Mean & Standard Deviation

Question, try, then observe

The same 2% mean: the same ups and downs?

Both six-month samples keep the same mean. Change B's spread, then add a balanced pair of extremes. Predict which numbers will change.

Illustrative data, not live market data. A monthly return of 2 = 2%; dispersion and return differences use percentage points, with no annualization.

01 · Make a predictionDoes the same mean imply the same dispersion?

B baseline return spread 3 pp

Balanced endpoint extremes 0 pp

Same mean, different return dispersion

Monthly return (%)

Illustrative month (1–6)

Asset A (circles)Asset B (squares)

Inspect observation

Observation 1 · Illustrative month (1–6): 1Asset A (circles): 1.5 · Asset B (squares): -1

Read the data table

6 observations; units are shown in the headings.
No.	Illustrative month (1–6)	Asset A (circles) · Monthly return (%)	Asset B (squares) · Monthly return (%)
1	1	1.5	-1
2	2	1.7	0.2
3	3	1.9	1.4
4	4	2.1	2.6
5	5	2.3	3.8
6	6	2.5	5

B mean monthly return (%)

x̄ = \frac{12}{6} \approx 2

B standard deviation (pp)

s = \sqrt{\frac{25.2}{5}} \approx 2.245

A mean monthly return2.00%n = 6

B mean monthly return2.00%n = 6

A · Sample SD0.37 pp

B · Sample SD2.24 pp

Both means stay at 2.00%. B's Sample SD is 2.24 pp. Balanced endpoint changes leave the mean unchanged but increase squared distances from it.

Square and add deviations, then divide by n−1 or n to obtain variance. Its square root is standard deviation. For these six observations, sample SD is population SD × √(6/5). Sample SD is undefined at n=1.

What did you just see?

Two assets can both average 2% a month while one stays near 2% and the other swings widely. The alone cannot show that difference.

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.

Another 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.

Show formal definition and formula

The arithmetic mean is the sum of observations divided by their count. Variance measures squared deviations from the mean; SD brings that measure back to the original unit.

Mean

x̄ = \frac{\sum_{i = 1}^{n} x_{i}}{n}

The mean is the sum of observations divided by n.

Sample variance and SD

s^{2} = \frac{\sum_{i = 1}^{n} {(x_{i} - x̄)}^{2}}{n - 1}; s = \sqrt{s^{2}}

Sample variance divides by n−1. Sample SD is its square root; n must exceed 1.

Population variance and SD

σ^{2} = \frac{\sum_{i = 1}^{N} {(x_{i} - μ)}^{2}}{N}; σ = \sqrt{σ^{2}}

Population variance divides by N. Population SD is its square root; μ is the mean of the complete population.

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.

Common mistakes

Low volatility does not guarantee profit. Multiplying an arithmetic monthly mean by 12 does not recover realized annual return. Sample SD from one observation is undefined, not zero.

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.

You can now explore…

Equal means can hide different volatility
Compare sample and population SD denominators

Term reference2 terms

Open a term when you need a reminder. Underlined words in the article work too.