# What is Variance? Definition, Formula, Calculation, and Example

By gobrain

Jun 29th, 2024

Variance is a statistical measurement that tells you how spread out a set of numbers is from their average value, or mean. In simpler terms, it reflects how much variation there is within your data.

## How to Calculate Variance

There are two formulas for calculating variance, depending on whether you're dealing with a population (the entire dataset) or a sample (a subset of the data).

Here are steps to calculate the variance of a dataset:

- Calculate the mean of the dataset
- Subtract the mean from each value in set
- Square the value found in the second step and sum all the values
- Divide the sum by the number of values in set

For the steps above, the formula is:

```
V = Σ(xi - μ)^2 / N
```

Where:

- V (variance) is the population variance
- Σ (capital sigma) represents the sum of
- xi (x-subscript-i) is each individual value in your data set
- μ (mu) is the population mean (average)
- N is the total number of values in the population

For example, A given [2,6,7] set

- mean = (2 + 6 + 7) / 3 = 5
- V = (2-5)^2 + (6-5)^2 + (7-5)^2 = 14
- V = (14 / 3)
- V = 4.66...

As mentioned above, for the sample data, there will be a small change in the formula as below

```
V = Σ(xi - μ)^2 / (N-1)
```

The new variance for the set above will be :

- V^2 = (14 / 2)
- V^2 = 7

## Standart Deviation vs Variance

If you look the formula of Standard Deviation, you can see the minor difference between their formulas and realize they are related to each other.

Variance and standard deviation are two related measures of dispersion or spread of data in a set of observations. There are, however, a few differences let's see them.

### Result in Different Unit

As you will notice, in the standard deviation, the square root of the value is taken so that the values in the dataset have the same unit of the result, which can provide a better interpretation.

For Example:

- [2cm,6cm, 7cm] -> SD = 2.160cm
- [2cm,6cm, 7cm] -> V = 4.66cm^2

### Deviation Value

In simpler terms, the standard deviation shows the general spread of the data set, while variance highlights the distribution of individual values (including outliers) within the data set.

In conclusion, both variance and standard deviation measure spread, but standard deviation is generally preferred for interpretation because it's in the same units as the data.

## Advantages and Disadvantages of Variance

Every statistical measure has advantages and disadvantages depending on a sample or population of a dataset. Let's see the advantages of disadvantages of variance:

### Advantages:

- Easy to calculate: Variance is a straightforward calculation that requires only a few basic arithmetic operations. As a result, it is easy to compute and understand.
- Sensitive to differences: Variance is highly sensitive to differences in the data points, which makes it useful for detecting outliers or unusual values in the data set.
- Provides a measure of spread: Variance provides a measure of how spread out the data points are from the mean, which can be useful for understanding the distribution of the data.

### Disadvantages:

- Sensitive to extreme values: Because variance is based on the squared differences between the data points and the mean, it is highly sensitive to extreme values or outliers in the data set. These outliers can distort the value of the variance and make it less useful as a measure of variability.
- Not robust: Variance is not a robust measure of variability, meaning that it can be heavily influenced by small changes in the data set. As a result, it may not be the best choice for data sets with a large number of outliers or extreme values.

## Frequently Asked Questions – FAQs

### What is the symbol of variance?

There are actually two common symbols for variance, depending on whether you're dealing with the entire population or a sample:

- Population variance: Represented by the Greek letter sigma squared (σ²).
- Sample variance: Represented by the lower-case Latin letter s squared (s²).

You might also see variance denoted by Var(X), where X is the variable you're analyzing.

### When is variance used?

Variance is used in many areas of statistics, including:

- Descriptive statistics: To describe the spread of data in a dataset.
- Hypothesis testing: It plays a crucial role in tests like Analysis of Variance (ANOVA) to compare groups. By analyzing the variance within and between groups, ANOVA helps determine if observed differences are likely due to chance or indicate a true difference between the groups..
- Statistical modeling: Statistical modeling: When building models to predict future outcomes, variance helps assess the model's accuracy. A model with low variance suggests its predictions are close to the actual values, while high variance indicates more scattered predictions.
- Quality control: In manufacturing, low variance is often desirable. It indicates a consistent production process with minimal variations in product quality.
- Financial risk analysis: Investors use variance to measure the risk associated with investments. High variance in stock prices suggests greater volatility and potential for larger losses or gains.

### How to convert standard deviation to variance?

Converting standard deviation (SD) to variance is quite straightforward! Since variance is the square of the standard deviation, you can obtain it simply by squaring the SD value.

Here's the formula:

Variance (σ²) = Standard Deviation (σ) ^ 2

### Can variance be equal to SD?

The variance can never be equal to the standard deviation except in the special case where the variance is 1. If the variance is 1

## Conclusion

In conclusion, variance in statistics, is a measure of how spread out your data is from its average value (the mean). Finally, keep in mind that understanding the dispersion of a dataset requires a combination of statistical measures, not just variance alone.

Thank you for reading.