Understanding the Variance Formula

Variance is a key statistical measure that quantifies the spread or dispersion of data points around their average value (the mean).

Analyzing the Options

Let's examine why other options are incorrect:

• Option 1: $Variance = $\frac{(\Sigma X - \Sigma X)^2}{N}$
  The term $\Sigma X - \Sigma X$ equals zero, leading to a variance of zero, which is incorrect for most datasets.
• Option 3: $Variance = $\frac{(\Sigma X-\bar{X})^2}{N^2}$
  The denominator should typically be $N$ (for population variance) or $N-1$ (for sample variance), not $N^2$. Also, the numerator structure is incorrect.
• Option 4: $Variance = $\frac{(\Sigma X^2-\bar{X}^2)}{N}$
  This formula does not represent the variance. Variance specifically deals with the squared differences *from the mean*.

The Correct Variance Formula

The correct formula for population variance is:

Variance = $\frac{\Sigma(X-\bar{X})^2}{N}$

Here's a breakdown:

• $X$ represents each individual data point.
• $\bar{X}$ represents the mean (average) of all data points.
• $(X-\bar{X})$ is the deviation of a single data point from the mean.
• $(X-\bar{X})^2$ is the squared deviation. Squaring ensures all values are positive and emphasizes larger deviations.
• $\Sigma(X-\bar{X})^2$ is the sum of all the squared deviations.
• $N$ is the total number of data points in the population.

This formula calculates the average of the squared differences between each data point and the mean, providing a measure of data variability.