Mean & Standard Deviation Calculator
Enter returns at one frequency and separate their mean from their dispersion.
2 means 2% · SD in percentage points · no automatic annualization
Results
Review or adjust your inputs, then calculate to see your results and interpretation.
Understand this calculation
The mean describes the center; standard deviation measures dispersion around it. Equal mean returns can have different volatility. Sample variance divides by n−1 and population variance by n. Sample standard deviation is undefined for one observation.
See calculation method and assumptions
Formula
Mean
Sample variance and SD
Population variance and SD
The mean is the sum of observations divided by n.
Sample variance divides by n−1. Sample SD is its square root; n must exceed 1.
Population variance divides by N. Population SD is its square root; μ is the mean of the complete population.
Variables, rates and periods
- xᵢ / yᵢ
- Observations from the same period. A return input of 2 means 2%, calculated directly in percentage units.
- n / x̄ / s
- n is sample size, x̄ the sample mean, and s the sample SD using n−1 in the variance denominator.
- N / μ / σ
- N is complete-population size, μ its mean, and σ its SD using N in the variance denominator.
- Units
- Mean return and bounds use %. SD, standard error, and margins use percentage points; variance uses squared percentage points. No annualization.
Units and frequency
- A return input of 2 means 2%, not twice the value.
- Use one consistent observation frequency.
- SD is shown in percentage points without automatic annualization.
- Demonstration data illustrate the concept only.
Squared deviations and extremes
- Deviations are squared, so extreme observations have a strong effect on variance.
- The n−1 denominator corrects sample variance's bias;
- it does not make sample SD itself exactly unbiased.
Analyze a return series through its mean, sample and population standard deviations, count and range.
Worked example
- Enter −2, 0, 2, 4 and 6 as percentage numbers; 2 means 2%.
- Calculate the mean and both SDs; confirm that the count is five.
- Compare the n−1 and n denominators: sample SD is slightly higher for the same data.
Frequently asked questions
Should I use sample or population SD?
Use sample SD when the observations are a sample from a larger population of interest. Population SD describes the complete listed set itself.
How do SD and variance differ?
SD is the square root of variance and uses the original data's unit. With percentage-number returns, SD is in percentage points and variance in squared percentage points.
Is the SD of monthly returns annualized volatility?
No. This tool keeps the observation frequency and does not silently apply an annualization factor. Annualization requires additional frequency and dependence assumptions.