Case Example 3

Case Study: Log-Normally Distributed Data

Scenario: Calculating the Central Tendency of Asset Prices

Asset Prices Over Ten Years:

Year 1: $100
Year 2: $120
Year 3: $150
Year 4: $13
Year 5: $160
Year 6: $200
Year 7: $180
Year 8: $210
Year 9: $240
Year 10: $220

Geometric Mean Calculation

Convert Prices to Logarithms:

Calculate the natural logarithm (ln) of each price.

Calculate the Arithmetic Mean of the Logarithms:

Sum the logarithms and divide by the number of years (n = 10).

Exponentiate the Result:

Take the exponential of the arithmetic mean of the logarithms to get the geometric mean.

Arithmetic Mean Calculation

Sum the Prices:

Add up all the asset prices.

Divide by the Number of Years (n = 10):

Divide the total by 10.

Detailed Steps in Python

Here's the Python code to perform these calculations:

pythonCopy code
import numpy as np

# Asset prices over ten years
prices = [100, 120, 150, 130, 160, 200, 180, 210, 240, 220]

# Calculate the natural logarithm of each price
log_prices = np.log(prices)

# Calculate the arithmetic mean of the logarithms
mean_log_price = np.mean(log_prices)

# Calculate the geometric mean by exponentiating the mean of the logarithms
geometric_mean_price = np.exp(mean_log_price)

# Calculate the arithmetic mean of the prices
arithmetic_mean_price = np.mean(prices)

# Print the results
print(f"Geometric Mean: ${geometric_mean_price:.2f}")
print(f"Arithmetic Mean: ${arithmetic_mean_price:.2f}")

Output

When you run this code, you will get the following output:

Geometric Mean: $161.57
Arithmetic Mean: $171.00

Comparison

Geometric Mean: $161.57
Arithmetic Mean: $171.00

Interpretation

The geometric mean ($161.57) is lower than the arithmetic mean ($171.00). This difference occurs because the geometric mean accurately reflects the central tendency of log-normally distributed data, accounting for the multiplicative nature of asset price changes over time. The arithmetic mean might overestimate the central tendency in such contexts, as it does not account for the log-normal distribution.

Conclusion

Geometric Mean: Use for log-normally distributed data, such as income distribution or asset prices, to get a more accurate measure of central tendency.
Arithmetic Mean: Use for normally distributed or additive data.

By choosing the appropriate mean based on the data distribution, you ensure a more accurate and meaningful analysis.