Grouped Frequency Distributions Revision Notes for Leaving Cert Mathematics

Grouped Frequency Distributions

Overview

A grouped frequency distribution is a method of organising data into intervals (called classes or bins) to summarise and display large data sets more effectively. Each interval is associated with a frequency, which indicates the number of data points that fall within that range.

Key Components of a Grouped Frequency Distribution

Class Intervals:

Non-overlapping ranges that divide the data.
Intervals are usually of equal width for simplicity.
Example: $10-20, 20-30, 30-40$

Frequency ( $f$ ):

The number of data points within each class interval.

Midpoint ( $x$ ):

The central value of a class interval:

\text{Midpoint} = \frac{\text{Lower Bound} + \text{Upper Bound}}{2}

Cumulative Frequency:

The running total of frequencies, showing how many data points fall below the upper limit of each interval.

Relative Frequency:

Proportion of data in each interval, expressed as a fraction or percentage:

\text{Relative Frequency} = \frac{\text{Frequency}}{\text{Total Frequency}}

Steps to Create a Grouped Frequency Distribution

Step 1: Determine Range:

Subtract the smallest data value from the largest.

Step 2: Choose the Number of Intervals:

Commonly, use between 5 and 10 intervals depending on the data set size.

Step 3: Calculate Interval Width:

\text{Interval Width} = \frac{\text{Range}}{\text{Number of Intervals}}

Step 4: Create the Intervals:

Start from the smallest data value and add the interval width to form consecutive intervals.

Step 5: Tally Frequencies:

Count how many data points fall into each interval.

Worked Examples

infoNote

Example 1: Creating a Grouped Frequency Table

Problem: The ages of 20 people are: $12, 13, 15, 16, 16, 17, 18, 19, 19, 20, 21, 22, 23, 24, 24, 25, 26, 27, 28, 30.$

Solution:

Step 1: Range:

\text{Range} = 30 - 12 = :highlight[18]

Step 2: Intervals:

Choose 5 intervals:

\text{Interval Width} = \frac{18}{5} \approx :highlight[4]

Step 3: Create Intervals:

12−15,16−19,20−23,24−27,28−31

Step 4: Tally Frequencies:

Count how many ages fall into each interval.

Grouped Frequency Table:

Interval	Frequency ( $f$ )	Midpoint ( $x$ )	Cumulative Frequency
12–15	3	13.5	3
16–19	5	17.5	8
20–23	4	21.5	12
24–27	6	25.5	18
28–31	2	29.5	20

infoNote

Example 2: Calculating Relative Frequency

Problem: Use the table above to find the relative frequency of each interval.

Solution:

Step 1: Calculate Total Frequency

$(\sum f): :highlight[20]$

Step 2: Calculate Relative Frequency for each interval:

$12−15: \frac{3}{20} = :highlight[0.15]$ (15%)
$16-19: \frac{5}{20} = :highlight[0.25]$ (25%)
$20−23: \frac{4}{20} = :highlight[0.20]$ (20%)
$24-27: \frac{6}{20} = :highlight[0.30]$ (30%)
$28-31: \frac{2}{20} = :highlight[0.10]$ (10%)

Relative Frequency Table:

Interval	Frequency ( $f$ )	Relative Frequency
12–15	3	0.15
16–19	5	0.25
20–23	4	0.20
24–27	6	0.30
28–31	2	0.10

Summary

A grouped frequency distribution organises data into intervals, making it easier to analyse large data sets.
Key components:
- Class Intervals, Frequencies, Midpoints, Cumulative Frequency, and Relative Frequency.
Steps:
1. Calculate the range and interval width.
2. Create class intervals.
3. Count and record frequencies.
4. Calculate midpoints and relative frequencies.
Grouped frequency distributions simplify patterns and trends in data.

Grouped Frequency Distributions (Leaving Cert Mathematics): Revision Notes