In mathematics, standard deviation and variance are two very important concepts. These concepts are popular in the fields of finance, investments and economics.
Both the values of standard deviation and variance are calculated using the mean of a certain group of numbers. Mean is the average value of the group of numbers. From this mean, the variance is determined. It is found out by calculating the degree of difference of each number from the mean value. Standard deviation is the square root of the variance value.
Let us delve deeper into the details of both concepts.
What is a Variance?
In the field of statistics, a variance is a measure that represents the spreading out of numbers in a group or collection. Variance determines the average degree of how the mean varies from each number in the group. Furthermore, it gives us an actual estimation of data volatility.
For calculating the variance, the mean of the number set is taken. The difference between the numbers and the mean is determined. The differences are then squared.
When the value is made positive. The final result is obtained by dividing the sum of squares by the total quantity of numbers in the set.
The variance of a set can be a small value. This indicates that these numbers are closer to the mean value. If the value is larger, then the numbers are dispersed from each other and the mean.
This value can also be negative. A negative variance value represents that the numbers in the data set are identical. The non zero variance values are positive.
Formula of Variance
The formula for variance is:
σ² = ∑ni=1 (xi − x¯ )² / n
- Here, xi is the number at the ith data point
- n is the total number of data points in the set
- x¯ is the mean of the data points
What is Standard Deviation?
Standard deviation is used to measure the dispersion of a set of data from their mean value. This value is calculated by determining the square root of the variance of the data set. In any data set, this is the measure of the absolute variability.
The data points may have high variability from the mean. This means that the variability is high within the data set. This will indicate a high value of standard deviation. Therefore, the data in the set will be more spread out.
Karl Pearson in 1893 introduced the concept of standard deviation. It is also called the root mean square division.
It can be determined using the following steps:
- The mean of the data set has to be figured. Adding all the numbers and dividing them by the total number of data points will give the mean.
- Then the variance has to be obtained. The mean value is subtracted from each number. The resultant values are squared. Then they are summed up and divided by the total number of data points that are lesser than 1.
- The standard deviation is the square root of the variance value.
Standard deviation is a very important tool used for developing trading and investment strategies.
Formula of Standard Deviation
The formula for standard deviation is:
Standard deviation = √∑ni=1 (xi − x¯ )² / (n-1)
- Here, xi is the number at the ith data point
- n is the total number of data points in the set
- x¯ is the mean of the data points
Standard Deviation vs Variance
Basis of comparison |
Standard Deviation |
Variance |
Definition |
It is used to measure the dispersion of values within a set against their mean |
It is used to measure the variability of the numbers in a data set from their mean |
Symbol used |
σ denotes standard deviation |
σ² denotes variance |
Indication |
This value indicated how numbers in a particular group are spread out |
This indicates how numbers in a group differ from their mean value |
Units |
This is expressed using the units of the numbers of the data set |
This is expressed using square units |
Calculation |
It can be calculated by the square root of the variance value of the data set |
This can be calculated by using the average value of the squared deviations of the values from their mean value |
Uses of Standard Deviation and Variance
Variance
As variance is the measure of variability, it is used by investors. They use it while buying security or making an investment. The variance value determines how risky or safe the decision will be. For investments, variance of returns on the assets is analysed. This is done to determine the best asset.
Variance is a measure of volatility. So the investors use this to measure risks associated with buying security. It is also useful in allocating assets. The variance value is measured alongside correlation. This helps investors to develop a portfolio that minimises risks. This enhances the chances of gaining better returns on their investments.
It is also useful for analysis of opinion polls. It is very difficult to collect data for a huge population. A criterion is used for sampling the data of the population. Variance is used there to find the variation between the mean and the data.
Standard deviation
For investment, a standard deviation value is applied to its rate of return. The historical volatility can be determined this way. If the value of deviation is large, the variance between the mean and price will be large. A volatile stock will have a high deviation value. But, a blue-chip stock will have a low deviation.
Financial analysts and investors use standard deviation to analyse possible risks. Investment companies use this measure to understand risks associated with mutual funds. The amount of dispersion shows the type of returns that they might get. They get an idea of how the returns vary from the expected return on that fund.
Some traders use standard deviation to perform a Monte Carlo simulation. This is done to assess the risks of the project.
Conclusion
Variance and standard deviation are two widely used statistical concepts affecting major decisions in finance and data analysis. Stock traders use standard deviation to assess market volatility. Risk assessment is another crucial function.
On the other hand, the standard deviation is used in finance as well as weather forecasting. It is useful to understand the variation of temperature. Variance provides an idea about the degree of uncertainty in a data set.
Variance can be used to analyse the volatility of a population data set.