Variance is a statistical measure that quantifies the dispersion of a set of data points around their mean. It is an essential concept in probability, statistics, and data analysis.

Formula

For a dataset with $n$ elements $x_1, x_2, \ldots, x_n$, and a mean $\mu$:

• Population Variance ($\sigma^2$): $\sigma^2 = \frac{1}{n} \sum_{i=1}^n (x_i - \mu)^2$

• Sample Variance ($s^2$): $s^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2$ Where:

  • $\mu$ or $\bar{x}$ is the mean of the data.
  • $x_i$ are the data points.
  • $n$ is the number of data points.
  • The denominator for the population variance is $n$, but for the sample variance, it is $n-1$ (to account for sample bias).

Interpretation

• High Variance: Data points are widely spread out from the mean.
• Low Variance: Data points are clustered closely around the mean.

Applications

1. Measuring risk in finance.
2. Analyzing variability in scientific experiments.
3. Assessing consistency in quality control.

Let me know if you’d like a worked-out example or additional clarification!

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Five Related Questions:

1. What is the relationship between variance and standard deviation?
2. How does variance differ for populations and samples?
3. Why is variance squared, rather than absolute values of deviations?
4. How can variance be used to compare different datasets?
5. What are the limitations of variance as a measure of dispersion?

Tip: Always check if you are dealing with a sample or population, as this changes the formula for variance.