## 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

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 portfolio 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 stocks 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

**Step 1:**First, the mean of the observations is calculated just like the average adding all the data points available in a data set and dividing it by the number of observations.**Step 2:**Then the variance from each data point is measured with the mean it can come as a positive or negative number then the value is squared and the result is subtracted by one.**Step 3:**The square of the variance which is calculated from step 2 is then taken to calculate the 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 = 0.33**

#### 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 root 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 classes 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 statistics 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 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

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