3.3:

3.3: Media geometrica

JoVE Core
Statistics

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JoVE Core Statistics

Geometric Mean

3.2: Arithmetic Mean

3.4: Harmonic Mean

3,406 Views

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01:15 min

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April 30, 2023

Overview

The mean is a measure of the central tendency of a data set. In some data sets, the data is inherently multiplicative, and the arithmetic mean is not useful. For example, the human population multiplies with time, and so does the credit amount of financial investment, as the interest compounds over successive time intervals.

In cases of multiplicative data, the geometric mean is used for statistical analysis. First, the product of all the elements is taken. Then, if there are n elements in the dataset, the n^th root of the products is defined as the geometric mean of the data set. It can also be expressed via the use of the natural logarithmic function.

For example, suppose money compounds at annual interest rates of 10%, 5%, and 2%. In that case, the average growth factor can be calculated by computing the geometric mean of 1.10, 1.05, and 1.02. Its value comes out to be 1.056, which means that the average growth rate is 5.6% per annum.

It can be shown that the geometric mean of a sample data set is always quantitatively less than or at most equal to the arithmetic mean of the sample.

Transcript

The geometric mean is used for the analysis of data related to economics or biology, where the values change exponentially. If n number of data values are given, their geometric mean is expressed as the nth root of the product.

For example, consider the following set of numbers. Since these numbers are changing exponentially, their arithmetic mean would be skewed towards larger values. So, calculating the geometric mean can help find the mean of such exponentially changing values.

Begin by multiplying all the given numbers. Since there are four numbers in the data set, take the 4th root of the product. The resulting value is the geometric mean of the data.

Alternatively, convert the data values into corresponding logarithmic numbers. Then, add up all the log numbers and divide them by the total number of values in the data set. Finally, take antilog to arrive at the geometric mean.

It is important to note that the geometric mean cannot be used if given data contains zero or negative value.

Key Terms and definitions

Geometric Mean - Central tendency measure used for multiplicative datasets.
Central Tendency - Evaluation of where the center of data lies.
Arithmetic Mean - Average of data, not useful for multiplicative datasets.
Growth Factor - multiplied value over successive time intervals.
Statistical Analysis - Evaluating, interpreting, and visualizing quantitative data.

Learning Objectives

Define Geometric Mean – Explanation and application (e.g., geometric mean).
Contrast Geometric vs Arithmetic Mean – Understand key differences (e.g., compounding).
Explore Examples of Growth Factors – How to apply a geometric mean (e.g., financial investments).
Explain the concept of Central Tendency – How a central value is calculated and interpreted.
Apply formulas in context – Understand implications and practical usage of this statistical method.

Questions that this video will help you answer

What is the geometric mean and what role does it play in statistics?
What is the central tendency of a data set, and how is it important?
How does the geometric mean differ from the arithmetic mean and why?

This video is also useful for

Students – Grasp the concept of geometric mean, its importance and application in real-world problems.
Educators – Provides a clear explanation and practical examples, aiding in teaching complex statistical concepts.
Researchers – Useful for analysing multiplicative data and allows for better interpretation of results.
Statistics enthusiasts – Offers insights into understanding statistical analysis, data interpretation and decision making.

Tags

Media Geometrica Tendenza Centrale Dati Moltiplicativi Analisi Statistica Set Di Dati Prodotto Di Elementi Radice N-Esima Fattore Di Crescita Media Interesse Composto Funzione Logaritmica Naturale Media Aritmetica Set Di Dati Campione