The Relationship Between Variance and Standard Deviation
When talking about risk, 'variance' and 'standard deviation' come up in turns. The two are like siblings—so exactly how are they connected?
Variance is the 'average of the squared deviations'
Variance is the value you get by squaring how much each data point deviates from the mean (the deviation) and averaging those.
Why square? Because deviations can be positive or negative, so simply adding them cancels them out to 0. So you square them to make everything positive, then take the average.
Squaring also has the effect of reflecting large deviations more heavily. In other words, variance reacts more sensitively to 'values far from the mean.'
Standard deviation is the square root of variance
Variance has one problem. Because the deviations were squared, the unit becomes squared too. Squaring a return (%) gives you a strange unit, 'percent squared.'
So standard deviation is variance with a square root applied to return the unit to its original form.
- Variance = the average of deviation² - Standard deviation = √variance
For example, if the variance is 0.0225 (= 0.15²), the standard deviation is 0.15, i.e., 15%. That's why, when people say 'volatility of 15%' in practice, they almost always mean the standard deviation. Variance is the calculation process, and standard deviation is the result value we read.
So why is variance also used separately?
Even though standard deviation is easier to read, variance survives because of its 'good additive property.'
When combining mutually independent assets, standard deviations cannot simply be added, but variances can. That's why the formulas for calculating a portfolio's risk (the covariance matrix) are all built from variances and covariances.
In short, do the calculations and theory with variance, and when showing it to people, apply the square root and speak of it as standard deviation. The two are the same information dressed in different clothes.
Both standard deviation and variance measure only 'scatter relative to the mean.' They cannot distinguish an upward surge from a downward collapse, and both share the same limitation of underestimating fat tails (extreme values).
常见问题
Q. If the variance is large, is the standard deviation always large?
Yes. Standard deviation is the square root of variance, so if the variance grows, the standard deviation must grow too. The two always move in the same order. However, because of the square root, the size of the increase differs—when the variance becomes 4 times as large, the standard deviation becomes 2 times as large.
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