Group Arithmetic Mean Calculator

What is Group Arithmetic Mean?

The group arithmetic mean, also known as the weighted arithmetic mean, is a measure of central tendency that takes into account both the values of a dataset and their frequencies or weights. It's particularly useful when dealing with grouped data or when certain values occur more frequently than others.

Formula and Its Meaning

The formula for the group arithmetic mean is:

\[\bar{x} = \frac{\sum_{i=1}^{n} x_i f_i}{\sum_{i=1}^{n} f_i}\]

Where:

• \(\bar{x}\) is the group arithmetic mean
• \(x_i\) are the individual values
• \(f_i\) are the corresponding frequencies or weights
• \(n\) is the number of distinct values

This formula calculates the average by giving more importance to values that occur more frequently.

Calculation Steps

1. Multiply each value by its frequency: \(x_i f_i\)
2. Sum all these products: \(\sum_{i=1}^{n} x_i f_i\)
3. Sum all the frequencies: \(\sum_{i=1}^{n} f_i\)
4. Divide the sum of products by the sum of frequencies

Example Calculation

Let's calculate the group arithmetic mean for the following data:

Value (\(x_i\))	Frequency (\(f_i\))
10	2
15	5
20	3

1. \(x_i f_i\): (10 × 2) + (15 × 5) + (20 × 3) = 20 + 75 + 60 = 155
2. \(\sum_{i=1}^{n} f_i\): 2 + 5 + 3 = 10
3. \(\bar{x} = \frac{155}{10} = 15.5\)

Visual Representation

This bar chart represents the example dataset. The height of each bar corresponds to the frequency of each value. The red dashed line indicates the group arithmetic mean (15.5).