Definition:
Variance is a measure of how spread out a set of data is from its mean (average). It measures the average of the squared differences from the mean.

Symbols:
• Population variance: \(\sigma^2\) (sigma squared)
• Sample variance: \(s^2\)
• Also written as: \(\text{Var}(X)\) or \(V(X)\)

• High variance: Data points are spread far from the mean
• Low variance: Data points are clustered close to the mean
• Variance is always non-negative (\(\sigma^2 \geq 0\))

\[ \sigma^2 = \frac{\sum_{i=1}^{N}(x_i - \mu)^2}{N} \]

Where:
• \(\sigma^2\) = population variance
• \(x_i\) = each individual value in the population
• \(\mu\) = population mean (average)
• \(N\) = total number of values in the population
• \(\sum\) = sum of all values

Use when: You have data for the entire population (all possible values)

\[ s^2 = \frac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n-1} \]

Where:
• \(s^2\) = sample variance
• \(x_i\) = each individual value in the sample
• \(\bar{x}\) = sample mean (average)
• \(n\) = number of values in the sample
• \(n-1\) = degrees of freedom (Bessel's correction)

Use when: You have data from a sample (subset of the population)
Why n-1? Using \(n-1\) provides an unbiased estimate of the population variance

Computational Formula (easier for calculations):

\[ \text{Var}(X) = E[X^2] - (E[X])^2 \]

This means: "Expected value of squares minus square of expected value"

For Population (expanded form):

\[ \sigma^2 = \frac{\sum_{i=1}^{N}x_i^2}{N} - \mu^2 \]

For Sample (expanded form):

\[ s^2 = \frac{\sum_{i=1}^{n}x_i^2 - n\bar{x}^2}{n-1} \]

Step 1: Calculate the Mean

Add all values and divide by the number of values:
\[ \bar{x} = \frac{\sum x_i}{n} \]

Step 2: Find Deviations from Mean

Subtract the mean from each value:
\[ (x_1 - \bar{x}), (x_2 - \bar{x}), ..., (x_n - \bar{x}) \]

Step 3: Square Each Deviation

Square each difference to make them positive:
\[ (x_1 - \bar{x})^2, (x_2 - \bar{x})^2, ..., (x_n - \bar{x})^2 \]

Step 4: Calculate Average of Squared Deviations

• For population: Divide by \(N\)
• For sample: Divide by \(n-1\)

Problem: Find the variance of the sample: 4, 8, 6, 5, 3, 2

Step 1: Calculate Mean
\[ \bar{x} = \frac{4+8+6+5+3+2}{6} = \frac{28}{6} = 4.67 \]

Step 2 & 3: Find and Square Deviations
(4 - 4.67)² = 0.4489
(8 - 4.67)² = 11.0889
(6 - 4.67)² = 1.7689
(5 - 4.67)² = 0.1089
(3 - 4.67)² = 2.7889
(2 - 4.67)² = 7.1289

Step 4: Calculate Variance
Sum of squared deviations = 23.3334
\[ s^2 = \frac{23.3334}{6-1} = \frac{23.3334}{5} = 4.67 \]

Answer: Sample variance = 4.67

Population Variance (Grouped Data):

\[ \sigma^2 = \frac{\sum f_i(M_i - \mu)^2}{N} \]

Sample Variance (Grouped Data):

\[ s^2 = \frac{\sum f_i(M_i - \bar{x})^2}{n-1} \]

Where:
• \(f_i\) = frequency of each class
• \(M_i\) = midpoint of each class interval
• \(\bar{x} = \frac{\sum f_i M_i}{\sum f_i}\) (mean for grouped data)

Property 1: Adding a constant doesn't change variance

\[ \text{Var}(X + c) = \text{Var}(X) \]

Property 2: Multiplying by a constant

\[ \text{Var}(cX) = c^2 \cdot \text{Var}(X) \]

Property 3: Linear transformation

\[ \text{Var}(aX + b) = a^2 \cdot \text{Var}(X) \]

Property 4: Variance of independent variables

\[ \text{Var}(X_1 + X_2 + ... + X_n) = \text{Var}(X_1) + \text{Var}(X_2) + ... + \text{Var}(X_n) \]

Property 5: Variance is always non-negative

\[ \text{Var}(X) \geq 0 \]

Relationship:
Standard deviation is the square root of variance

\[ \sigma = \sqrt{\sigma^2} \quad \text{or} \quad s = \sqrt{s^2} \]

Key Differences:
• Variance: Measured in squared units
• Standard Deviation: Measured in same units as original data
• Standard deviation is easier to interpret

Example: If data is in meters and variance = 16 m²
Then standard deviation = √16 = 4 meters

✓ Finance: Measuring investment risk and volatility

✓ Quality Control: Measuring consistency in manufacturing

✓ Research: Analyzing data spread in scientific studies

✓ Weather: Understanding temperature variability

✓ Education: Measuring consistency in test scores

❌ Using \(n\) instead of \(n-1\) for sample variance
✅ Always use \(n-1\) for samples to get unbiased estimate

❌ Forgetting to square the deviations
✅ Must square each \((x_i - \bar{x})\) before summing

❌ Confusing variance with standard deviation
✅ Variance = squared units, Standard deviation = original units

❌ Calculating mean incorrectly
✅ Double-check mean calculation before finding variance

Population Variance:

\[ \sigma^2 = \frac{\sum(x_i - \mu)^2}{N} \]

Sample Variance:

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

Standard Deviation:

\[ \sigma = \sqrt{\sigma^2} \quad \text{or} \quad s = \sqrt{s^2} \]

Remember:
• Variance measures spread of data
• Always non-negative
• Use \(n-1\) for sample variance
• Square root gives standard deviation