Mean, Mode and Median of a Frequency Distribution Revision Notes for Leaving Cert Mathematics

Mean, Mode and Median of a Frequency Distribution

Overview

To summarise a frequency distribution, measures of central tendency—mean, median, and mode—are often calculated. These measures help to identify the typical or central value in the data.

Mean

The mean of a frequency distribution is calculated using the formula:

\text{Mean} = \frac{\sum (f \times x)}{\sum f}

Where:

$f$ : Frequency of each class.
$x$ : Midpoint of each class (for grouped data).
$\sum f$ : Total frequency.

Steps:

Find the midpoint ( $x$ ) for each class:

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

Multiply the midpoint ( $x$ ) by the frequency ( $f$ ).
Sum all $f \times x$ values.
Divide by the total frequency ( $\sum f$ ).

Median

The median is the value below which $50$ % of the data lies. For a grouped frequency distribution, it is calculated using the formula:

\text{Median} = L + \left(\frac{\frac{N}{2} - CF}{f_m}\right) \times c

Where:

$L$ : Lower boundary of the median class.
$N$ : Total frequency ( $\sum f$ ).
$CF$ : Cumulative frequency before the median class.
$f_m$ : Frequency of the median class.
$c$ : Class width.

Steps:

Identify the median class:

The median class is the class where the cumulative frequency reaches or exceeds $\frac{N}{2}$

Use the formula to calculate the median.

Mode

The mode is the value or class with the highest frequency. For grouped data, it is calculated using the formula:

\text{Mode} = L + \left(\frac{f_m - f_{1}}{(f_m - f_{1}) + (f_m - f_{2})}\right) \times c

Where:

$L$ : Lower boundary of the modal class.
$f_m$ : Frequency of the modal class.
$f_{1}$ : Frequency of the class before the modal class.
$f_{2}$ : Frequency of the class after the modal class.
$c$ : Class width.

Steps:

Identify the modal class (class with the highest frequency).
Use the formula to calculate the mode.

Worked Examples

infoNote

Example 1: Mean of Grouped Data

Data: Find the mean for the frequency distribution below.

Class Interval	Frequency ( $f$ )
10–20	4
20–30	6
30–40	10
40–50	8
50–60	2

Solution:

Step 1: Find midpoints ( $x$ ):

$15, 25, 35, 45, 55$

Step 2: Calculate $f \times x$

4 \times 15, 6 \times 25, 10 \times 35, 8 \times 45, 2 \times 55

Total: $f \times x = 60 + 150 + 350 + 360 + 110 = :highlight[1030]$

Total: $f = 4 + 6 + 10 + 8 + 2 = :highlight[30]$

Step 3: Calculate the mean:

\text{Mean} = \frac{1030}{30} = :success[34.33]

Answer: $34.33$

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Example 2: Median of Grouped Data

Data: Use the same table as above.

Solution:

Total frequency

$N=30$ , so $\frac{N}{2} = :highlight[15]$

Median class:

$30–40$ (cumulative frequency reaches $20$ here).

Formula inputs:

L=30, CF = 10, f_m = 10, c = 10

Median:

\text{Median} = 30 + \left(\frac{15 - 10}{10}\right) \cdot 10 = 30 + 5 = :success[35]

Answer: $35$

infoNote

Example 3: Mode of Grouped Data

Data: Use the same table as above.

Solution:

Modal class:

$30-40$ (highest frequency, $f_m = 10$ )

Formula inputs:

L=30, f_{1} = 6, f_{2} = 8, c = 10

Mode:

\text{Mode} = 30 + \left(\frac{10 - 6}{(10 - 6) + (10 - 8)}\right) \times 10

= 30 + \frac{40}{6} \approx :success[36.67]

Answer: $36.67$

Summary

Mean: Average of the data, calculated as:

\text{Mean} = \frac{\sum (f \times x)}{\sum f}

Median: Middle value of data, using:

\text{Median} = L + \left(\frac{\frac{N}{2} - CF}{f_m}\right) \times c

Mode: Most frequent value, using:

\text{Mode} = L + \left(\frac{f_m - f_{1}}{(f_m - f_{1}) + (f_m - f_{2})}\right) \times c

These measures provide a comprehensive understanding of a frequency distribution's central tendency.

Mean, Mode and Median of a Frequency Distribution (Leaving Cert Mathematics): Revision Notes