Simple Definition:

Variance measures how far each number is from the average, and gives us a single number that tells us how spread out the data is.

Real-Life Examples:

• Heights of students in a class
• Scores on a test
• Number of pets students have
• Daily temperatures in a month

Basic Steps to Calculate Variance:

1. Find the mean (average) of the numbers
2. Find how far each number is from the mean (the difference)
3. Square each difference (multiply it by itself)
4. Find the average of the squared differences

Variance = Average of the squared differences from the mean

Simple Example:

Let's find the variance of the numbers: 4, 6, 8, 10, 12

Standard Deviation:

Standard deviation is simply the square root of the variance. It is often more useful because it's in the same units as our original data.

Standard Deviation = \(\sqrt{\text{Variance}}\)

In our example: Standard Deviation = \(\sqrt{8}\) ≈ 2.83

Population Variance:

When we have data for the entire population, we use the population variance formula.

\(\sigma^2 = \frac{\sum (X_i - \mu)^2}{N}\)

• \(\sigma^2\) = population variance
• \(X_i\) = each value in the population
• \(\mu\) = population mean
• \(N\) = total number of values in the population

Sample Variance:

When we only have data from a sample of the population, we use the sample variance formula.

\(s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1}\)

• \(s^2\) = sample variance
• \(x_i\) = each value in the sample
• \(\bar{x}\) = sample mean
• \(n\) = sample size

Notice: We divide by (n-1) instead of n. This is called "Bessel's correction" and it gives us a better estimate of the population variance when using a sample.

Example: Sample Variance

Calculate the sample variance for the test scores: 75, 82, 90, 68, 95

Population Variance Computational Formula:

\(\sigma^2 = \frac{\sum X_i^2}{N} - \mu^2\)

This formula means: Take the average of the squared values and then subtract the square of the mean.

Sample Variance Computational Formula:

\(s^2 = \frac{\sum x_i^2 - \frac{(\sum x_i)^2}{n}}{n-1}\)

Mathematical Properties:

1. Variance is always non-negative. It equals zero only when all values are identical.
2. Adding or subtracting a constant from all values does not change the variance.
3. Multiplying or dividing all values by a constant multiplies the variance by the square of that constant.

Variance of Combined Variables:

For independent random variables X and Y:

Var(X + Y) = Var(X) + Var(Y)

For variables that are not independent, we need to consider their covariance:

Var(X + Y) = Var(X) + Var(Y) + 2 × Cov(X, Y)

Variance of Common Distributions:

Coefficient of Variation:

The coefficient of variation (CV) is the ratio of the standard deviation to the mean. It's useful for comparing the variation of distributions with different means.

CV = \(\frac{\sigma}{\mu}\) × 100%