Geometric Mean Formula with Worked Examples

Key Points

• The geometric mean formula is $\text{GM} = \sqrt[n]{x_1 x_2 \cdots x_n}$, the $n$th root of the product of $n$ values.

• It is the correct average for quantities that multiply, such as growth rates, ratios and index numbers.

• The US Bureau of Labor Statistics uses a geometric mean index formula for most item strata of the Consumer Price Index.

• The geometric mean never exceeds the arithmetic mean, and the two are equal only when every value is identical.

• All values must be positive, because a zero makes the product zero and a negative value can make the root undefined.

What Is the Geometric Mean Formula?

The geometric mean formula averages values by multiplying rather than adding:

$$\text{GM} = \sqrt[n]{x_1 x_2 \cdots x_n} = \left( \prod_{i=1}^{n} x_i \right)^{1/n}$$

For two values it reduces to $\sqrt{x_1 x_2}$. The US Bureau of Labor Statistics states that most item strata of the Consumer Price Index use a geometric mean index formula, applied to price ratios with expenditure weights.

Why Multiplying Is the Right Move for Growth

An arithmetic mean assumes the quantities add. Growth does not add, it compounds, so averaging growth factors by adding them gives an answer that does not reproduce the actual outcome.

The geometric mean is built to reproduce it. If you replace every growth factor with the geometric mean and compound it $n$ times, you land on exactly the same final value as the original sequence.

That property is why index numbers use it. Averaging a set of price ratios arithmetically would overstate inflation, because a ratio of 2 and a ratio of 0.5 cancel multiplicatively but average to 1.25 additively.

Variable Key

Examples of the Geometric Mean Formula

Example 1

Find the geometric mean of 4 and 9.

$$\text{GM} = \sqrt{4 \times 9} = \sqrt{36} = 6$$

Final answer: $\text{GM} = 6$

Example 2

An investment grows 10 per cent, then 21 per cent, then falls 10 per cent. Find the average annual growth rate by taking the arithmetic mean.

The three changes are $+10$, $+21$ and $-10$ per cent, so the arithmetic mean is $\frac{10+21-10}{3} = 7$ per cent. Test it by compounding: $100 \times 1.07^3 = 122.50$.

The real sequence gives $100 \times 1.10 \times 1.21 \times 0.90 = 119.79$, so 7 per cent per year overstates the outcome. Growth factors multiply, so they must be averaged multiplicatively.

Using the growth factors 1.10, 1.21 and 0.90:

$$\text{GM} = \sqrt[3]{1.10 \times 1.21 \times 0.90} = \sqrt[3]{1.1979} \approx 1.0619$$

Final answer: about 6.19 per cent per year, and $100 \times 1.0619^3 = 119.79$ as required.

Example 3

Find the geometric mean of 2, 8 and 32.

$$\text{GM} = \sqrt[3]{2 \times 8 \times 32} = \sqrt[3]{512} = 8$$

Final answer: $\text{GM} = 8$

These values form a geometric sequence, and the geometric mean of three consecutive terms is always the middle term.

Example 4

Find the geometric mean of 1, 3, 9, 27.

$$\text{GM} = \sqrt[4]{1 \times 3 \times 9 \times 27} = \sqrt[4]{729} = \sqrt[4]{3^6} = 3^{1.5} \approx 5.196$$

Final answer: about $5.196$

The arithmetic mean of the same values is 10, which sits well above the geometric mean, as it must.

Example 5

A city's population multiplies by 1.5, then 0.8, then 1.25 over three decades. Find the average decade factor.

$$\text{GM} = \sqrt[3]{1.5 \times 0.8 \times 1.25} = \sqrt[3]{1.5} \approx 1.1447$$

Final answer: about $1.1447$, or 14.47 per cent growth per decade.

Geometric Mean, Common Confusions Cleared Up

The ordering is fixed: the harmonic mean never exceeds the geometric mean, which never exceeds the arithmetic mean, with equality only when all values are the same.

Mistake 1: Averaging growth rates arithmetically

Where it slips in: Percentage changes look like ordinary numbers, so they get added and divided.

Don't do this: Average $+10$, $+21$ and $-10$ per cent to get 7 per cent.

The correct way: Convert to factors 1.10, 1.21 and 0.90, take the geometric mean, then convert back. Test any answer by compounding it and checking you land on the true final value.

Students who always convert a percentage change into a multiplier before averaging stop making this error, because the multiplier makes the compounding visible.

Mistake 2: Including a zero or a negative value

Where it slips in: A data set of returns contains a total loss, or a reading of zero.

Don't do this: Apply the formula to 4, 0 and 9, which gives a geometric mean of zero.

The correct way: The geometric mean needs strictly positive values. A zero collapses the product, and negative values can make an even root undefined, so use a different average or shift the data.

Mistake 3: Taking the wrong root

Where it slips in: The root must match the count of values, and the count is easy to misread when the data is listed inline.

Don't do this: Take a square root of a product of three values.

The correct way: Count the values first and let that set the root. A geometric mean outside the range of the data is the signature of a wrong root.

Key Takeaways

• The geometric mean formula is $\text{GM} = \sqrt[n]{x_1 x_2 \cdots x_n}$, the $n$th root of the product.

• It is the right average whenever quantities compound, which is why price indices such as the CPI use it.

• Compounding the geometric mean $n$ times reproduces the true final value, and the arithmetic mean does not.

• Every value must be strictly positive, and the root must match the count of values.

• To work through averages with a teacher, explore an 8th grade math tutor or math tutoring.

Sharpen Your Geometric Mean

Find the geometric mean of 5 and 20, then of 1.2, 0.9 and 1.4. Check the second answer by compounding it three times against the true product.

Want a live Bhanzu trainer to walk through more averages problems? Book a free demo class.

Read More

• Weighted Mean Formula

• Mean Median Mode Formula

• Average Formula

• Geometric Sequence Formula

• What Is Ratio