Grouped Frequency Distributions Revision Notes for Leaving Cert Mathematics

Grouped Frequency Distributions

What are grouped frequency distributions?

When working with large amounts of data, it becomes impractical to list every individual value. Grouped frequency distributions organise data into classes or groups to make analysis more manageable.

For example, instead of recording each person's exact age, we might group ages into ranges like 0-9 years, 10-19 years, and so on. This approach is particularly useful when dealing with continuous data or large datasets.

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Grouped frequency distributions are especially valuable when:

You have more than 20-30 data points
The data spans a wide range of values
You want to identify general patterns rather than specific values
The data is continuous (like heights, weights, or time measurements)

The table above shows how student test marks can be grouped into class intervals, making it easier to see patterns in the data.

Finding the mean of grouped data

Mid-interval values

Since we don't know the exact values within each class, we use the mid-interval value (also called the class midpoint) to represent each group.

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Mid-interval value formula:

$\text{Mid-interval value} = \frac{\text{Lower boundary} + \text{Upper boundary}}{2}$

For example:

Class 1-5: Mid-interval value = $(1 + 5) ÷ 2 = 3$
Class 6-10: Mid-interval value = $(6 + 10) ÷ 2 = 8$

Mean formula for grouped data

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Mean formula for grouped data:

$\text{Mean} = \frac{\sum fx}{\sum f}$

Where:

f = frequency of each class
x = mid-interval value of each class
Σfx = sum of (frequency × mid-interval value) for all classes
Σf = total frequency

Worked example: calculating the mean

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Worked Example: Calculating the Mean from Grouped Data

Using the student marks data:

Class	Mid-interval (x)	Frequency (f)	fx
1-5	3	11	33
6-10	8	12	96
11-15	13	15	195
16-20	18	9	162
21-25	23	3	69
Totals		50	555

Step 1: Calculate mid-interval values for each class

Step 2: Multiply each mid-interval value by its frequency (fx)

Step 3: Sum all fx values: $\sum fx = 555$

Step 4: Sum all frequencies: $\sum f = 50$

Step 5: Apply the formula: $\text{Mean} = \frac{555}{50} = 11.1$

The estimated mean mark is 11.1 out of 25.

Finding the mode and median in grouped data

Modal class

For grouped data, we cannot find an exact mode. Instead, we identify the modal class - the class interval with the highest frequency.

In this phone sales data, the modal class is 15-19 phones because it has the highest frequency (9).

Median class

We cannot find an exact median value from grouped data, but we can identify the median class - the class interval containing the median position.

Steps to find the median class:

Calculate the total frequency (n)
Find the median position = $(n + 1) ÷ 2$
Count cumulative frequencies until you reach the median position
The class containing this position is the median class

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Worked Example: Finding the Median Class

Using the phone sales data:

Class	Frequency	Cumulative frequency
0-4	5	5
5-9	8	13
10-14	4	17
15-19	9	26
20-24	3	29

Step 1: Total frequency = 29

Step 2: Median position = $(29 + 1) ÷ 2 = 15$ th value

Step 3: Build cumulative frequency column

Step 4: The 15th value lies in the 10-14 class (since cumulative frequency reaches 17 at this class)

Therefore, the median class is 10-14 phones.

Worked example: complete analysis

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Worked Example: Complete Statistical Analysis

Table

Let's analyse this data showing number ranges and people:

Finding the modal class: The class 4-6 has the highest frequency (17), so this is the modal class.

Finding the mean:

Class	Mid-interval (x)	Frequency (f)	fx
1-3	2	3	6
4-6	5	17	85
7-9	8	2	16
10-12	11	8	88
Totals		30	195

Mean = $\frac{195}{30} = 6.5$

Finding the median class:

Total frequency = 30
Median position = $(30 + 1) ÷ 2 = 15.5$ th value
Cumulative frequencies: 3, 20, 22, 30
The 15.5th value lies in the 4-6 class

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Exam Tips:

Always use mid-interval values when calculating the mean for grouped data
Remember that you can only identify modal and median classes, not exact values
Check your arithmetic carefully - errors in $\sum fx$ calculations are common
Show all working clearly, including the mid-interval value calculations
Don't forget to divide by the total frequency when finding the mean

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Key Points to Remember:

Grouped data organises large datasets into class intervals for easier analysis
Use mid-interval values to represent each class when calculating statistics
Mean formula: $\frac{\sum fx}{\sum f}$ where x = mid-interval value and f = frequency
Modal class is the interval with the highest frequency
Median class contains the middle value position, found using cumulative frequencies

Grouped Frequency Distributions (Leaving Cert Mathematics): Revision Notes