Mean, Median and Mode of a Frequency Distribution

When you have a frequency distribution, calculating the mean, median, and mode is a bit different than when you just have a list of numbers. Let's break down how to find each one using simple steps and a worked example.

What is a Frequency Distribution?

A frequency distribution is a way to organise data into a table, showing how often each value occurs. This makes it easier to spot patterns and calculate key statistics, like the mean, median, and mode.

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Example: Let's say you're looking at the number of people in different households on your street. You've collected data, and you organise it into a frequency distribution table:

Number in Household	2	3	4	5	6	7
Number of Households	6	8	14	11	4	1

This table shows that:

6 households have 2 people,
8 households have 3 people, and so on.

1. Mode of a Frequency Distribution

What is it?

The mode is the value that appears most often. In a frequency distribution, you can find the mode by looking for the highest frequency.

Steps:

Find the highest frequency.

Why? The mode is the number that appears the most, so we look for the largest number in the "Number of Households" row.
The highest frequency is 14.

Look at the corresponding value in the "Number in Household" row.

Why? The mode is the number linked to the highest frequency.
The value corresponding to the highest frequency (14) is 4. Answer: The mode is 4, meaning most households have 4 people.

2. Mean of a Frequency Distribution

What is it?

The mean (average) is the total number of people divided by the total number of households. In a frequency distribution, you calculate the mean by multiplying each value by its frequency, adding those results together, and then dividing by the total frequency.

Formula:

$\text{Mean} = \frac{\text{(Value} \times \text{Frequency) for each row}}{\text{Total Frequency}}$

Steps:

Multiply each value by its frequency.

Why? This step helps us calculate the total number of people in each group. For example, there are 6 households with 2 people, so $2 \times 6 = 12$ .

Add all these results together.

Why? This gives us the total number of people across all households.

Add up the total frequency (total number of households).

Why? We divide the total number of people by the total number of households to find the mean.

Divide the total number of people by the total number of households.

Why? Dividing gives us the mean, or the average number of people per household.

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Worked Example: Using the same table:

Number in Household	2	3	4	5	6	7
Number of Households	6	8	14	11	4	1

Step 1: Multiply each value by its frequency:

$2 \times 6 = 12$

$3 \times 8 = 24$

$4 \times 14 = 56$

$5 \times 11 = 55$

$6 \times 4 = 24$

$7 \times 1 = 7$

Step 2: Add these results together:

$12 + 24 + 56 + 55 + 24 + 7 = :highlight[178]$

Step 3: Add the total number of households:

$6 + 8 + 14 + 11 + 4 + 1 = :highlight[44]$

Step 4: Divide the total number of people by the total number of households:

$178 \div 44 \approx :highlight[4.05]$

Answer: The mean number of people per household is 4.05 (rounded to two decimal places).

3. Median of a Frequency Distribution

What is it?

The median is the middle value when all the data is arranged in order. In a frequency distribution, you find the median by locating the middle position in the data.

Steps:

Add up the total number of households (frequency).

Why? We need to know how many values there are to find the middle one.

Find the position of the median.

Why? The median is the middle value, so if there are 44 households, the median will be between the 22nd and 23rd values.

Determine which group contains the 22nd and 23rd values.

Why? We need to find the range in which these values fall by adding up frequencies until we reach the median position.

Identify the median.

Why? The median is the value that lies in the middle of the dataset.

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Worked Example: Using the same table:

Number in Household	2	3	4	5	6	7
Number of Households	6	8	14	11	4	1

Step 1: Add up the total number of households:

$6 + 8 + 14 + 11 + 4 + 1 = :highlight[44]$

Step 2: Find the position of the median:

The median is between the 22nd and 23rd values.

Step 3: Determine which group contains the 22nd and 23rd values:

The first 6 values are in the 2-person households.
The next 8 values (positions 7 to 14) are in the 3-person households.
The next 14 values (positions 15 to 28) are in the 4-person households.

Step 4: The 22nd and 23rd values fall within the 4-person households, so the median is 4.

Answer: The median is 4.

Summary

Mode: Look for the highest frequency and find the corresponding value in the table.
Mean: Multiply each value by its frequency, add the results, and divide by the total frequency.
Median: Find the total frequency, locate the middle position, and determine which group contains the median.

Mean, Median and Mode of a Frequency Distribution Simplified Revision Notes for Junior Cycle Mathematics

Mean, Median and Mode of a Frequency Distribution

What is a Frequency Distribution?

1. Mode of a Frequency Distribution

2. Mean of a Frequency Distribution

Steps:

3. Median of a Frequency Distribution

Steps:

Summary

500K+ Students Use These Powerful Tools to Master Mean, Median and Mode of a Frequency Distribution For their Junior Cycle Exams.

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