Standard deviation is a measure of the amount of variation or dispersion of a set of values in statistics. It tells you, on average, how far each value lies from the mean of the set. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wider range. Standard deviation is calculated as the square root of the variance. It is expressed in the same unit as the data, which makes it a useful measure of statistical dispersion if the center of the data is measured about the mean.
Standard deviation is often used in finance as a measure of the risk associated with price fluctuations of a given asset or the risk of a portfolio of assets. It is also a useful measure of spread for normal distributions, where data is symmetrically distributed with no skew. The mean and the standard deviation of a set of data are descriptive statistics usually reported together.
To calculate the standard deviation, you first need to calculate the variance by finding the average of the squared differences from the mean, and then take the square root of the variance. The formula for standard deviation is:
$$\sigma = \sqrt{\frac{\sum_{i=1}^{n}(x_i - \mu)^2}{n-1}}$$
where:
- $\sigma$ is the standard deviation
- $x_i$ is the value of the $i$th point in the data set
- $\mu$ is the mean value of the data set
- $n$ is the number of data points in the data set
Standard deviation is a more robust and accurate measurement compared to other observations because it is all-inclusive of observations, and each data point is included in the analysis.
