Statistics: Variance Formula

Variance

Definition

• Variance is a measure of the spread or dispersion of a set of data from its mean value.
• It represents how much the individual data points deviate from the mean.

Formula

• The population variance (σ²) is calculated as:
  • σ² = (Σ(xi - μ)²) / N
  • where xi is each data point, μ is the population mean, and N is the total number of data points.
• The sample variance (s²) is calculated as:
  • s² = (Σ(xi - x̄)²) / (n - 1)
  • where xi is each data point, x̄ is the sample mean, and n is the sample size.

Properties

• Variance is always non-negative (≥ 0).
• A small variance indicates that the data points are close to the mean, while a large variance indicates that they are spread out.
• Variance is sensitive to outliers, meaning that a single extreme data point can greatly affect the variance.

Interpreting Variance

• A variance of 0 indicates that all data points have the same value (no spread).
• A small variance (e.g., 0.1) indicates that the data points are relatively close to the mean.
• A large variance (e.g., 10) indicates that the data points are widely spread out.

Relationship with Standard Deviation

• The standard deviation (σ or s) is the square root of the variance.
• σ = √(σ²) and s = √(s²)
• Standard deviation is often used in place of variance, as it is easier to interpret and has the same units as the data.

Variance

Definition and Purpose

• Measures the spread or dispersion of a set of data from its mean value.
• Represents how much individual data points deviate from the mean.

Calculating Variance

Population Variance

• Formula: σ² = (Σ(xi - μ)²) / N
• Where xi is each data point, μ is the population mean, and N is the total number of data points.

Sample Variance

• Formula: s² = (Σ(xi - x̄)²) / (n - 1)
• Where xi is each data point, x̄ is the sample mean, and n is the sample size.

Properties of Variance

• Always non-negative (≥ 0).
• Small variance indicates data points are close to the mean.
• Large variance indicates data points are spread out.
• Sensitive to outliers, meaning a single extreme data point can greatly affect the variance.

Interpreting Variance

• Variance of 0 indicates no spread (all data points have the same value).
• Small variance (e.g., 0.1) indicates data points are relatively close to the mean.
• Large variance (e.g., 10) indicates data points are widely spread out.

Relationship with Standard Deviation

• Standard deviation (σ or s) is the square root of the variance.
• σ = √(σ²) and s = √(s²)
• Standard deviation is often used in place of variance as it is easier to interpret and has the same units as the data.