Types of mean in statistics

In statistics, several types of means are used to summarize data, each with its own significance and use cases. The most common types include:

1. Arithmetic Mean
2. Geometric Mean
3. Harmonic Mean
4. Weighted Mean
   

Let's explore each of these means and see how they can be computed using Python.

1. Arithmetic Mean

The arithmetic mean is the most common type of mean, often referred to simply as the "average." It is calculated by summing all the values and dividing by the number of values.

Formula:

$$\text{Arithmetic Mean} = \frac{\sum_{i=1}^{n} x_i}{n}$$

Python Example:

import numpy as np

data = [10, 20, 30, 40, 50]
arithmetic_mean = np.mean(data)
print(f"Arithmetic Mean: {arithmetic_mean}")
 

2. Geometric Mean

The geometric mean is useful when dealing with data that involves multiplication or percentages, such as growth rates. It is calculated by multiplying all the values together and then taking the nth root, where n is the number of values.

Formula:

$$\text{Geometric Mean} = \left( \prod_{i=1}^{n} x_i \right)^{\frac{1}{n}}$$

Python Example:

from scipy.stats import gmean

data = [10, 20, 30, 40, 50]
geometric_mean = gmean(data)
print(f"Geometric Mean: {geometric_mean}")
 

3. Harmonic Mean

The harmonic mean is useful when dealing with rates or ratios, such as speed or density. It is calculated as the reciprocal of the arithmetic mean of the reciprocals of the data values.

Formula:

$$\text{Harmonic Mean} = \frac{n}{\sum_{i=1}^{n} \frac{1}{x_i}}$$

Python Example:

from scipy.stats import hmean

data = [10, 20, 30, 40, 50]
harmonic_mean = hmean(data)
print(f"Harmonic Mean: {harmonic_mean}")
 

4. Weighted Mean

The weighted mean is an average that takes into account the relative importance (weight) of each value. It is useful when different data points contribute differently to the overall mean.

Formula:

$$\text{Weighted Mean} = \frac{\sum_{i=1}^{n} w_i \cdot x_i}{\sum_{i=1}^{n} w_i}$$

Where $w_i$ is the weight for each value $x_i$.

Python Example:

data = [10, 20, 30, 40, 50]
weights = [1, 2, 3, 4, 5]  # Weights corresponding to each data point
weighted_mean = np.average(data, weights=weights)
print(f"Weighted Mean: {weighted_mean}")
 

Summary of Outputs:

• Arithmetic Mean: $30.0$
• Geometric Mean: Approximately $26.379$
• Harmonic Mean: Approximately $21.818$
• Weighted Mean: $40.0$

Explanation:

• Arithmetic Mean is a simple average and is widely used for data where each observation is equally important.
• Geometric Mean is more appropriate for data that involves products, such as growth rates.
• Harmonic Mean is particularly useful for average rates, such as speed or density.
• Weighted Mean adjusts the mean by giving different importance to different data points, useful in scenarios where some data points have more significance than others.