Finding Averages (Edexcel GCSE Maths): Revision Notes
Finding averages from frequency tables
What are frequency tables?
A frequency table is a simple way to organise data that shows how many in each category. The word frequency simply means "how many", so these tables help us see patterns in data by grouping similar values together and counting how often each one appears.
Frequency tables are particularly useful because they condense large amounts of data into a clear, organised format that makes it easy to spot patterns and calculate statistical measures.
You can find averages and ranges from frequency tables using the same principles you've learned before, but the data is now organised in a structured table format.
The four key measures
When working with frequency tables, there are four important statistical measures you can calculate: mode, range, median, and mean. Each one tells us something different about the data.
Mode
The mode is the easiest measure to find from a frequency table. It's simply the category that appears most often - in other words, the category with the highest frequency.
To find the mode, look down the frequency column and identify which category has the largest number next to it. This category is your mode.
The mode is particularly useful for categorical data where you want to know the most popular choice or most common response.
| Number of sisters | Frequency |
|---|---|
| 0 | 7 |
| 1 | 15 |
| 2 | 12 |
| 3 | 8 |
| 4 | 4 |
| 5 | 0 |
Range
The range shows us how spread out our data is. To calculate the range from a frequency table, you need to look at the first column (which shows the different categories or values) and find the difference between the highest and lowest values.
The range is calculated as:
Remember that range only looks at the extreme values - it doesn't tell us anything about how the data is distributed between these extremes.
Median
The median represents the middle value when all the data is arranged in order. With frequency tables, you don't need to write out all the individual values - you can work out which category contains the middle value.
First, add up all the frequencies to find the total number of data points. Then, find the position of the middle value (if there's an even number of data points, you'll need the average of the two middle positions). Finally, count through the frequency column to see which category contains this middle position.
| Number of sisters (x) | Frequency (f) | No. of sisters × Frequency (f × x) |
|---|---|---|
| 0 | 7 | 0 |
| 1 | 15 | 15 |
| 2 | 12 | 24 |
| 3 | 8 | 24 |
| 4 | 4 | 16 |
| 5 | 0 | 0 |
| Total | 46 | 79 |
Mean
The mean requires a bit more work because you need to create a third column. This third column shows the result of multiplying each category value by its frequency (value × frequency).
Key Formula for Mean from Frequency Tables:
This gives you the total of all individual values divided by the total number of data points.
Worked example step-by-step
Worked Example: Finding All Four Measures
Let's work through a complete example using data about the number of sisters people have.
Step 1: Finding the Mode The mode is found by looking for the category with the highest frequency. In this case, the category "1 sister" has a frequency of 15, which is the highest, so the mode is 1.
Step 2: Calculating the Range For the range, we look at the first column and find the difference between the highest value (4 sisters) and the lowest value (0 sisters).
Step 3: Determining the Median To find the median, we need to identify the middle position. With 46 total responses, the middle position falls between the 23rd and 24th values. When we count through the frequencies, we find that both of these positions fall in the "2 sisters" category, so the median is 2.
Step 4: Calculating the Mean For the mean, we create a third column by multiplying each number of sisters by its frequency, then add up this column to get 79. We then divide by the total number of people (46):
Practice and application
Understanding how to find averages from frequency tables is essential for analysing real-world data. You might encounter similar tables showing survey results, test scores, or other collected information.
Real-world Applications:
- Survey data analysis
- Quality control in manufacturing
- Academic performance tracking
- Market research findings
- Scientific data collection
| Number of times sport played | Frequency |
|---|---|
| 0 | 8 |
| 1 | 15 |
| 2 | 17 |
| 3 | 6 |
| 4 | 4 |
| 5 or more | 0 |
The key is to remember that each measure tells us something different about the data: the mode shows the most common value, the range shows how spread out the data is, the median shows the middle value, and the mean gives us the overall average.
Key Points to Remember:
- Mode: Look for the category with the highest frequency - it's the most common value
- Range: Find the difference between the highest and lowest values in the first column
- Median: Calculate the middle position, then identify which category it falls into
- Mean: Create a third column (value × frequency), add it up, then divide by the total frequency
- Always check: Make sure your calculations make sense by comparing the different measures