**The most important information about the standard deviation at a glance**

**Definition: **Key statistic of the stochastic to measure the spread of a sample around the arithmetic mean

**Means:** Average distance of measured data sets from the mean

**Calculation: **The arithmetic mean and the variance must be available for the determination

**Normal distribution**: Most of the values are around the average value; the “two-thirds rule” applies

**Interpretation: **The higher the standard deviation, the greater the spread of the data

**Stock Analysis:** In share analysis, the ratio is used to determine the volatility of share prices

## Definition: What is the standard deviation?

that **standard deviation of a population** is denoted by the sign sigma “**σ**“ (Sigma) abbreviated. It is derived from the code for a **sample** estimated and in this case as “**p**” designated. It is the most important key figure in the statistics for determining the **Spread around the mean**.

The standard deviation is a concept from statistics, more precisely: stochasticity. It is one of the most important key figures for statistical analysis. The value is used in probability calculation and determines how strong the spread of the values in a data set is around a mean value.

As a statistical key figure for the spread of values in a sample, it is one of the** spread measure** and forms **mean square deviation of individual sample values from the statistical mean** (**arithmetic mean**) away. It indicates how far data from a sample deviates from the mean. So it is used when you want to analyze how much data points differ from each other.

**To determine the standard deviation of a value, the mean and variance must first be calculated.** The average value is called the “arithmetic mean” in statistics.

The standard deviation is calculated as follows:

**calculate the average value****calculate variance****Calculate the standard deviation**

## Standard Deviation: Formula and Determination

on a **example** the calculation of the standard deviation should be clarified: An online company would like to know how satisfied the customers are with the webshop. You start a poll. One of the questions is: “*What is the likelihood that you will visit our website again?” *The possible answers go on a scale from 0-10. Zero means “very unlikely” and ten means “very likely”. If we assume that all customers answered the survey with a value of 5, then the mean would be 5. There would be no spread around the mean.

**In this case, the standard deviation would be zero.** There would be no need for statistical analysis. Statistics aims to explain and analyze differences between data sets. In the case of a standard deviation of zero, there would be nothing to explain.

However, one is more likely **Normal distribution**: Most of the values are then around the average value. Values that deviate slightly from the mean are more common than values that deviate significantly. Once an average value is determined, the standard deviation can be used to see how much the values differ from each other. But first she has to **variance** be calculated.

We assume that we have the following data set as a result of the survey:

customer |
assessment |

1 | 5 |

2 | 3 |

3 | 4 |

4 | 7 |

5 | 8 |

6 | 7 |

7 | 5 |

**Arithmetic mean (average)**

X(A) = (5 + 3 + 4 + 7 + 8 + 7 + 5) / 7 = 5.6

The mean value of X(A) is 5.6.

**Calculation of the variance**

To get the variance, we average the sum of the squares of the deviation from the arithmetic mean:

S²(A) = [(5-5,6)² + (3-5,6)² + (4-5,6)² + (7-5,6)² + (8-5,6)² + (7-5,6)² + (5-5,6)²] / 7 = 2.8

The variance S²(A) is 2.8.

The variance is not in the same units as the original data set. Therefore, variance lends itself to an interpretation of volatility **Not appropriate**. In order to be able to interpret the spread, the standard deviation is calculated in the third step.

**Calculation of the standard deviation**

The standard deviation is calculated as the square root of the variance:

S(A) = S² √ (A) = √ 2.8 = 1.67

The standard deviation S(A) is 1.67.

## Interpretation of standard deviation: A rule of thumb

**There is a rule of thumb for the normal distribution:** If there is a normally distributed data set (*Gaussian bell curve*), the standard deviation from the mean can easily be read. Then probability theory helps to make better interpretations with the standard deviation. With a normal distribution of the data, about 68% of the data set is within one standard deviation of the arithmetic mean. About 95% of the data are within two standard deviations and about 99.7% are within three standard deviations.

We are talking about **“68-95-99.7 rule”**.

This is also called the “two-thirds rule” of probability theory.

**With normally distributed data, you can derive the standard deviation by looking at the mean. It can be guessed in which range most data moves. **

The higher the standard deviation, the greater the volatility of the data. The standard deviation is only interesting to statisticians in combination with the mean.

## Standard deviation: volatility of stock markets

Investors who invest in the financial market and in stocks in particular always come up with the concept **volatility** in touch. Unfortunately, the term is rarely defined, leaving many investors at a loss as to what **Stock market volatility** has on himself. The volatility is right **essential** for investment success. Investors therefore not only meet at **securities trading** on this indicator. Also at **Investment funds**, **raw material** (*raw materials*) and especially with **currency markets** there is no getting around volatility. But what really lies behind it?

### Volatility: what does the swing range say?

The code gives one **range of fluctuations** indicates what a particular course takes in a particular period of time. This increases both the fluctuation range (*price gain*) as well as below (*price loss*) calculated. To calculate the key figure, a **period of one year** firm. In technical analysis, however, it is in principle possible to change the period at will. But the extent to which larger time intervals or the volatility in very short sections are meaningful depends on the strategy.

Volatility describes this fluctuation range around a previously determined one **dimension value**. The measured value is **Average value of the price in the same period to be analyzed** taken. Afterwards **Distance of course from this benchmark** considered. For this find **standard deviation **their application again.

### volatility and standard deviation

For example, if you take a stock that rose 1 percentage point in month 1 (x) but lost 1 percentage point (y) the next month, the standard around which volatility unfolds is zero percent (z). The total period used for the volatility is therefore two months (n).

that **standard deviation** is therefore calculated with the square root: 1/n*((xz)²+(yz)²).

In this example, the volatility is the square root of the following formula: ½ * ((1%-0%)²+(-1%-0)²)

As a result, you get a **volatility of one percent** – that is, during the observed two-month period **The yield deviated from the standard by one percent**. In reality, the calculation of these values is, of course, much more complex. This is because larger time periods are considered with a real number base and more fluctuations. In practice, these calculations are therefore left to the IT systems. use investors **volatility dependent indicators** on their charts or look at the experts’ results, for example when evaluating an ETF.

Before making a purchase decision, a consideration of the fluctuation range is one of them **typical tasks** from investors. Because the greater the volatility, the greater it is **Risk of a losing transaction**. But since the fluctuation range is also larger upwards, it will **Possibility of greater returns**. Based on the individual risk profile, a **Opportunity-risk assessment** occur.

### The greater the volatility, the greater the risk

Different securities exhibit different levels of volatility. For example, company X may issue a share with a calculated volatility value of 8%, while company Y’s share has a volatility of 12%. based on **own risk profile and type of investor** now it’s time to decide which stock fits better in the portfolio.

While the shares of X in this example are likely to be much less risky to trade, the profit is relatively low. For securities from company Y, however, this development is exactly the opposite.

However, newcomers to the financial arena should aim for the lowest possible volatility to begin with and only venture into riskier investments as they gain experience.

## Portfolio Selection Theory: An Example

The more the price of a share fluctuates, the higher the risk associated with the security. The expected return from the security is all the more **fraught with risk**. The standard deviation measures the volatility in stock trading and thus the risk to which the return is exposed. It is determined by how much the return on the share fluctuates around the mean value within a period.

**In stock analysis, the mean is the expected return (expected value).**

Our example warehouse has achieved the following results over the past three years:

Year |
the share price at the beginning of the year |
share price at the end of the year |
yield |

2016 |
45 | 47 | 4.4% |

2017 |
47 | 53 | 12.8% |

2018 |
53 | 55 | 3.8% |

For the expected value of return, we assume an average of 2.6% for the years 2016-2018. The standard deviation for the stocks is then:

S = √ [(4,4-2,6%)² + (12,8-2,6%)² + (3,8-2,6%)²] / 3 = 6.02%

The expected annual standard deviation from its mean of 2.6% is 6.02% for the stock. Comparing the standard deviation of different stocks can support the investor in his buying decision.

## Conclusion on volatility

As an important indicator of stochastics dealing with changes in economic variables, the standard deviation has been successfully used in stock analysis. The fluctuations in stock prices over time are for stock market participants **of great importance**.

**Standard deviation is a way of estimating a stock’s earning potential independently of the rest of the stock market**. It should be noted that the historical value development of the data set is assumed for the future.

The volatility of financial products is thus measured as the standard deviation of changes in a share price.

**It is used as a risk measure of future profit and loss potential.**