Calculate Log-Geometric Mean
Online calculator to compute the log-geometric mean of a data series
Log-Geometric Mean Calculator
The Log-Geometric Mean
The log-geometric mean is the arithmetic mean of the logarithms of the values. It equals the logarithm of the geometric mean: log(G).
Log-Geometric Mean Concept
The log-geometric mean is the arithmetic mean of the logarithms.
It is numerically more stable than the geometric mean.
■ Input Values ■ Logarithms ■ Log-Geometric Mean
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What is the Log-Geometric Mean?
The log-geometric mean is a logarithmic representation of the geometric mean:
- Definition: Arithmetic mean of the logarithms
- Calculation: log(G) = (1/n)Σlog(xᵢ)
- Relationship: log(G) equals the logarithm of the geometric mean
- Advantage: Numerically more stable for very large/small values
- Application: Growth rates, probability theory
- Conversion: G = exp(log(G)) = 10^(log(G))
Calculating the Log-Geometric Mean
The calculation follows four steps:
1. Logarithmize
Create logarithms:
log(x₁), log(x₂), ..., log(xₙ)
2. Sum
Add logarithms:
S = Σlog(xᵢ)
3. Count
Determine count:
n = Number of Values
4. Divide
Divide S by n:
log(G) = S / n
Applications of the Log-Geometric Mean
The log-geometric mean has applications in specialized areas:
Numerical Computation
- Avoiding overflow with large numbers
- Avoiding underflow with small numbers
- More stable calculations than direct geometric mean
- Important in machine learning and probability theory
Scientific Applications
- Information theory: Entropy calculations
- Statistics: Log-normal distributions
- Biology: pH values, population growth
- Finance: Continuous returns
Formulas for the Log-Geometric Mean
Log-Geometric Mean
Arithmetic mean of the logarithms
Extended Form
Explicit representation
Relationship to Geometric Mean
Conversion: Apply exponential function
Logarithm Rules
Important computational rules for logarithms
Symbol Explanations
| \(\log(G)\) | Log-Geometric Mean |
| \(G\) | Geometric Mean |
| \(x_i\) | Individual Data Value |
| \(n\) | Number of Values |
| \(\log\) | Logarithm (Base 10) |
| \(\ln\) | Natural Logarithm |
Example Calculations for the Log-Geometric Mean
Example 1: Basic Calculation
Calculate: Log-geometric mean of 3 values
1. Logarithmize
Logarithm base 10
2. Sum & Divide
Average of logarithms
3. Geometric Mean
Optional: Convert back to G
Example 2: Numerical Stability with Large Numbers
Problem: Direct product causes overflow
Direct Method (Problematic)
⚠️ Risk of overflow with very large numbers!
Log Method (Stable)
✓ No overflow problems!
Important Advantage
The log-geometric mean works with additions instead of multiplications. This prevents overflow with very large numbers and underflow with very small numbers. In practice, calculations are often done with logarithms and transformed back only at the end.
Example 3: Growth Rates Over Multiple Periods
Calculate average growth factor
Log Method
Result
Average Growth:
12.3% per period
Interpretation
The geometric mean (calculated via logarithms) gives the average multiplicative growth. With growth factors of 1.1, 1.2, 1.05, and 1.15, the average growth is 12.3% per period. This is the correct method for growth rates over multiple periods!
Mathematical Foundations of the Log-Geometric Mean
The log-geometric mean is a logarithmic representation of the geometric mean with important practical advantages in numerical computations.
Properties of the Log-Geometric Mean
The log-geometric mean has characteristic mathematical properties:
- Equivalence: log(G) = (1/n)Σlog(xᵢ) corresponds to G = ⁿ√(Πxᵢ)
- Linearity: Adds logarithms instead of multiplying
- Numerical Stability: Prevents overflow/underflow
- Logarithm Rules: log(a·b) = log(a) + log(b)
- Base Independence: Works with ln, log₁₀, or log₂
Why Use Logarithms?
Advantages
- Numerical Stability: No overflow problems with large numbers
- No Underflow: No precision loss with small numbers
- Simpler Arithmetic: Addition instead of multiplication
- Better Conditioning: More stable with extreme values
- Efficiency: Faster computation
Practical Applications
- Machine Learning: Log-likelihood functions
- Statistics: Log-normal distributions
- Information Theory: Entropy calculations
- Finance: Continuous returns
- Biology: pH values, population dynamics
Relationship to Other Concepts
Log-Normal Distribution
If X is log-normally distributed, then log(X) is normally distributed. The log-geometric mean is the expected value of log(X): E[log(X)] = log(G)
Continuous Returns
In finance, the log-geometric mean of growth factors corresponds to the average continuous return: r = log(G)
Entropy
In information theory, entropy is closely related to the log-geometric mean. Average information corresponds to the negative log-geometric mean of probabilities.
Choosing a Logarithm Base
Natural Logarithm
ln(x) = log_e(x)
Base e ≈ 2.718
For continuous processes
Common Logarithm
log₁₀(x) = log(x)
Base 10
For pH values, decibels
Binary Logarithm
log₂(x) = lb(x)
Base 2
For computer science, bits
Practical Example: Machine Learning
Log-Likelihood
In machine learning, probabilities are often multiplied: P = p₁ · p₂ · ... · pₙ
Problem: With many small probabilities (e.g., 0.01), P becomes extremely small → underflow
Solution: Use log-likelihood:
log(P) = log(p₁) + log(p₂) + ... + log(pₙ)
This corresponds to the log-geometric mean multiplied by n.
Summary
The log-geometric mean is a practical and numerically stable alternative to the direct geometric mean. It transforms multiplications into additions, preventing overflow and underflow problems with extreme values. The method is indispensable in numerical mathematics, especially in machine learning (log-likelihood), finance (continuous returns), and information theory (entropy). The choice of logarithm base (ln, log₁₀, log₂) depends on the application context but does not change the principle.
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