Standard Deviation vs. Variance: An Overview
Standard deviation and variance are two basic mathematical concepts that have an important place in various parts of the financial sector, from accounting to economics to investing. Both measure the variability of figures within a data set using the mean of a certain group of numbers. They are important to help determine volatility and the distribution of returns. But there are inherent differences between the two. While standard deviation measures the square root of the variance, the variance is the average of each point from the mean.
Key Takeaways
- Standard deviation and variance are two key measures commonly used in the financial sector.
- Standard deviation is the spread of a group of numbers from the mean.
- The variance measures the average degree to which each point differs from the mean.
- While standard deviation is the square root of the variance, variance is the average of the squared difference of each data point from the mean.
- The two concepts are useful and significant for traders, who use them to measure market volatility.
Standard Deviation
Standard deviation is a statistical measurement that looks at how far a group of numbers is from the mean. Put simply, standard deviation measures how far apart numbers are in a data set.
This metric is calculated as the square root of the variance. This means you have to figure out the variation between each data point relative to the mean. Therefore, the calculation of variance uses squares because it weighs outliers more heavily than data that appears closer to the mean. This calculation also prevents differences above the mean from canceling out those below, which would result in a variance of zero.
But how do you interpret standard deviation once you figure it out? If the points are further from the mean, there is a higher deviation within the data. But if they are closer to the mean, there is a lower deviation. So the more spread out the group of numbers are, the higher the standard deviation.
As an investor, make sure you have a firm grasp on how to calculate and interpret standard deviation and variance so you can create an effective trading strategy.
Variance
A variance is the average of the squared differences from the mean. To figure out the variance, calculate the difference between each point within the data set and the mean. Once you figure that out, square and average the results. Using software like Excel can help you in this process.
For example, if a group of numbers ranges from one to 10, you get a mean of 5.5. If you square the differences between each number and the mean and find their sum, the result is 82.5. To figure out the variance:
- Divide the sum, 82.5, by either N (population variance) or N-1 (sample variance)
- In this example N is 10 and N-1 would be 9.
- The result is a sample variance of 82.5/9 = 9.17 and a population variance of 8.25.
Note that the standard deviation is the square root of the variance so the standard deviation is about 3.03 for a sample data set. The standard deviation for a population data set would be 2.87. The sample formula is used when the data set represents a random sample from the entire population in question. The population formula is used when there is data from the entire population being studied or considered.
The mean is the average of a group of numbers, and the variance measures the average degree to which each number is different from the mean. The extent of the variance correlates to the size of the overall range of numbers, which means the variance is greater when there is a wider range of numbers in the group, and the variance is less when there is a narrower range of numbers.
Key Differences
Other than how they’re calculated, there are a few other key differences between standard deviation and variance. Here are some of the most basic ones.
- Standard deviation measures how far apart numbers are in a data set. Variance, on the other hand, gives an actual value to how much the numbers in a data set vary from the mean.
- Standard deviation is the square root of the variance and is expressed in the same units as the data set. Variance can be expressed in squared units or as a percentage (especially in the context of finance).
- Standard deviation can be greater than the variance since the square root of a decimal is larger (and not smaller) than the original number when the variance is less than one (1.0 or 100%).
- The standard deviation is smaller than the variance when the variance is more than one (e.g. 1.2 or 120%).
The table below summarizes some of the key differences between standard deviation and variance.
| Key Differences Between Standard Deviation and Variance | ||
|---|---|---|
| Standard Deviation | Variance | |
| What Is it? | The square root of the variance | The average squared differences from the mean |
| What Does it Indicate? | The spread between numbers in a data set | The average degree to which each point differs from the mean |
| How Is it Expressed? | The same as the units in the data set | In squared units or as a percentage |
| What Does it Mean? | A low standard deviation (spread) means low volatility while a high standard deviation (spread) means higher volatility | The degree to which returns vary or change over time |
Standard Deviation and Variance in Investing
These two concepts are of paramount importance for both traders and investors. That’s because they are used to measure security and market volatility, which plays a large role in creating a profitable trading strategy.
Standard deviation is one of the key methods that analysts, portfolio managers, and advisors use to determine risk. When the group of numbers is closer to the mean, the investment is less risky. But when the group of numbers is further from the mean, the investment is of greater risk to a potential purchaser.
Securities that are close to their means are seen as less risky, as they are more likely to continue behaving as such. Securities with large trading ranges that tend to spike or change direction are riskier.
Risk in and of itself isn’t necessarily a bad thing in investing. That’s because riskier investments tend to come with greater rewards and a larger potential for payout.
Example of Standard Deviation vs. Variance
To demonstrate how both principles work, let’s look at an example of standard deviation and variance.
Suppose you have a series of numbers and you want to figure out the standard deviation for the group. The numbers are 4, 34, 11, 12, 2, and 26. We need to determine the mean or the average of the numbers. In this case, we determine the mean by adding the numbers up and dividing it by the total count in the group:
(4 + 34 + 18 + 12 + 2 + 26) ÷ 6 = 16
So the mean is 16. Now subtract the mean from each number then square the result:
- (4 – 16)2 = 144
- (34 – 16)2 = 324
- (18 – 16)2 = 4
- (12 – 16)2 = 16
- (2 – 16)2 = 196
- (26 – 16)2 = 100
Now we have to figure out the average or mean of these squared values to get the variance. This is done by adding up the squared results from above, then dividing it by the total count in the group:
(144 + 324 + 4 + 16 + 196 + 100) ÷ 6 = 130.67
This means we end up with a variance of 130.67. To figure out the standard deviation, we have to take the square root of the variance, then subtract one, which is 10.43
What Does Variance Mean?
The simple definition of the term variance is the spread between numbers in a data set. Variance is a statistical measurement used to determine how far each number is from the mean and from every other number in the set. You can calculate the variance by taking the difference between each point and the mean. Then square and average the results.
What Does Standard Deviation Mean?
Standard deviation measures how data is dispersed relative to its mean and is calculated as the square root of its variance. The further the data points are, the higher the deviation. Closer data points mean a lower deviation. In finance, standard deviation calculates risk so riskier assets have a higher deviation while safer bets come with a lower standard deviation.
What Is Variance Used for in Finance and Investing?
Investors use variance to assess the risk or volatility associated with assets by comparing their performance within a portfolio to the mean. For instance, you can use the variance in your portfolio to measure the returns of your stocks. This is done by calculating the standard deviation of individual assets within your portfolio as well as the correlation of the securities you hold.
What Are the Shortcomings of Variance?
The variance of an asset may not be a reliable metric. Calculating variance can be fairly lengthy and time-consuming, especially when there are many data points involved. Variance doesn’t account for surprise events that can eat away at returns. And variance is often hard to use in a practical sense not only is it a squared value, so are the individual data points involved.
The Bottom Line
The standard deviation and variance are two different mathematical concepts that are both closely related. The variance is needed to calculate the standard deviation. These numbers help traders and investors determine the volatility of an investment and therefore allows them to make educated trading decisions.