Standard Deviation Formula

What is Standard Deviation Formula?

Standard Deviation (SD) is a popular statistical tool that is represented by the Greek letter ‘σ’ and is used to measure the amount of variation or dispersion of a set of data values relative to its mean (average), thus interpret the reliability of the data. If it is smaller then the data points lies close to the mean value, thus shows reliability. But if it is larger then data points spreads far from the mean.

The formula of standard deviation is given below

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For eg:
Source: Standard Deviation Formula (wallstreetmojo.com)

Where:

• xi = Value of each data point
• x̄ = Mean
• N = Number of data points

• Standard deviation is most widely used and practiced in portfolio management services, and fund managers often use this basic method to calculate and justify their variance of returns in a particular portfolio.
• A high standard deviation of a portfolioStandard Deviation Of A PortfolioPortfolio standard deviation refers to the portfolio volatility calculated based on three essential factors: the standard deviation of each of the assets present in the total portfolio, the respective weight of that individual asset, and the correlation between each pair of assets of the portfolio.read more signifies the there is a large variance in a given number of stocks in a particular portfolio, whereas, on the other hand, a low standard deviation signifies a less variance of stock among themselves.
• A risk-averse investor will only be willing to take any additional risk if he or she is compensated by an equal or a larger amount of return in order to take that particular risk.
• A more risk-averse investor may not be comfortable with his standard deviation and would want to add in safer investment such government bonds or large-cap stocksLarge-cap StocksLarge-cap stocks refer to stocks of large companies with value, also known as the market capitalization of 10 billion dollars or more, and these stocks are less risky than others and are stable. They also pay a good dividend and return, and it is the safest option to invest.read more in its portfolio or mutual funds for that matter in order to diversify the risk of the portfolio and its standard deviation and variance.
• The variance and the closely-related standard deviation are measures of how spread out a distribution is. In other words, they are measures of variability.

Steps to Calculate Standard Deviation

Examples

Example 1

The data points are given 1,2, and 3. What is the standard deviation of the given data set?

Solution:

Use the following data for the calculation of the standard deviation.

So, the calculation of variance will be –

Variance = 0.67

The calculation of standard deviation will be –

Standard Deviation = 0.82

Example #2

Find the standard deviation of 4,9,11,12,17,5,8,12,14.

Solution:

Use the following data for the calculation of the standard deviation.

The calculation of mean will be –

First, find the mean of the data point 4+9+11+12+17+5+8+12+14/9

Mean = 10.22

So, the calculation of variance will be –

The variance will be –

Variance = 15.51

The calculation of standard deviation will be –

Standard Deviation = 3.94

Variance = Square rootSquare RootThe Square Root function is an arithmetic function built into Excel that is used to determine the square root of a given number. To use this function, type the term =SQRT and hit the tab key, which will bring up the SQRT function. Moreover, this function accepts a single argument.read more of standard deviation.

Example #3

Use the following data for the calculation of the standard deviation.

So, the calculation of variance will be –

Variance = 132.20

The calculation of standard deviation will be –

Standard Deviation = 11.50

This type of calculation is frequently being used by portfolio managers to calculate the risk and return of the portfolio.

Relevance and Uses

• Standard deviation is helpful is analyzing the overall risk and return a matrix of the portfolio and being historically helpful. It is widely used and practiced in the industry. The standard deviation of the portfolio can be impacted by the correlation and the weights of the stocks of the portfolio.
• As the correlation of the two asset classesAsset ClassesAssets are classified into various classes based on their type, purpose, or the basis of return or markets. Fixed assets, equity (equity investments, equity-linked savings schemes), real estate, commodities (gold, silver, bronze), cash and cash equivalents, derivatives (equity, bonds, debt), and alternative investments such as hedge funds and bitcoins are examples.read more in a portfolio reduces the risk of the portfolio, in general, reduces it is however not necessary all the time that equally weighted portfolio provides the least risk among the universe.
• A high Standard Deviation may be a measure of volatility, but it does not necessarily mean that such a fund is worse than one with a low Standard Deviation. If the first fund is a much higher performer than the second one, the deviation will not matter much.
• Standard deviation is also used in statisticsStatisticsStatistics is the science behind identifying, collecting, organizing and summarizing, analyzing, interpreting, and finally, presenting such data, either qualitative or quantitative, which helps make better and effective decisions with relevance.read more and is widely taught by professors among various top universities in the world however, the formula for standard deviation is changed when it is used to calculate the deviation of the sample.
  • The equation for SD in Sample = just the denominator is reduced by 1

Recommended Articles

This has been a guide to Standard Deviation Formula. Here we learn how to calculate standard deviation using its formula along with practical examples and a downloadable excel template. You can learn more about financial modeling from the following articles –

• Examples of the Correlation
• Standard Deviation Excel Formula
• Formula of Sample Standard Deviation
• Formula of Relative Standard Deviation