Standard Deviation — Technical Indicators

20 — Volatility

Standard Deviation#

Statistical1 min read

◆ The PatternHow widely values spread around their mean, the single number most of risk measurement is built on

Standard Deviation — the statistical building block of volatility measurement.

σ = √(Σ(x − μ)² / N)
where μ = mean, N = number of observations

Higher σ = more volatility = prices are spread far from the mean.

In markets:
• Historical Volatility = σ of returns × √252 (annualized)
• For normal data, 1σ covers ~68% and 2σ ~95% → the basis of Bollinger Bands
• 3σ events are "rare" but happen more often in markets than normal distributions predict (fat tails)

Worked example

Day 15: the last five closes (104, 102, 99, 101, 104) average 102.0. The deviations are 2, 0, −3, −1 and 2; squared, 4, 0, 9, 1 and 4; their mean is 3.6, and its square root 1.90. Measured on daily returns instead of prices it is about 2.6% a day — returns are what risk models use.

The example data: 15 days, used in every indicator here

Day	High	Low	Close	Volume
1	101	99	100	50,000
2	103	100	102	62,000
3	103	100	101	48,000
4	105	101	104	70,000
5	108	104	107	85,000
6	108	105	106	60,000
7	110	106	109	90,000
8	110	107	108	72,000
9	108	104	105	80,000
10	105	102	103	66,000
11	105	102	104	55,000
12	104	101	102	75,000
13	102	98	99	95,000
14	102	99	101	64,000
15	105	101	104	71,000

Where it misleads

It treats moves up and down alike, so a sharp rally counts as “risk” the same as a fall.
It assumes the spread is stable. Volatility clusters, so after a calm spell yesterday’s figure understates tomorrow’s.
Returns have fat tails: extreme days come far more often than a bell curve with this standard deviation predicts.

In code

df['sd5'] = df['close'].rolling(5).std(ddof=0)
df['ret_sd5'] = df['close'].pct_change().rolling(5).std()   # the same idea on returns

Python, in your browser — edit the code and run it again

Assumes import numpy as np, import pandas as pd, and a DataFrame df with columns high, low, close and volume — the 15 days above.

Sources

B. Mandelbrot, “The Variation of Certain Speculative Prices”, Journal of Business 36(4), 1963 — fat tails in prices
J. C. Hull, Options, Futures, and Other Derivatives, Pearson — estimating volatility from returns

Last reviewed October 2026 · How this site checks its content · Report a mistake

Pattern bridge: This is standard deviation from statistics, applied to price returns. In ML, it’s the denominator in batch normalization and the width of Gaussian distributions.

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