Overview
This lecture covers the key descriptive statistics used to summarize data sets: mean, variance, skewness, and kurtosis, collectively known as the four moments of the mean.
The Mean
- The mean (arithmetic average) is the sum of all data values divided by the number of data points, denoted as x̄ (x-bar).
- The formula for the mean is x̄ = (ΣX) / n, where ΣX is the sum of data values and n is the number of data points.
Variance and Deviation
- Variance measures how much data values spread around the mean.
- Deviation is the distance of each data point from the mean (x - x̄).
- Variance formula: s² = Σ(x - x̄)² / (n - 1); uses n - 1 for degrees of freedom because the mean is already known.
- Degrees of freedom reduce as more statistics (like the mean) are calculated from the data set.
Skewness
- Skewness measures the asymmetry of the data distribution.
- Calculated by cubing the deviation: Skewness = Σ(x - x̄)³ / [(n - 1)(s³)], where s is the standard deviation.
- Skewness can be negative (left) or positive (right), indicating the direction of the tail.
Kurtosis
- Kurtosis measures the "peakedness" or "flatness" of a distribution.
- Calculated by raising deviations to the fourth power: Kurtosis = Σ(x - x̄)⁴ / [(n - 1)(n - 2)s⁴].
- Higher kurtosis means a taller, narrower peak; lower kurtosis means a flatter, wider shape.
The Four Moments of the Mean
- Mean (first moment): central location.
- Variance (second moment): spread of data.
- Skewness (third moment): asymmetry.
- Kurtosis (fourth moment): shape of the distribution's peak.
- Together, these describe the key features of a data set and allow further statistical analysis.
Key Terms & Definitions
- Mean (x̄) — arithmetic average of all data values.
- Variance (s²) — average squared deviation from the mean.
- Deviation — difference between a data value and the mean.
- Degrees of Freedom — number of independent values in a calculation after statistics like the mean are determined.
- Skewness — measure of distribution's asymmetry.
- Kurtosis — measure of distribution's peakedness.
Action Items / Next Steps
- Review the formulas for mean, variance, skewness, and kurtosis.
- Practice calculating each statistic using sample data.
- Read the textbook section on descriptive statistics and degrees of freedom.