Geometric mean
About this app
Mean growth rate the right way: production 1000, 1200, 1500, 900. Guess first, then three paths grow with a constant factor. The mean of the rates (+1.67%) ends at 1050.8, while production ends at 900. Only the geometric mean (−3.45% per year) hits the end value. Factors, product, n-th root, plus portfolio, inflation, bacteria and ±50%.
With explain mode: an AI tutor walks you through the app, points out what matters in the scene and answers your questions.
The statistics
For a time series of values B₀, B₁, …, Bₙ, xᵢ = Bᵢ/Bᵢ₋₁ is the growth factor and rᵢ = xᵢ − 1 the growth rate of period i. The mean growth rate is r̄ = x̄_geom − 1 with the geometric mean x̄_geom = ⁿ√(x₁ · x₂ ⋯ xₙ). Since the product of the factors telescopes to Bₙ/B₀, start and end values are enough: x̄_geom = ⁿ√(Bₙ/B₀). The arithmetic mean of the rates is wrong here, because growth acts multiplicatively. It is always at least as large as the geometric mean and so overstates growth.
What you see
The starting point is the lecture example: a company produces 1000, 1200, 1500 and 900 units. The rates are +20%, +25% and −40%, their arithmetic mean 1.67% per year. After revealing, three paths grow with a constant factor: the 1.67% path ends at 1050.8, the geometric-mean path hits the 900. The gap of 150.8 lights up as apparent growth. Below are the rates of each period with both means, and a panel works out factors, product, cube root and check with numbers: ∛0.9 = 0.9655, i.e. −3.45% per year.
Things to try
First guess the mean growth rate with the slider or the dotted line, then reveal: your guess grows as its own path, and the feedback tells you where it ends. Drag the bars to another height and watch the gap grow when the rates fluctuate more. More examples: a share with +50% and −50% (0% on average, yet €100 becomes €75), a portfolio averaging +3.75% that still loses money, inflation 2021 to 2024 where both means are almost equal, and a bacterial culture.
Frequently asked questions
- How do you compute the average growth rate?
- Convert the rates into factors (+20% → 1.2), multiply them, take the n-th root and subtract one: r̄ = ⁿ√(x₁ ⋯ xₙ) − 1. Equivalently r̄ = ⁿ√(Bₙ/B₀) − 1 from start and end value. For 1000 → 900 over three years that gives ∛0.9 − 1 ≈ −3.45% per year.
- Why can growth rates not be averaged arithmetically?
- Growth rates act multiplicatively: each rate refers to the value of the previous period. A loss hits an amount that has grown before and therefore weighs more than a gain of the same size. +50% and −50% average to 0% arithmetically, yet €100 becomes €75. The arithmetic mean overstates growth.
- When are the arithmetic and geometric mean equal?
- For positive values x̄_geom ≤ x̄ always holds, with equality only when all values are equal. If the rates are close together, the means hardly differ: the inflation rates 2021 to 2024 give 4.53% vs 4.51%. The more the rates fluctuate, the larger the gap.
- What can I change in the geometric mean app?
- You guess the mean growth rate, reveal and watch three paths grow. The values can be dragged as bars, five examples are available, and the Working panel shows the table, the steps via factors or via start and end, the check and the counter-calculation.
Subject: Statistics
More from Statistics
- Anscombe's quartet
- Bias-variance / overfitting
- Law of large numbers
- Histogram workshop
- Hypothesis test & p-value
- Confidence interval
- Measures of location & spread
- Measures of location & spread 2.0
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