The standard deviation is the most commonly used measure of dispersion. It is calculated by summing up the squares of the differences of the observations from the mean and then dividing by the number of observations. It can be calculated using the formula, \text{Standard Deviation }= \frac{\sum_{i=1}^{n}(x_i-\bar{x})^2}{n}. WE now list out some of the advantages and disadvantages of standard deviation as a measure of dispersion.

**Advantages of Standard Deviation:**

- It is the most well-known measure of dispersion and is applicable in a wide variety of situations.
- The standard deviation takes all observations into consideration. This is in contrast to other measures of dispersion such as range which are not based on all observations.
- Since the formula involves squaring the error, the numerator \sum_{i=1}^{n}(x_i-\bar{x})^2 can be further simplified on expanding the square. This makes the formula amenable to further algebraic treatment. In fact, on expanding the square we can obtain an alternative formula as follows,\begin{align*}\text{Standard Deviation }&= \frac{\sum_{i=1}^{n}(x_i-\bar{x})^2}{n} \\ &= \frac{\sum_{i=1}^{n}(x_i)^2-\sum_{i=1}^{n}2x_i\bar{x}+\sum_{i=1}^{n}(\bar{x})^2}{n} \\ &= \frac{\sum_{i=1}^{n}(x_i)^2-2\bar{x}\sum_{i=1}^{n}x_i+(\bar{x})^2\sum_{i=1}^{n}1}{n} \\ &= \frac{\sum_{i=1}^{n}(x_i)^2-2\bar{x}\times n\bar{x}+(\bar{x})^2\times n}{n} \\ &= \frac{\sum_{i=1}^{n}(x_i)^2-2n(\bar{x})^2+n(\bar{x})^2}{n} \\ &= \frac{\sum_{i=1}^{n}(x_i)^2-n(\bar{x})^2}{n} \end{align*}
- The sample standard deviation is an unbiased estimator of the population standard deviation. The standard deviation is not affected by the fluctuations of sampling and therefore it can be used to estimate the standard deviation of the population.
- The standard deviation is used to calculate the skewness and kurtosis of the data which tells us about the symmetry and shape of the data.
- If we are given the standard deviation of two data sets, we can find the combined standard deviation by the combined standard deviation formula. There are no such formulas that give combined values for the other measures of dispersion.

**Disadvantages of Standard Deviation:**

- The standard deviation does not exactly measure the actual distance of each observation from the mean but the square of the differences of the observations from the mean.
- When squaring the differences, outliers will contribute a large value to the numerator because squaring large values makes them even larger. This means that the standard deviation gives more weightage to extreme values. Therefore the standard deviation is not resistant to the effect of outliers.
- The standard deviation ignores the actual sign of the error term (x_i-\bar{x}).
- The calculation is very cumbersome to carry out by hand since it involves extracting square roots. Although this drawback is easily taken care of by using calculators.

**Further Reading:**

- Advantages and Disadvantages of Mean in Statistics.
- Properties of Mode – Uses, Advantages, and Disadvantages.
- Advantages and Disadvantages of Histogram.