If the differences between each value and the mean (the deviations) are simply averaged, the positive and negative ones cancel out to zero, so the spread is measured by the variance, the average of the squared deviations (explained here in the form that takes the average over all the data)1. The variance is a squared quantity and so has a different unit from the original data. Taking its positive square root gives the standard deviation, which expresses the spread in the same unit as the original values. The larger the variance, the larger the standard deviation.

Consider a hypothetical example in which an asset's closing prices over the past five days were 94 yen, 98 yen, 100 yen, 102 yen and 106 yen. The mean is 100 yen, and the deviations are −6, −2, 0, 2, 6-6,\ -2,\ 0,\ 2,\ 6 yen. The variance, the average of the squared deviations, is (36+4+0+4+36)÷5=16(36 + 4 + 0 + 4 + 36) \div 5 = 16. The standard deviation is its square root, 4 yen.

In derivatives, the standard deviation of the rate of change of a price, expressed as an annual rate, is used as historical volatility2. Volatility is the size of price movement expressed by this standard deviation. A normal distribution is determined in shape by its mean and standard deviation3, and a value's position can be expressed by how many standard deviations it lies from the mean4. A large standard deviation only shows that values tend to move far from the mean. It does not show whether they go up or down.

Notes

  1. Statistics Bureau of Japan, "データの散らばりを見る" (Looking at the spread of data) (Naruhodo Statistics Gakuen, in Japanese), section on standard deviation and variance. ↩

  2. Japan Exchange Group, "ヒストリカル・ボラティリティ（HV）" (Historical volatility (HV)) (glossary, in Japanese). Described as the standard deviation of the underlying asset's daily rate of change, converted to an annual rate. ↩

  3. NIST/SEMATECH e-Handbook of Statistical Methods, "Normal Distribution" (1.3.6.6.1). The probability density function is determined by the location parameter μ (the mean) and the scale parameter σ (the standard deviation). ↩

  4. Statistics Bureau of Japan, "推測統計" (Inferential statistics) (Naruhodo Statistics Gakuen, in Japanese), section "正規分布" (Normal distribution): for normally distributed data, the mean and standard deviation let one read off, almost exactly, where a value sits within the whole. ↩