Grouped Frequency Tables Revision Notes for Edexcel GCSE Maths

Grouped frequency tables

What are grouped frequency tables?

Grouped frequency tables are a way to organise data into different ranges or categories called classes. They look similar to ordinary frequency tables, but instead of individual values, they show ranges of values. This makes it easier to handle large amounts of data by grouping similar values together.

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Grouping data is particularly useful when dealing with continuous data or large datasets where listing every individual value would be impractical and difficult to interpret.

Setting up non-overlapping classes

When creating grouped frequency tables, it's essential to use inequality symbols correctly to ensure all possible values are covered without any overlap between classes.

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Essential Rules for Creating Classes:

Each class must be set up so that:

No value can fall into more than one class
All possible values within the data range are covered
The classes don't leave any gaps

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For example, if you have classes like $5 \< h ≤ 10$ and $10 \< h ≤ 15$ , notice how the value 10 would go in the first class (since $h ≤ 10$ ), whilst values greater than 10 would go in the second class.

Calculating mid-interval values

Since grouped data doesn't show exact values, we need to find the middle point of each class interval, called the mid-interval value. This is calculated by adding the two endpoints of the class and dividing by 2.

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

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Worked Example: Finding Mid-interval Value

For a class from 5 to 10: $\text{Mid-interval value} = \frac{5 + 10}{2} = 7.5$

These mid-interval values are crucial for estimating averages from grouped data.

Finding the modal class

The modal class is the class interval that contains the most frequently occurring values. Unlike finding the mode in ordinary frequency tables, you cannot determine an exact modal value from grouped data.

Here's the table written out clearly:

Weight (w kg)	Frequency
30 < w ≤ 40	8
40 < w ≤ 50	16

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Steps to identify the modal class:

Look at the frequency column
Find the class with the highest frequency
This class is your modal class

Locating the median class

The median class is the class interval that contains the middle value when all data points are arranged in order. For grouped data, you need to:

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Finding the Median Class:

Calculate the position of the median (for $n$ values, this is the $\frac{n+1}{2}$ position)
Count through the frequencies to find which class contains this position
This class becomes your median class

Estimating the mean from grouped data

Unlike ordinary frequency tables where you know exact values, grouped data requires estimation of the mean. This involves creating additional columns to help with calculations.

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The Process for Estimating Mean:

Adding a third column for mid-interval values
Adding a fourth column for frequency × mid-interval value
Calculating the total of the fourth column
Dividing this total by the total frequency

Here's how this works in practice:

$\text{Mean} = \frac{\sum(f \times \text{mid-interval value})}{\sum f}$

Length (l cm)	Frequency
15.5 ≤ l < 16.5	12
16.5 ≤ l < 17.5	18
17.5 ≤ l < 18.5	23
18.5 ≤ l < 19.5	8

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The calculation gives you an estimate rather than an exact mean because you're using the mid-interval values to represent all data points within each class.

Worked example

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Worked Example: Complete Analysis of Weight Data

Let's look at a complete example using weight data for 60 school children:

For the modal class: Look for the highest frequency in the table. The class $50 \< w ≤ 60$ has the highest frequency (18), so this is the modal class.

For the median class: With 60 values, the median lies between the 30th and 31st values. Counting through the frequencies shows both these positions fall in the $50 \< w ≤ 60$ class.

For the mean estimation: Create additional columns for mid-interval values and frequency × mid-interval value. The calculation becomes: $\text{Mean} = \frac{3220}{60} = 53.7 \text{ kg (to 3 significant figures)}$

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Key Points to Remember:

Grouped frequency tables organise data into class intervals rather than individual values
Use inequality symbols carefully to create non-overlapping classes that cover all possible values
Mid-interval values are found by adding class endpoints and dividing by 2
The modal class is simply the class with the highest frequency
The median class contains the middle value(s) when data is arranged in order
Mean estimation requires additional columns and gives an approximate rather than exact result

Grouped Frequency Tables (Edexcel GCSE Maths): Revision Notes