1. What is Variance for Grouped Data?

Variance measures how far a set of grouped data points are spread out from their mean value. For grouped data, we calculate variance using frequency and midpoint of each class interval.

2. How Does the Calculator Work?

The calculator uses the formula for sample variance of grouped data:

Where:

• \( f \) — Frequency of each class
• \( x \) — Midpoint of each class
• \( \mu \) — Mean of the grouped data
• \( N \) — Total number of observations (sum of frequencies)

Explanation: The formula calculates the average of squared deviations from the mean, weighted by frequency, and adjusted for sample bias (N-1).

3. Importance of Variance Calculation

Details: Variance is a fundamental measure of dispersion in statistics. It's used in statistical tests, quality control, risk assessment, and many other fields to understand data variability.

4. Using the Calculator

Tips: Enter pairs of frequency and midpoint values, one per line. For example:
5 25 (frequency=5, midpoint=25)
10 35 (frequency=10, midpoint=35)
The calculator will compute both the mean and variance of your grouped data.

5. Frequently Asked Questions (FAQ)

Q1: Why use N-1 in the denominator?
A: Using N-1 (Bessel's correction) gives an unbiased estimate of population variance when calculating from a sample.

Q2: What's the difference between population and sample variance?
A: Population variance divides by N, sample variance divides by N-1 to correct for sampling bias.

Q3: How is grouped data variance different from ungrouped?
A: Grouped variance uses class midpoints and frequencies, while ungrouped uses each individual data point.

Q4: What are common applications of variance?
A: Quality control, financial risk analysis, experimental design, and any field requiring dispersion measurement.

Q5: How does variance relate to standard deviation?
A: Standard deviation is the square root of variance, providing dispersion in original units.