Finding the Mean from a Table

Understanding how to calculate the mean of a set of data is an important skill in GCSE Maths. The mean, also known as the average, is a measure of central tendency, which gives us an idea of the 'middle' or 'typical' value in a dataset. This guide will walk you through how to find the mean from a table.

What is the Mean?

The mean is calculated by adding all the values together and then dividing by the number of values. Mathematically, it is expressed as:

$$\text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}}$$

When you’re given data in a table, particularly one with frequency, you must take account of how many times each value occurs. Here's how you do it.

This video explains Averages in more detail.

Step-by-Step Guide to Finding the Mean from a Frequency Table

Let’s break down the process with a clear example:

Example Table:

Steps:

Multiply Each Value by Its Frequency: For each row, multiply the value $x$ by its frequency $f$. This gives the total of that value occurring multiple times.

$$2 \times 4 = 8 \quad \text{(total for the value 2)}$$

$$5 \times 3 = 15 \quad \text{(total for the value 5)}$$

$$7 \times 2 = 14 \quad \text{(total for the value 7)}$$

$$10 \times 1 = 10 \quad \text{(total for the value 10)}$$

Add Up the Total Sums: Now, add all the products you calculated together:

$$8 + 15 + 14 + 10 = 47$$

Add Up the Frequencies: Next, add up all the frequencies to find how many data points there are in total:

$$4 + 3 + 2 + 1 = 10$$

Calculate the Mean: Finally, to find the mean, divide the total sum by the total frequency:

$$\text{Mean} = \frac{47}{10} = 4.7$$

Formula for Finding the Mean from a Table:

The formula for calculating the mean from a frequency table is:

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

Where:

• $x$ is each value,
• $f$ is the frequency of that value,
• $∑(x×f)\sum (x \times f)$ is the sum of all the value-frequency products,
• $∑f$ is the total frequency.

Key Tips:

• Always check that you have correctly multiplied the values by their frequencies before adding them.
• Double-check that the frequencies add up correctly to the total number of values in your dataset.
• If there are a lot of values or a complex table, break it down into smaller parts and work through step by step.

Practice Example:

Here’s another example for you to try on your own:

Steps:

1. Multiply the values by their frequencies:
   • $3 \times 6 = 18$
   • $6 \times 4 = 24$
   • $8 \times 2 = 16$
2. Add the totals together: $18 + 24 + 16 = 58$
3. Add up the frequencies: $6 + 4 + 2 = 12$
4. Calculate the mean: $\frac{58}{12} = 4.83$

Final Answer: The mean is 4.83.

Finding the mean from a table is a simple process if you follow the steps carefully. Remember to multiply the values by their frequencies, add them up, then divide by the total frequency. Practise with different tables to become more confident, and soon it will feel like second nature.