Understanding Variance and Standard Deviation

Overview

This lecture explains how to calculate and interpret variance, standard deviation, typical values, and outliers in a dataset with a normal (bell-shaped) distribution.

Variance and Standard Deviation

• Variance is calculated as the average of squared deviations from the mean and is the square of the standard deviation.
• Standard deviation is the square root of the variance and represents average distance from the mean.
• In the brick weight data example, the standard deviation was approximately 14.7 kilograms.
• Standard deviation is rounded to one more decimal place than the original data.

Identifying Typical Values

• Typical values fall within one standard deviation above or below the mean.
• For normal data, about 68% of values are within one standard deviation from the mean.
• Calculation: Mean ± Standard Deviation (e.g., 36.1 kg ± 14.7 kg yields 21.4 kg to 50.8 kg for typical brick weights).
• Visually, these are the middle data points in the plot.

Identifying Outliers

• Outliers are values more than two standard deviations away from the mean.
• High outliers: ≥ Mean + 2 × Standard Deviation (top 2.5%).
• Low outliers: ≤ Mean – 2 × Standard Deviation (bottom 2.5%).
• In the example, cutoffs were 65.5 kg (high) and 6.7 kg (low).
• Bricks weighing 70 kg (high) and 3 kg (low) were considered outliers.

Data Distribution Zones

• Data is not just typical values and outliers; there is a "could happen" zone between them.
• Most data (middle 68%) is typical; outliers are only a small portion at the extremes.
• Not all values are typical or outliers.

Data Analysis Report Summary

• The dataset was normal (bell-shaped) with a mean of 36.1 kg and a standard deviation of 14.7 kg.
• Typical values ranged from 21.4 kg to 50.8 kg.
• There were two outliers: 70 kg (high) and 3 kg (low).

Key Terms & Definitions

• Variance — The average of squared differences from the mean; symbolized as s².
• Standard Deviation — The square root of variance; measures spread from the mean.
• Mean — The average of all values in a dataset.
• Outlier — A value more than two standard deviations from the mean.
• Typical Value — A value within one standard deviation of the mean.

Action Items / Next Steps

• Practice calculating variance and standard deviation with sample data.
• Identify typical values and outliers in a given dataset.
• Prepare for discussion on the empirical rule (68-95-99.7%) in future lectures.