Exam 4: Measures of Central Tendency and Variability

Correct Answer:

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Variance is a statistical measure that represents the degree of spread or dispersion of a set of values. Specifically, it quantifies how much the numbers in a data set deviate from the mean (average) of the set. In other words, variance measures how far each number in the set is from the mean and thus from every other number in the set.

Mathematically, variance is calculated by taking the average of the squared differences between each data point and the mean. The formula for variance (σ²) for a population (the entire set of values) is:

σ² = Σ (xi - μ)² / N

where:
- Σ is the summation symbol, indicating that you sum over all data points.
- xi represents each data point in the population.
- μ is the mean (average) of the population.
- N is the number of data points in the population.

For a sample (a subset of values from the population), the formula is slightly different to correct for the bias that comes from estimating the population variance from a sample:

s² = Σ (xi - x̄)² / (n - 1)

where:
- s² is the sample variance.
- xi represents each data point in the sample.
- x̄ is the sample mean.
- n is the number of data points in the sample.
- The denominator (n - 1) is known as the degrees of freedom.

Variance is a foundational concept in statistics and probability theory, and it is crucial for various analytical techniques, including hypothesis testing, ANOVA, regression analysis, and in finance for portfolio theory and risk management. However, because variance is in squared units of the original data, it is often more intuitive to use the standard deviation, which is the square root of the variance, to understand the variability in the same units as the original data.