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 calculated as 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
• \( f \) — Frequency of each value
• \( x \) — Individual data values
• \( \mu \) — Population mean
• \( \sum \) — Summation of all values

Explanation: The formula calculates the mean first, then sums the squared differences from the mean weighted by their frequencies, and divides by the total frequency.

3. Importance of Population Variance

Details: Population variance is a fundamental measure of dispersion in statistics. It quantifies how spread out the data points are from the mean value. A higher variance indicates that data points are more spread out.

4. Using the Calculator

Tips: Enter your values and their corresponding frequencies as comma-separated lists. Both lists must be of equal length. Example:
Values: 10, 20, 30
Frequencies: 3, 5, 2

5. Frequently Asked Questions (FAQ)

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

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

Q3: What does a variance of zero mean?
A: All values in the dataset are identical (no variability).

Q4: How is variance related to standard deviation?
A: Standard deviation is the square root of variance, providing dispersion in original units.

Q5: When should I use population vs sample variance?
A: Use population variance when you have data for the entire population, sample variance when working with a sample.