Variance

Published Sep 8, 2024

Definition of Variance

Variance is a statistical measurement that describes the spread or dispersion of a set of data points around their mean value. It illustrates how much the data points differ from the average value (mean) and hence from each other. More specifically, the variance is calculated as the average of the squared differences between each data point and the mean. It is an important concept in probability theory and statistics, often used to quantify the degree of variation within a data set.

Example

Consider a small dataset representing the ages of five students in a class: 10, 12, 14, 16, and 18. To calculate the variance:

1. First, find the mean by summing up all the ages and dividing by the number of students: (10 + 12 + 14 + 16 + 18) / 5 = 14.
2. Next, subtract the mean from each age to find the deviation from the mean:
   • 10 – 14 = -4
   • 12 – 14 = -2
   • 14 – 14 = 0
   • 16 – 14 = 2
   • 18 – 14 = 4
3. Then, square each deviation to eliminate negative values and weight outliers:
   • (-4)^2 = 16
   • (-2)^2 = 4
   • 0^2 = 0
   • 2^2 = 4
   • 4^2 = 16
4. Sum these squared deviations: 16 + 4 + 0 + 4 + 16 = 40.
5. Finally, divide this sum by the number of data points (N) to find the variance: 40 / 5 = 8.

In this case, the variance of the ages is 8, indicating the level of dispersion or variability in the ages around the mean of 14 years.

Why Variance Matters

Variance is crucial for several reasons:

• Understanding Data Distribution: Variance helps to understand how spread out the data is around the mean. A high variance indicates that the data points are widely dispersed, while a low variance suggests that the data points are close to the mean.
• Risk Assessment: In finance, variance is used to measure the risk associated with an investment. Higher variance in the returns of an asset indicates higher risk.
• Quality Control: In manufacturing and production, variance can help assess the consistency of processes and identify variations that might indicate issues with quality.
• Economic Forecasting: Economists use variance to analyze economic data series and forecast future trends by understanding the variability and stability over time.

Frequently Asked Questions (FAQ)

What is the difference between variance and standard deviation?

Variance and standard deviation both measure the spread of data points, but they do so in slightly different ways. Variance is the average of the squared deviations from the mean, whereas standard deviation is the square root of the variance. This means standard deviation is expressed in the same units as the original data, making it more interpretable as it reflects the average distance between each data point and the mean.

Why do we square the deviations when calculating variance?

By squaring the deviations, we eliminate negative values which ensures that positive and negative deviations do not cancel each other out. Squaring gives greater weight to larger deviations, thus emphasizing outliers. This helps in accurately representing the spread of the data around the mean.

Can variance be negative?

No, variance cannot be negative. Since variance is calculated as the average of the squared differences from the mean, and squaring any real number results in a positive value or zero, the variance will always be zero or a positive number.

How is variance used in portfolio diversification?

In finance, variance is used to assess the risk of individual assets within a portfolio. By understanding the variance in returns of different assets, investors can diversify their portfolio to minimize risk. A well-diversified portfolio contains assets with varying degrees of variance, so that the combined risk is lower than the risk of individual assets.

What are the limitations of variance as a measure of variability?

There are a few limitations to using variance as a measure of variability:

• Units Squared: Because variance involves squaring the deviations, the result is in units squared, which can be difficult to interpret relative to the original data values.
• Influenced by Outliers: Variance is sensitive to outliers since squaring the deviations weights larger differences more heavily. This can sometimes distort the measure of variability for skewed distributions.
• Not Always Intuitive: For those unfamiliar with statistical concepts, variance can be less intuitive compared to other measures of variability like range or standard deviation.