What is the arithmetic mean?
– Definition: The arithmetic mean (often called the simple average) is the sum of a set of numbers divided by how many numbers there are. It is a measure of central tendency that summarizes a data series with one representative value.
– Formula: Arithmetic mean = (x1 + x2 + … + xn) / n
– Where xi are the individual values and n is the count of values.
How to compute it (step-by-step)
1. Add every number in the set to get the total (sum).
2. Count how many numbers are in the set (n).
3. Divide the sum by n.
4. Report the result in the same units as the inputs (dollars, percent, points, etc.).
Quick example (simple numbers)
– Numbers: 34, 44, 56, 78
– Sum = 34 + 44 + 56 + 78 = 212
– Count (n) = 4
– Arithmetic mean = 212 / 4 = 53
How to compute in common tools
– Excel / Google Sheets: =AVERAGE(range)
– Calculator: add the numbers, then divide by n.
When the arithmetic mean is useful (finance examples)
– Averaging analyst earnings estimates: sum the analysts’ forecasts and divide by the number of analysts.
– Average closing price over a month: sum the daily closes and divide by the number of trading days.
– Summary statistics for large, roughly symmetric datasets where compounding or serial dependence is not central.
Limitations — when the arithmetic mean can mislead
– Sensitivity to outliers: a single extreme value can pull the mean away from what most observations look like. In skewed data, the median may better reflect a “typical” value.
– Compounding and serial correlation: for returns that build on previous periods (compounded returns), the arithmetic mean overstates the typical compounded growth. When observations are correlated over time (serial correlation), the arithmetic mean does not capture that dynamic.
– Cash-flow discounting: present-value and future-value calculations use compounding and rates that should be handled with appropriate time-value formulas rather than a simple arithmetic average.
Arithmetic vs. geometric mean — short definition
– Geometric mean: takes the product of values (or 1 plus each period return), then
then takes the nth root of that product (where n is the number of periods or observations) and — if you started from returns expressed as “1 + return” — subtracts 1 to convert back to a periodic rate. The geometric mean therefore measures the constant compounded rate that would produce the same cumulative result as the observed sequence.
Formulae (clear and ready)
– Arithmetic mean (for values x1…xn):
– Arithmetic mean = (x1 + x2 + … + xn) / n
– Geometric mean (for positive values x1…xn):
– Geometric mean = (x1 * x2 * … * xn)^(1/n)
– Geometric mean for periodic returns R1…Rn (Ri expressed as decimals, e.g., 10% = 0.10):
– Geometric mean return = (Π(1 + Ri))^(1/n) − 1
– Equivalent using log returns (often numerically stable):
– Average log return = (1/n) Σ ln(1 + Ri)
– Geometric mean return = exp(average log return) − 1
Worked numeric example (two-year return)
– Year 1 return = +50% (R1 = 0.50)
– Year 2 return = −20% (R2 = −0.20)
Step 1 — arithmetic mean:
– AM = (0.50 + (−0.20)) / 2 = 0.15 = 15.0%
Step 2 — geometric mean:
– Cumulative growth factor = (1 + 0.50) × (1 − 0.20) = 1.5 × 0.8 = 1.2
– Geometric mean return = (1.2)^(1/2) − 1 = sqrt(1.2) − 1 ≈ 0.09545 = 9.545%
Interpretation: the arithmetic mean (15%) overstates the typical compounded annual growth; the portfolio actually grew at about 9.545% per year on a compounded basis.
Key relationships and rules of thumb
– AM ≥ GM for any nonnegative set of numbers; equality holds only when all observations are equal. (This is the arithmetic–geometric mean inequality.)
– Use the arithmetic mean when you want the expected value of a single, randomly chosen period outcome (e.g., the expected one-period return for forecasting when periods are independent).
– Use the geometric mean for multi-period compounded growth (e.g., multi-year investment returns, average growth rates of populations, or repeated multiplicative changes).
– If returns are volatile, the geometric mean will be noticeably lower than the arithmetic mean because volatility reduces compound growth.
Practical checklist — which mean to use
– If you want to estimate long-term compound growth → use geometric mean (or average log return).
– If you want the expected value of one random draw from the distribution → use arithmetic mean.
– If observations are skewed or include outliers → check median and trimming; the mean can be misleading.
– If returns are correlated over time (serial correlation) or you need present-value calculations → use proper time-value formulas (discounting/compounding) rather than a simple arithmetic mean.
– If sample size is small → treat the arithmetic mean’s estimate of the population mean with caution; compute standard error = s/√n (s = sample standard deviation) and consider confidence intervals.
Numerical stability tip
– For many small returns or long series, compute the geometric mean via average log returns:
– average log = (1/n) Σ ln(1 + Ri)
– geometric mean = exp(average log) − 1
This avoids underflow/overflow and gives the same result.
Common pitfalls for traders and students
– Applying the arithmetic mean to multi-period returns produces a misleading “average” growth rate.
– Failing to account for compounding or for cash inflows/outflows and rebalancing can distort both arithmetic and geometric assessments.
– Using sample means without checking independence, stationarity (statistical properties not changing over time), or data quality leads to unreliable inferences.
– Ignoring the difference between returns on capital (percentage changes) and absolute changes (dollar amounts). Arithmetic averaging of
Arithmetic averaging of dollar amounts and percentage returns can produce inconsistent or misleading figures if you mix units or ignore compounding. Always ensure you’re averaging like with like: either average percentages (returns) or average dollar changes, and treat multi‑period growth with geometric methods (see worked examples below).
Key properties and formulas (clean reference)
– Arithmetic mean (sample): x̄ = (1/n) Σ xi. This is the usual “average” and is an unbiased estimator of the population mean if observations are independent and identically distributed (i.i.d.).
– Sample variance: s^2 = (1/(n−1)) Σ (xi − x̄)^2. Use n−1 for an unbiased estimator of population variance.
– Standard error of the sample mean: SE = s / √n. If returns are independent, SE quantifies sampling uncertainty about the mean.
– 95% confidence interval (approximate for moderate n): x̄ ± t(n−1,0.975) × SE, where t is from Student’s t distribution.
– Geometric mean of returns: g = (Π (1 + Ri))^(1/n) − 1. Equivalent and numerically more stable: g = exp[(1/n) Σ ln(1 + Ri)] − 1.
Worked numeric examples
1) Basic arithmetic mean and standard error
Returns: R1 = 5%, R2 = −2%, R3 = 8%, R4 = 1%.
– Arithmetic mean: x̄ = (0.05 − 0.02 + 0.08 + 0.01)/4 = 0.03 = 3.0% per period.
– Sample variance: compute deviations: (0.05−0.03)=0.02, (−0.02−0.03)=−0.05, (0.08−0.03)=0.05, (0.01−0.03)=−0.02. Squared sum = 0.0004 + 0.0025 + 0.0025 + 0.0004 = 0.0058. s^2 = 0.0058/(4−1) ≈ 0.001933. s ≈ 4.40%.
– Standard error: SE = 0.0440 / √4 = 0.0220 = 2.2 percentage points. A rough 95% CI ≈ 3.0% ± 2×2.2% ≈ (−1.4%, 7.4%) using large‑sample normal approximation; for small n use t‑critical value (t3 ≈ 3.182), CI ≈ 3.0% ± 3.182×2.2% ≈ (−4.0%, 10.0%).
2) Arithmetic vs geometric mean for multi‑period growth
Returns: +50%, −50% (R1 = 0.5, R2 = −0.5).
– Arithmetic mean: (0.5 + (−0.5))/2 = 0%.
– Holding $1 through two periods: final wealth = (1 × 1.5) × 0.5 = 0.75 → total two‑period return = −25% → per‑period compound growth = (0.75)^(1/2) − 1 ≈ −13.4%.
– Geometric mean: g = sqrt(1.5 × 0.5) − 1 ≈ −13.4%, which correctly reflects compounding. The arithmetic mean (0%) is misleading for long‑run growth here.
When to use arithmetic mean vs geometric mean
– Use arithmetic mean when estimating the expected value for a single upcoming independent period (e.g., “expected return next month” under IID assumption).
– Use geometric mean when describing