Overview
This lecture discusses the normal distribution (bell curve), how to identify if data follows it, and how to use its properties for quick, practical decisions—like managing shoe inventory.
The Normal Distribution
- The normal distribution (bell curve) describes how data like shoe sizes or goat heights commonly spread around a mean.
- Most values cluster around the average, with fewer extreme values at the edges.
- Many real-world data sets approximate a normal distribution due to the central limit theorem.
Central Limit Theorem
- The central limit theorem states that averages of many independent, identically-distributed events will form a normal distribution, even if individual events are not perfectly random or independent.
- The closer the data conditions match the theorem's assumptions, the more "normal" the distribution appears.
- Minor dependencies in real data still often yield an approximately normal curve.
Identifying Normal Distributions
- Cumulants describe the distribution's shape: mean, variance, skewness (tail direction), and kurtosis (tail thickness).
- Skewness measures data symmetry; kurtosis measures "peakedness" and the presence of outlier extremes.
- For real data, if skewness is between -1 and 1 and kurtosis is between -6 and 6, the data can be treated as approximately normal.
Practical Tools & Analysis
- Software functions (e.g., Google Sheets’ SKEW and KURT) quickly calculate skewness and kurtosis.
- In a perfectly normal distribution, both skewness and kurtosis are zero.
- Small skewness and kurtosis values indicate the data approximates normality.
The Empirical Rule (1-2-3 Rule)
- In a normal distribution:
- About 68% of data falls within 1 standard deviation of the mean.
- About 95% falls within 2 standard deviations.
- About 99.7% falls within 3 standard deviations.
- Knowing the mean and standard deviation allows for quick estimation of coverage and outliers.
Real-World Example: Shoe Inventory
- If men's shoe sizes average 10.5 with a standard deviation of 1.5, 95% of sales will be sizes between 7.5 and 13.5.
- To cover 68% of sales, only stock sizes 9 to 12.
- Extreme sizes (beyond 3 standard deviations) are rare and can be managed differently.
Key Terms & Definitions
- Normal Distribution — Symmetrical, bell-shaped data distribution centered around the mean.
- Central Limit Theorem — Principle that averages of independent events tend to be normally distributed.
- Cumulants — Quantitative descriptors of a distribution’s shape: mean, variance, skewness, kurtosis.
- Skewness — Measure of asymmetry in data.
- Kurtosis — Measure of data’s peakedness and tail heaviness.
- Empirical Rule — Predicts data proportions within 1, 2, and 3 standard deviations in a normal distribution.
Action Items / Next Steps
- Practice using SKEW and KURT functions on sample data sets.
- Memorize the empirical rule percentages (68%, 95%, 99.7%).
- Prepare examples where the normal distribution helps in real-world decision-making.