A measure of variation between a set of values.

**Scenario 1:** A normal distribution is a return distribution that is symmetrical about its mean. In other words, the data is centered and not skewed to either the right or left and the mean is equal to the median and that is equal to the mode.

So Imagine a perfect and evenly distributed bell curve with 5 evenly divided intervals. 99% of the observations fall within three standard deviations of the mean, the extreme left, and right of the curve account for .5%.

Those intervals between set values are standard deviations.

**Scenario 2:** If you were to graph the average income of individuals in the U.S. using the variables of age and education level, you would find a median income for certain age brackets. From this information, you could use statistical deviation to find percentiles within the data set.

Those percentiles could be used to identify earnings information for certain groups within the data set.

## Definition

To clearly grasp the idea and further define statistical deviation let’s use a more in-depth example and walk through some more data which also incorporates the concept of normal distribution.

Consider the population of a typical town in U.S. The average height of an adult male is 70 inches with a standard deviation 6 inches and average height of an adult female is 65 inches with a standard deviation of 3.5 inches.

Average is the sum of height’s of all the male or female population in the town divided by the number of male or female residents in the town. Standard Deviation is a measure of dispersion, i.e. a measure of distribution of sample heights away from the mean.

According to the theory of normal distribution, popular for it’s bell shaped curve, 68% of the population will be 1 standard deviation away from the mean (i.e. 68% of the male population in the town will have a height in the range of 69 to 71 inches ). 95% of the population will be 2 standard deviation away from the population mean (i.e. 95% of the male population in the town will be in the range of 68 to 72 inches ). Finally 99.7% of the population will be three standard deviation away from the population mean (i.e. 99.7% of the male population in the town will be in the range of 67 to 73 inches )

In this context, it is clear where standard deviation can be applied in real life, and how normal distribution helps us confine data and theorize more accurate expectations.

## Deep Analysis

Standard deviation measures the dispersion of data relative to its mean. That being said, the standard deviation is a mathematical necessity and inherent for businesses that involve investing, trading, and measuring market volatility.

Standard deviation is relied upon to hypothesize trends, and capitalize off the performance of the market or even its decline. A standard deviation is a tool used to mitigate and understand risk and tolerance for investment objectives.

Ultimately standard deviation is a useful mathematical tool, not limited to census and stock markets, but applicable to anything in the world with data that needs refinement.

### Drawbacks

Standard deviation does have its downside. Data in a set can always be affected by extreme figures on the periphery of data sets. Standard deviation works with normal distribution, but values which are anomalies can be observed as risky.

### Practical Applications

Inversely, Standard deviation can ascertain and solve absolute ranges, it is also used to postulate degrees of uncertainty.

In physical science, for example, the reported standard deviation of a group of repeated measurements gives the precision of those measurements. When deciding whether measurements agree with a theoretical prediction, the standard deviation of those measurements is of crucial importance: if the mean of the measurements is too far away from the prediction (with the distance measured in standard deviations), then the theory being tested probably needs to be revised.

For example, in industrial applications, the weight of products coming off a production line may need to comply with a legally required value. By weighing some fraction of the products an average weight can be found, which will always be slightly different from the long-term average. By using standard deviations, a minimum and maximum value can be calculated that the averaged weight will be within some very high percentage of the time (99.9% or more). If it falls outside the range then the production process may need to be corrected.