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Learn the key differences between variance and covariance, and how they are used to measure risk and relationships in data.
Variance and covariance are two statistical concepts used to understand data spread and the relationships between different variables. While they share some similarities, they have different purposes and applications. In this article, we will explain what variance and covariance are, their formulas, how they differ, and their real-world applications.
Variance is a measure of how much a set of data points deviate from the mean (average) of that dataset. In simple terms, it tells us how spread out the numbers are. The higher the variance, the more spread out the numbers in the dataset are. A lower variance indicates that the numbers tend to be closer to the mean.
The formula for calculating variance is:
Variance (σ²) = Σ (xi - μ)² / N
Where:
xi = each individual data point
μ = the mean (average) of the data
N = the number of data points
Σ = sum of the squared differences
This formula calculates how far each data point is from the mean, squares that difference, and then averages it.
Covariance is a measure of the relationship between two variables. It indicates whether two variables move in the same direction (positive covariance) or in opposite directions (negative covariance). If the covariance is zero, it indicates that the variables are independent of each other.
The formula for calculating covariance is:
Cov(X, Y) = Σ (xi - μX) * (yi - μY) / N
Where:
xi = data points of variable X
yi = data points of variable Y
μX = mean of variable X
μY = mean of variable Y
N = number of data points
Σ = sum of the products of differences from their means
This formula calculates the average of the product of the differences between corresponding data points in both variables.
The following table summarises the key differences between variance and covariance:
| Metric | Variance | Covariance |
|---|---|---|
Definition |
Measures the spread of a single variable |
Measures the relationship between two variables |
Purpose |
Shows how much data points deviate from the mean |
Shows how two variables move together |
Sign |
Always non-negative |
Can be negative, zero, or positive |
Units |
Square of the units of the data |
Product of the units of both variablesUse in Statistics |
Use in Statistics |
Used to understand the variability of data |
Used to understand the relationship between variables |
Variance is essentially a special case of covariance. While variance measures the spread of a single variable, covariance measures how two variables move together. In other words, covariance extends the concept of variance to two variables, showing how they interact with each other.
For example, variance tells us how a single stock's price moves over time, while covariance tells us whether two stocks move together—whether they rise and fall at the same time or do the opposite.
The following examples illustrate both metrics:
Variance Example:
Suppose we have a dataset representing the heights (in cm) of five people: [170, 175, 180, 185, 190].
The mean height is 180 cm. The variance will tell us how much the heights deviate from the mean.
Covariance Example:
Consider two variables, X (daily temperature) and Y (ice cream sales). If both tend to increase together (higher temperatures lead to higher sales), the covariance will be positive. If one increases while the other decreases (for example, higher temperatures lead to lower sales), the covariance will be negative.
Variance and covariance are widely used in finance, statistics, and risk management:
Finance: In portfolio theory, variance and covariance are used to assess the risk and diversification benefits of combining different assets.
Statistics: These metrics are used to quantify the variability and relationships within datasets.
Economics: Economists use variance and covariance to study the relationships between various economic variables like inflation and unemployment.
While variance and covariance are useful, they also have limitations:
Variance Limitations: Variance does not provide a clear indication of whether the data points are large or small, as the squared differences can exaggerate the effect of outliers.
Covariance Limitations: Covariance alone does not provide a clear sense of the strength of the relationship between two variables. A high covariance value does no’t necessarily mean a strong relationship.
Variance and covariance are important statistical tools that help investors, analysts, and statisticians understand data variability and relationships between variables. While variance measures the spread of data within a single variable, covariance reveals how two variables move in relation to each other. Understanding both concepts helps explain data analysis and risk management practices.
This content is for informational purposes only and the same should not be construed as investment advice. Bajaj Finserv Direct Limited shall not be liable or responsible for any investment decision that you may take based on this content.
Variance measures how widely data points of a single variable are spread around the mean. Covariance measures how two variables move together, indicating whether their values tend to increase or decrease in relation to each other.
Variance cannot be negative because it is calculated using squared differences from the mean. Squaring removes negative signs, ensuring the final variance value is always zero or a positive number.
Covariance can be negative when two variables move in opposite directions. This means that as one variable increases, the other tends to decrease, indicating an inverse relationship between the two variables.
Covariance and correlation both measure relationships between variables, but they are not the same. Correlation standardises covariance, making it easier to interpret the strength and direction of the relationship consistently.
Variance is always positive because it is based on squared deviations from the mean. Since squared values cannot be negative, the calculated variance remains zero or positive regardless of data direction.
Covariance indicates whether the relationship between two variables is positive or negative, but it does not clearly show strength. The magnitude depends on the data scale, which limits direct interpretation of relationship strength.
Variance and covariance are affected by the scale of the variables because changes in measurement units alter their values. This scale dependence is why correlation is often preferred for comparing relationships across datasets.
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Posted on 19 Dec
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