How to Find a Geometric Mean When Values Are Negative

The standard geometric mean is designed for positive values. If your data include negatives, the usual logarithmic calculation does not apply, and a workaround can change what the result means. The right choice depends on why the values are negative: returns, signed measurements, and values that are negative only because of an arbitrary reference point need different treatment.

Why negative values cause a problem

For positive observations $x_1,\ldots,x_n$, the geometric mean is

The logarithm formula requires every $x_i$ to be positive. The product-and-root formula can sometimes produce a real number for negative inputs, but that does not make it the standard geometric mean of signed data. For example, a negative product has no real square root, while a product of two negative values is positive and has a positive square root. That positive result does not preserve the signs of the original observations.

The conventional geometric mean is most useful for quantities that combine multiplicatively, such as growth factors, ratios, and positive measurements. It is not a general-purpose average for any list of numbers.

Choose a method that matches the data

• Returns or growth rates: Convert each rate to its growth factor, average those factors geometrically, and convert back. This works when every factor is positive.
• Signed measurements: Use an arithmetic mean, median, or a model appropriate to the measurement. These retain the sign without inventing a geometric-mean convention.
• Values negative relative to a chosen baseline: A shift may be meaningful if the baseline has a real interpretation. Otherwise, the result depends on an arbitrary choice.
• Magnitude is the main interest: Calculate the geometric mean of the absolute values, and report sign information separately. This answers a magnitude question, not the signed-data question.

There is no universally accepted signed geometric mean for arbitrary real observations. If you define a custom version, state its formula and interpretation rather than presenting it as the ordinary geometric mean.

Worked example: averaging negative returns

Suppose an investment loses 20% in one period and gains 50% in the next. The returns are $-0.20$ and $0.50$, but the amounts by which the investment is multiplied are $0.80$ and $1.50$.

The average per-period growth factor is

Subtract 1 to express that factor as a return: $1.0954-1=0.0954$, or about 9.54% per period. This is the compound average return. It is not the geometric mean of the two negative-and-positive percentage numbers themselves.

The result checks out over both periods: $1.0954^2\approx1.20$, matching the total growth factor $0.80\times1.50=1.20$.

What if you shift the values?

A common suggestion is to add a constant to every observation to make them positive, calculate the geometric mean, and perhaps subtract the constant afterward. This can be done as a defined transformation, but it does not give a unique geometric mean: different constants produce different answers.

For example, take $-4, 9, 16$. Adding 5 gives $1,14,21$, whose geometric mean is $\sqrt[3]{294}\approx6.65$. Subtracting 5 gives about $1.65$. Instead, adding 10 gives $6,19,26$, whose geometric mean is $\sqrt[3]{2964}\approx14.36$; subtracting 10 gives about $4.36$. Both shifts make the inputs positive, yet they produce different adjusted answers. Use shifting only when the offset is meaningful and disclose it.

Common mistakes

• Taking logs of negative values: The real logarithm $\ln x$ is undefined for $x<0$. Taking the logarithm of absolute values changes the question to one about magnitudes.
• Using the root of the product without checking signs: A real root may exist for some lists, but existence alone does not make the result a suitable average.
• Adding an arbitrary constant: The geometric mean is not translation-invariant; the chosen shift affects the answer.
• Averaging percentage returns directly: Compound growth comes from multiplying growth factors, not averaging the return percentages geometrically. Convert each decimal return $r_i$ to $1+r_i$ first. A return of $-100\%$ gives a zero factor, so the investment does not have a positive geometric growth factor across that period; returns below $-100\%$ are not valid for an ordinary investment value that cannot fall below zero.

When in doubt, state what the numbers represent and what question the average should answer. That usually determines whether to use growth factors, a conventional signed-data statistic, or a clearly labeled transformed measure.