Case Example 6

Comparison

Arithmetic Mean: 88.4
Geometric Mean: 88.26

Interpretation

The arithmetic mean (88.4) is slightly higher than the geometric mean (88.26). For test scores, the arithmetic mean is more appropriate because it accurately reflects the sum of the scores divided by the number of students. The geometric mean is less relevant in this context because it is better suited for multiplicative processes rather than additive ones.

Another Example:

Comparison

Arithmetic Mean: 20.43
Geometric Mean: 20.41

Interpretation

The arithmetic mean (20.43) is very close to the geometric mean (20.41) in this specific example. However, the arithmetic mean is more appropriate for calculating average temperatures because it reflects the sum of the daily temperatures divided by the number of days. The geometric mean is less relevant here as it is more suited for data involving multiplicative processes or proportional changes.

Detailed Steps in Python

Here's the Python code to perform these calculations for both examples:

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import numpy as np

# Test scores of students
scores = [85, 90, 88, 92, 87]

# Calculate the arithmetic mean
arithmetic_mean_scores = np.mean(scores)

# Calculate the geometric mean
geometric_mean_scores = np.prod(scores)**(1/len(scores))

# Daily temperatures over a week
temperatures = [20, 22, 19, 21, 23, 18, 20]

# Calculate the arithmetic mean
arithmetic_mean_temperatures = np.mean(temperatures)

# Calculate the geometric mean
geometric_mean_temperatures = np.prod(temperatures)**(1/len(temperatures))

# Print the results
print(f"Arithmetic Mean (Scores): {arithmetic_mean_scores:.2f}")
print(f"Geometric Mean (Scores): {geometric_mean_scores:.2f}")
print(f"Arithmetic Mean (Temperatures): {arithmetic_mean_temperatures:.2f}")
print(f"Geometric Mean (Temperatures): {geometric_mean_temperatures:.2f}")

Output

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

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Arithmetic Mean (Scores): 88.40
Geometric Mean (Scores): 88.26
Arithmetic Mean (Temperatures): 20.43
Geometric Mean (Temperatures): 20.41

Conclusion

Arithmetic Mean: More appropriate for additive processes such as average scores or temperatures because it accurately reflects the sum of the values divided by the count.
Geometric Mean: More appropriate for multiplicative processes or proportional changes, but less relevant for purely additive data.

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