} = 5.5\).

2. Calculator check: If your "estimated mean" is a number that isn't even inside your table's range, you've made a calculation error! The mean must be somewhere between your smallest and largest class values.

3. Show your working: In Paper 2 especially, you get marks for showing the \(\sum fx\) and the \(\sum f\) even if your final division is wrong.

4. Don't overcomplicate: You do not need to draw cumulative frequency graphs or calculate moving means for this specification. Stay focused on the table!

Key Takeaway Checklist

- [ ] Midpoint \(x = (\text{start} + \text{end}) \div 2\)
- [ ] Estimated Mean \(= \sum fx \div \sum f\)
- [ ] Modal Class = Highest frequency
- [ ] Median Class = Class containing the \(\frac{n}{2}\) position

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is grouped data and why is it used?

Grouped data is data organized into "bins" or "classes" (class intervals) to make large datasets easier to handle and analyze.

How do you find the midpoint (\(x\)) of a class interval?

The midpoint is found by adding the start and end values of the class interval together and dividing by 2: \(x = \frac{\text{start} + \text{end}}{2}\).

What is the formula for the estimated mean of grouped data?

The formula is: \(\text{Estimated Mean} = \frac{\sum fx}{\sum f}\), where \(f\) is frequency and \(x\) is the midpoint.

How do you identify the modal class in a grouped frequency table?

The modal class is the class interval that has the highest frequency (\(f\)).

What is a common mistake when identifying the modal class?

The most common mistake is writing down the frequency number instead of the actual class interval (e.g., writing '12' instead of '\(10 < x \le 20\)').

How do you determine which class contains the median?

Find the middle position (\(\frac{n}{2}\)) and use a running total of frequencies to identify the class where that position falls.

What is a quick way to check if your estimated mean is likely correct?

Check if the calculated mean falls within the range of the smallest and largest values in your data table.

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