Mean, Mode and Median of a Frequency Distribution (Junior Cert Mathematics): Revision Notes
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.
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:
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 .
- 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.
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:
Step 2: Add these results together:
Step 3: Add the total number of households:
Step 4: Divide the total number of people by the total number of households:
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.
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:
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.