Pythagorean Means: Three Ways to Average#
The arithmetic mean is not the only way to compute an average. There are three classical types of averages, known as the Pythagorean means:
Arithmetic Mean
Geometric Mean
Harmonic Mean
Each is appropriate for a different type of data. Choosing the wrong mean can lead to incorrect conclusions.
1. Geometric Mean#
The geometric mean multiplies all values together and then takes the \(n\)-th root. It is used when values change multiplicatively rather than additively.
Common uses:
Growth rates
Investment returns
Population growth
Ratios
Why Use Geometric Mean (NOT Arithmetic Mean) for Growth?#
Growth compounds over time, so we must multiply changes instead of adding them.
Example (Investment Growth): Suppose an investment changes by:
+10%
+20%
−5%
We must convert percentages into multipliers (growth factors):
+10% → 1.10
+20% → 1.20
−5% → 0.95
What Happens if We Use Arithmetic Mean?
This is incorrect because it ignores compounding.
Apply Geometric Mean (Correct Method)
Multiply growth factors:
Now, take the cube root (since there are 3 periods):
Convert back to percentage growth:
The geometric mean is smaller because it correctly accounts for compounding effects.
Key Idea: When values multiply over time, always use the geometric mean.
Remember:
It is less sensitive to extreme large values than the arithmetic mean
All values must be positive
It is appropriate when data represents relative change (like growth rates or ratios)
Harmonic Mean#
The harmonic mean is another measure of central tendency, but it is designed specifically for averaging rates, ratios, and “per-unit” quantities.. Formula:
Where:
\(n\) = number of observations
\(x_i\) = each value
When Should We Use the Harmonic Mean?
Use the harmonic mean when averaging:
Speeds
Rates (e.g., km/hour, requests/second)
Ratios
“Per unit” quantities
Why? Because when dealing with rates, time or denominator matters more than magnitude. Slower rates dominate total time, so they should influence the average more strongly.
Example 1: Average Speed: Imagine you drive:
First half of the distance at 60 km/h
Second half of the distance at 40 km/h
What is your average speed? Most students instinctively compute:
But that’s incorrect. Why? Because speed is distance divided by time (speed = distance ÷ time). You spend more time traveling at 40 km/h (the slower speed) than at 60 km/h. Slower speeds should affect the average more (dominate total time). The harmonic mean correctly accounts for this:
This gives:
Remember: When averaging speeds over equal distances:
Slower speeds dominate total time
The harmonic mean gives more weight to smaller values
Example 2: Server Response Time Suppose two servers process requests at:
Server A: 100 requests/sec
Server B: 10 requests/sec
Even if workload is evenly distributed, the slower server becomes the bottleneck. The harmonic mean reflects this by penalizing smaller rates more heavily.
Why Not Arithmetic Mean?#
The arithmetic mean assumes “Quantities combine additively”. But rates combine through time accumulation, not addition. That’s the difference.
Important Properties:
Gives more weight to smaller values
Always less than or equal to the arithmetic mean
Undefined if any value equals zero
Only valid for positive numbers
Remember: Use the harmonic mean when averaging anything that is measured per unit (per hour, per second, per distance, etc.).
Big Picture Comparison#
Mean Type |
Best For |
Sensitive To |
|---|---|---|
Arithmetic Mean |
Additive quantities |
Large values |
Geometric Mean |
Multiplicative growth processes |
Relative scale |
Harmonic Mean |
Rates and ratios |
Small values |