Arithmetic Mean

What it is

The arithmetic mean (commonly called the simple average) is the sum of a set of numbers divided by the count of those numbers.

Formula:
* Arithmetic mean = (x1 + x2 + … + xn) / n

Example:
* For the numbers 34, 44, 56, and 78: mean = (34 + 44 + 56 + 78) / 4 = 212 / 4 = 53

Uses in finance

The arithmetic mean is widely used because it is simple to calculate and easy to interpret. Typical applications include:
* Average analyst earnings estimates (sum of estimates ÷ number of analysts)
* Average daily or monthly closing prices (sum of prices ÷ number of trading days)
* Simple measures of central tendency for large, roughly symmetric datasets

Limitations

The arithmetic mean has important drawbacks:
* Sensitive to outliers: a single extreme value can distort the average and make it unrepresentative of the group.
* Misleading for compounded processes: it does not account for compounding and therefore overstates long-term average returns when returns vary over time.
* Unsuitable for present- or future-value calculations and many cash-flow analyses where multiplicative effects matter.

When data are skewed or include outliers, the median or a trimmed mean often provides a more robust central value.

Arithmetic vs. geometric mean

For series that compound over time (most investment returns), the geometric mean is generally more appropriate because it accounts for period-to-period compounding.

How the geometric mean is calculated:
* Geometric mean = (Π (1 + ri))^(1/n) − 1
where ri are periodic returns expressed as decimals.

Key differences:
* Arithmetic mean adds and averages values; geometric mean multiplies and roots.
* Geometric mean is always ≤ arithmetic mean for a set of non-identical positive numbers.
* Use arithmetic mean for one-period expected values or independent observations; use geometric mean for multi-period compounded growth rates.

Example: stock returns

Five annual returns: 20%, 6%, −10%, −1%, 6%

• Arithmetic mean = (20 + 6 − 10 − 1 + 6) / 5 = 21 / 5 = 4.2% per year
• Geometric mean = (1.20 × 1.06 × 0.90 × 0.99 × 1.06)^(1/5) − 1 ≈ 3.74% per year

The geometric mean (3.74%) better reflects the actual compounded growth over the five years; the arithmetic mean (4.2%) overstates it.

Alternatives and when to use them

• Median — use when data are skewed or have outliers.
• Trimmed mean — removes extreme values before averaging; commonly used in economic measures (e.g., trimmed CPI, trimmed PCE).
• Geometric mean — use for compounded growth rates and serially correlated financial returns.
• Harmonic mean — use when averaging ratios or rates where the denominator is meaningful (e.g., average price per unit when quantities vary).

Key takeaways

• The arithmetic mean is the simple average and useful for many straightforward summaries.
• It is sensitive to outliers and inappropriate for compounded or serially correlated data.
• For multi-period investment returns use the geometric mean; for skewed distributions use the median or trimmed mean; for certain rate averages use the harmonic mean.