# Standard deviation

Standard deviation is a measure of how widely observations, such as daily stock returns, vary around their average, expressed in the same units as those observations.

## What it means

For an investor, the useful question is often how much a stock’s returns have fluctuated around their historical average. Standard deviation puts a number on that dispersion. If it is calculated from daily percentage returns, its unit is percentage points per day; if returns are expressed as decimals, the result is a decimal return per day.

A larger standard deviation means returns in the measured period were more spread out. It does not say whether the next return will be positive or negative. Standard deviation of returns is commonly used as a measure of historical volatility, but it describes the data and period chosen—not a fixed property of a stock.

## How it is calculated

For a sample of returns, use `s = sqrt[Σ(rᵢ − r̄)² / (n − 1)]`.

- `s` is the sample standard deviation, in the same units as the returns.
- `rᵢ` is return number `i`; `r̄` is the arithmetic mean of the sampled returns.
- `Σ` means add the squared differences for all returns; `n` is the number of returns.
- `n − 1` is the sample denominator, used when the observed returns are treated as a sample. If the data represent the entire population being described, divide by `n` instead.

For daily stock returns, a simple daily return can be calculated as `(today’s price / previous day’s price) − 1`. Be consistent about whether prices include dividend and stock-split adjustments. The calculation needs at least two returns when using the sample formula.

## Worked example

Suppose a hypothetical stock has five daily returns: −2%, −1%, 0%, 1% and 2%. These are illustrative numbers, not observed market data.

1. Find the mean: `(−2% − 1% + 0% + 1% + 2%) / 5 = 0%`.
2. Subtract that mean from each return: −2, −1, 0, 1 and 2 percentage points.
3. Square and add the differences: `4 + 1 + 0 + 1 + 4 = 10` squared percentage points.
4. Divide by `n − 1 = 4`: the sample variance is `10 / 4 = 2.5` squared percentage points.
5. Take the square root: `sqrt(2.5) ≈ 1.58` percentage points per day.

The result describes dispersion across those five returns. It is not a prediction that the next day’s return will fall within 1.58 percentage points of zero. Using `n = 5` instead would give a different result: a population standard deviation of about 1.41 percentage points.

## Why it matters before you trade

A daily return standard deviation helps put a proposed position’s possible day-to-day movement in context. For example, a hypothetical US$1,000 position and a daily return standard deviation of 1.58% correspond to US$15.80 of position-value variation for a one-standard-deviation return, assuming the position size is unchanged. That amount is a scale measure, not a loss limit or a forecast.

A pre-trade analysis may consider volatility alongside position size, liquidity and execution costs. Standard deviation alone omits trading frictions and does not capture every risk: sudden gaps, extreme losses and changes in volatility can make realized outcomes very different from a recent historical estimate.

## How PreTrAIde uses it

PreTrAIde estimates daily volatility using blended estimators: Parkinson, close-to-close, Yang-Zhang and simplified GARCH(1,1). Close-to-close return volatility relates to the standard deviation concept described here; the blended estimate should not be assumed to equal the simple sample calculation in the example.

In its pre-trade analysis, PreTrAIde also calculates 1-day 95% value at risk and a 0–100 volatility score for each stock and the basket. It returns a GO, NO-GO or MODIFY verdict with a risk assessment and plain-language reasoning. These outputs support a decision before an order is placed; they are not guarantees or recommendations.

## Common mistakes

- **Mixing units:** A standard deviation of 0.0158 in decimal-return terms equals 1.58% in return terms, or 1.58 percentage points around a percentage mean.
- **Confusing price and return dispersion:** The standard deviation of prices is measured in currency units; the standard deviation of returns measures proportional changes. They answer different questions.
- **Treating it as a worst-case loss:** Large moves can exceed one standard deviation, and standard deviation counts upside and downside deviations alike.
- **Comparing unlike periods:** Daily and annualized figures are not interchangeable. Multiplying a daily figure by the square root of the number of trading days is a convention that relies on assumptions about returns.
- **Assuming normal returns:** A standard deviation does not, by itself, establish probabilities for future gains or losses.

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Source: https://pretraide.com/glossary/standard-deviation