1. What is Population Variance?

Population variance (σ²) measures how far each number in the set is from the mean (average) and thus from every other number in the set. It's the average of the squared differences from the Mean.

2. How Does the Calculator Work?

The calculator uses the population variance formula:

Where:

• \( \sigma^2 \) — Population variance
• \( x_i \) — Each individual value in the population
• \( \mu \) — Population mean
• \( N \) — Total number of values in the population

Explanation: The formula calculates the average of the squared differences from the Mean, giving more weight to extreme values.

3. Importance of Population Variance

Details: Variance is fundamental in statistics, used in probability distributions, hypothesis testing, and statistical modeling. It measures dispersion in a dataset.

4. Using the Calculator

Tips: Enter numeric values separated by commas (e.g., 5, 7, 8, 9). The calculator will ignore any non-numeric entries.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between population and sample variance?
A: Population variance divides by N, while sample variance divides by N-1 (Bessel's correction) to account for sample bias.

Q2: Why square the differences?
A: Squaring ensures all differences are positive and gives more weight to larger differences.

Q3: What are the units of variance?
A: Variance is in squared units of the original data (e.g., if data is in meters, variance is in meters²).

Q4: When should I use population vs sample variance?
A: Use population variance when you have all data points; use sample variance when working with a sample of a larger population.

Q5: How does variance relate to standard deviation?
A: Standard deviation is the square root of variance, bringing units back to the original scale.