Lecture Notes: Variance and Standard Deviation

Introduction

• Focus on variance and standard deviation.
• Importance of these calculations for understanding data variability.
• Use of all data points for better measure of dispersion.

Dispersion in Data

• Range: Measures dispersion using max and min values.
• Quartiles: Uses more data points, but still limited.
• Variance/Standard Deviation: Uses all data, providing a better measure of dispersion around central tendency.

Sum of Squares

• Formula: Square each deviation from the mean and sum them.
• Larger values indicate greater dispersion.
• Useful in multiple statistical contexts.

Variance Calculation

• Variance = Sum of squared deviations / Number of observations.
• Indicates how much data varies from the mean.
• Population variance uses Greek symbols (e.g., μ and σ).

Standard Deviation

• Square root of variance.
• Returns measurement to original units.
• Provides average variability from the mean.

Variability

• Large variance/standard deviation = More variability.
• Small variance/standard deviation = Less variability.
• Variance and standard deviation range from 0 to large values.

Importance of Sample Size

• Larger sample size = Decreased variability.
• Smaller sample size = Increased variability.
• Sample size impacts variance and standard deviation calculations.

Population vs. Sample

• Population Variance/Standard Deviation: Uses population mean.
• Sample Variance/Standard Deviation: Uses sample mean and n-1 in the denominator for a conservative estimate.

Calculating Variance and Standard Deviation

1. Calculate Deviations: Data point minus mean.
2. Square Deviations: Eliminates negative values, results in positive values.
3. Sum of Squares: Sum squared deviations.
4. Variance: Divide sum of squares by n-1.
5. Standard Deviation: Square root of variance.

Reporting Results

• Mean ± Standard Deviation.
• Reflects data variability above and below the mean.

Further Reading & Practice

• Refer to section 4.3.3 of the text.
• Practice with exercise 4.1.

Next Lecture

• Focus on standard error and coefficient of variation.