# What is Standard Deviation? Formula for calculating standard deviation

Standard Deviation is a statistical tool that measures the dispersion of a data set against its mean and is calculated as the square root of the variance.

## What is the standard deviation?

Standard deviation is a statistical and financial measure applied to the annual rate of return of an investment, to reveal fluctuations in the history of that investment.

The larger the standard deviation of a security, or the greater the variance between its share price and its average value, indicates a wider range of prices. For example, a volatile stock has a high standard deviation, while a stable blue-chip stock’s standard deviation is usually quite low.

The standard deviation is calculated as the square root of the variance, calculated by determining the difference between each data point from the mean. If a data point is far from the mean, the point has a high deviation in the data set, the more spread out the data is, the higher the standard deviation.

## Standard deviation formula

Inwhich:

xi is the value of point i in the data set

x̄ is the value of the data set

n is the total number of observations in the data set

The mean x value is calculated by summing all observations and dividing by the number of observations.

The variance for each data point is calculated by subtracting the value of the observation from the mean. The result is then squared and divided by the number of observations minus one.

Square root of variance to find the standard deviation.

## Usability of standard deviation

Standard deviation is a particularly useful tool in either investment strategy or trading as it measures the volatility of markets and stocks, and ultimately predicts the performance of the investment.

For example, investors need to consider that positive growth funds often have a higher standard deviation than stock indices, as their portfolio managers bet a greater risk of risk. higher than average returns.

A lower standard deviation is not necessarily better, it all depends on the investment the investor has and whether or not they are willing to take risks. When there is volatility in a portfolio, investors should consider their individual tolerance to this volatility and their overall investment goals.

Risk-averse investors may be comfortable with strategies to invest in assets with higher than average volatility, while conservative (or risk-averse) investors do not.

Standard deviation is one of the main fundamental risk measures used by analysts, portfolio managers, and financial advisories. A large difference shows that the profitability of a fund is much different from the expected return. Due to its easy-to-understand nature, this statistical tool is frequently used for reporting to clients and investors.

## Standard deviation and variance

Variance is calculated by subtracting the mean of the observations from the average value, then squaring each of these results and finally averaging these results. The standard deviation is the square root of the variance.

Variance helps determine the spread of the observations when compared to the mean. Large variance indicates more variation in data set values ​​and there may be greater gaps between the values ​​of observations. If all the observations were close together, the variance would be smaller. However, this concept is much more difficult to understand than the standard deviation, since the variance denotes a squared result.

Standard deviations are usually easier to visualize and apply. Standard deviation is expressed in the same unit of measure as the data, using standard deviations, statisticians can determine whether the data has a normal distribution or a different mathematical relationship.

If the data had a normal distribution, then 68% of the observations would be within a standard deviation range to the median or mean. The variance due to squaring up causes many data points to fall outside the standard deviation, also known as outliers. The smaller variance results in more data close to the mean.

The biggest limitation of using the standard deviation is that it can be affected by outliers and negative values. The standard deviation assumes the normal distribution and considers all uncertainty to be risky, even if it benefits the investor, for example when returns are above average.

## An example of a standard deviation

Suppose we have the observations 5, 7, 3, and 7, totaling 22. Then, you would divide 22 by the number of observations, in this case 4 to get 5.5. We have the mean: x̄ = 5,5 and N = 4.

The variance is determined by subtracting each observation for the mean, giving the results -0.5, 1.5, -2.5, and 1.5, respectively. Each of these values ​​is then squared, equal to 0.25, 2.25, 6.25, and 2.25. Adding the squares then dividing by N minus 1, equals 3, gives a variance of approximately 3.67.