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2 min readTapeboard Staff

What is Standard Deviation? Definition, Formula, and Example

Standard deviation measures how far a stock's returns scatter around their average, and it is the base unit of risk used in volatility, position sizing, and options pricing.

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What is Standard Deviation?

Standard deviation is the statistical measure of how widely a set of values disperses around its mean. In trading, the values are a security's periodic returns — daily, weekly, or monthly — and the result quantifies volatility: a stock with a 2% daily standard deviation swings twice as hard as one with a 1% daily standard deviation. It is the single most-used risk number in finance. Implied volatility in options, the bands in Bollinger Bands, Value at Risk, and the Sharpe ratio are all built directly on standard deviation.

How Standard Deviation Is Calculated

For a series of n returns r₁…rₙ with mean μ:

σ = √[ Σ(rᵢ − μ)² / (n − 1) ]

Steps:

1. Compute the mean return μ over the lookback window.

2. Subtract the mean from each return and square the difference.

3. Average the squared differences (dividing by n−1 for a sample).

4. Take the square root. The result is in the same units as the returns.

Traders annualize it for comparability: annualized σ = daily σ × √252 (252 trading days). A 1.5% daily standard deviation annualizes to roughly 23.8%. Under a normal distribution, ~68% of returns fall within ±1σ of the mean, ~95% within ±2σ, and ~99.7% within ±3σ — the basis of Bollinger Band interpretation and options one-standard-deviation moves.

Worked Example

Take five daily returns for NVDA: +2.0%, −1.0%, +3.0%, −0.5%, +1.5%.

  • Mean μ = (2.0 − 1.0 + 3.0 − 0.5 + 1.5) / 5 = 1.0%
  • Squared deviations: (1.0)², (−2.0)², (2.0)², (−1.5)², (0.5)² = 1, 4, 4, 2.25, 0.25 → sum = 11.5
  • Variance = 11.5 / (5 − 1) = 2.875
  • σ = √2.875 ≈ 1.70% daily, or 1.70% × √252 ≈ 27% annualized

Compare with a utility like DUK running a 0.6% daily σ (~9.5% annualized): NVDA carries roughly three times the return dispersion, which is exactly why NVDA option premiums are richer.

When Traders Use Standard Deviation

  • Position sizing: risk-parity and vol-targeting frameworks size positions inversely to σ — a 2σ-vol stock gets half the capital of a 1σ stock.
  • Stops and targets: ATR-style stops are often set at 1.5–2× the standard deviation of daily moves to avoid noise stop-outs.
  • Options context: comparing implied volatility to the stock's historical (realized) standard deviation tells you whether options are priced rich or cheap.
  • Mean-reversion entries: a close beyond 2σ from a moving average flags a statistically stretched move.

Limitations and Common Misconceptions

Standard deviation treats upside and downside dispersion identically — a stock that gaps up violently has high σ even though holders profited. It also assumes returns are roughly normal; equity returns have fat tails, so "3-sigma events" happen far more often than the 0.3% the normal curve implies. Finally, σ is backward-looking: a quiet stock can have low trailing standard deviation the day before a binary event like an FDA ruling. It measures past volatility, not future risk.