What is the difference between variance and standard deviation?

The variance and the standard deviation are two terms of great importance, both in statistics and in all branches of science and engineering. Both are measures of dispersion with respect to a central value, but depending on the context in which they are used, they can be defined in different ways.

In the fields of statistics and probability, variance and standard deviation measure how far the values ​​of a random variable (almost always represented by the letter X) differ from their mean value.

However, when these terms are used in science or engineering, the variance and standard deviation refer to the dispersion of a data series, either of an entire population or a sample, around the population or sample mean. . The standard deviation of a series of repetitive measurements using the same measuring instrument is also often used to give an idea of ​​the level of precision of said instrument.

In the first case, the variance and the standard deviation measure the variability of a random variable, while in the second, they measure the dispersion of experimental data. In either case, a variance or standard deviation of zero indicates no variation at all (the random variable is actually constant, or the data is all exactly the same), while a high value indicates the opposite.

These two terms are closely related and can sometimes be confused with each other, however there are key differences between the two that we’ll get to right away.

Differences Between Variance and Standard Deviation

1. They have different definitions

The first difference between these two statistical terms is their definition:

Definition of variance

In statistics, variance is defined as the expected value of the square of the difference between the value of a random variable and its mean value.

Mathematically, this is written as:

In a slightly less formal way, it can also be defined as the average of the squares of the differences between the individual data of a data series (population or sample) and its mean value.

Standard Deviation Definition

Regardless of the context in which it is used, the standard deviation, also known as the standard deviation, is defined as the positive square root of the variance.

Mathematically, this is written as:

2. They are represented with different symbols

Variance and standard deviation are represented in different ways both in statistics texts and in formulas and equations:

Variance:

• σ ² when referring to the population variance
• S ² when referring to the sample variance
• Var(X) when referring to the variance of a random variable, in this case X.

Standard deviation:

• σ when referring to the population standard deviation
• S when referring to the sample standard deviation
• SD(X) when referring to the standard deviation of a random variable, in this case X.

3. They have different formulas

For both the variance and the standard deviation, there are two formulas, depending on whether the data series for which the variance or standard deviation is being calculated are data from a population or from a sample.

Population variance formula (σ ² )

In either of the two formulas for the population variance, μ represents the population mean, X _i represents the ith population data value, and N represents the size of the population or the total number of data points.

Sample variance formula (S ² )

Here, x-bar represents the mean of the sample data (sample mean), x _i represents the value of the ith sample data, and n represents the size or total number of data in the sample.

Population Standard Deviation Formula (σ)

In the case of the standard deviation, it can be calculated in three different ways:

Sample standard deviation formula (s)

Here, too, one of three different ways can be used:

A note must be made with respect to the last two formulas. It is common that, when calculating the standard deviation, the variance is calculated first and then the square root is taken. The standard deviation is rarely determined using the latter equations without calculating the variance first, so the former almost always precedes the latter.

4. They have different units

Both the units of the variance and the standard deviation depend on the nature and units of the data or the random variable to which they refer, however, the units are different in each case.

The standard deviation has the same units as the original data or the random variable, while the variance comes in these units squared.

Example:

If you have the data of the weights in kilograms (kg) of a sample of 8th grade students in a certain educational institution, then the variance of said data will have units of kg 2 while the standard deviation will come in ^kg .

5. They differ in their interpretation

For both the variance and the standard deviation, the interpretation is the same as that already mentioned: if they are worth zero, then there is no dispersion and all the data are exactly equal to each other; if they are small values ​​then there will be little scatter and if they are large there will be a lot of scatter.

However, when understanding what it means to be a large or small value, standard deviation values ​​are much easier to interpret than variance values, since they are in the same units as the data. This is not so simple in the case of variance.

6. They differ in their sensitivity to extreme values

As measures of dispersion, both the variance and the standard deviation suffer from sensitivity to the existence of extreme values ​​(either very high or very low). This means that when describing a data series in which all the data are very similar except for one that is much larger or smaller than the others, neither the variance nor the standard deviation will represent well the spread of the data (both will give values large despite the fact that the vast majority of the data show very little dispersion).

However, when comparing the variance to the standard deviation, the variance is much more sensitive to these outliers since all deviations are squared, while the standard deviation is not.

7. They differ in their mathematical properties

The last difference we’ll look at actually encompasses several much deeper differences that are important primarily to statisticians (or those studying statistics).

As mathematical functions, variance and standard deviation differ in terms of the effect of multiplying the data by a constant, the effect of adding constants, adding random variables together, raising to powers, and so on.

These differences, however, are outside the scope of this article.

Variance and Standard Deviation Calculation Example

Suppose that a sample of 12 bulls from a local producer was weighed. The weights, in kilos, are presented below:

You are asked to determine the variance and standard deviation of this sample.

SOLUTION

As mentioned above, when having a data series, it is convenient to first determine the variance and then the standard deviation.

Calculation of the sample variance (S ² )

We will use the second sample variance formula, as it is more practical. To do this, the following steps are followed:

• Step 1: A vertical list is made of all the data
• Step 2: The square of each data is calculated and written next to it in a new column.
• Step 3: All the data are added and the result is recorded at the end of the first column.
• Step 4: Add up all the squares and write down the result at the end of the second column.

These first 5 steps are summarized in the following table:

• Step 5: The formula is used to calculate the variance:

So the sample variance is approximately S ² = 50 kg ² .

Calculation of the sample standard deviation (S)

Now that we have the variance, calculating the standard deviation is as simple as taking the square root of the first one:

As can be seen, the comparison of the standard deviation, which is 7 kilos, with the average weight of the bulls, which is 502.25 kilos (calculated separately), allows us to conclude that this sample has a low dispersion, since which is only 1.4% of the average weight of the bulls.

References

Espinoza, CI, & Echecopar, AL (2020). Statistical Applications using MS Excel with Step-by-Step Examples (Spanish Edition) (1st ^ed .). Lima, Peru: Luis Felipe Arizmendi Echecopar and Duo Negocios SAC.

Investopedia. (2021, April 16). Learn How Standard Deviation Is Determined By Using Variance. Retrieved July 24, 2021, from https://www.investopedia.com/ask/answers/021215/what-difference-between-standard-deviation-and-variance.asp

Lopez, JF (November 18, 2017). Variance . Retrieved from https://economipedia.com/definiciones/varianza.html

National Institute of Standards and Technology. (nd). Basic definitions of uncertainty. Retrieved July 24, 2021, from https://physics.nist.gov/cuu/Uncertainty/basic.html

Webster, A. (2001). Statistics Applied to Business and the Economy (Spanish Edition) . Toronto, Canada: Irwin Professional Publishing.