Overview
The lecture explains key measures of variability (range, average deviation, variance, and standard deviation), including their formulas, step-by-step calculations, and the meaning of each measure.
Measures of Variability
- Measures of variability assess the spread of values around the mean.
- Smaller variability suggests greater consistency and reliability in data.
Range
- Range is the difference between the highest and lowest scores.
- Formula: ( r = H - L ), where ( H ) is highest and ( L ) is lowest score.
- Example: Scores 43, 46, 40, 39, 45, 42, 47, 50, 49, 39: ( r = 50 - 39 = 11 ).
Average Deviation
- Average deviation gives a better approximation of dispersion than range.
- Formula: ( AD = \frac{\sum |x - \bar{x}|}{N} )
- Steps: Calculate mean (( \bar{x} )), subtract from each score, take absolute value, sum, and divide by number of scores.
- Example: Total absolute deviations sum to 34; ( AD = \frac{34}{10} = 3.4 ).
Variance
- Variance measures how far each value is from the mean, on average, squared.
- Formula: ( \sigma^2 = \frac{\sum (x - \bar{x})^2}{N} )
- Steps: Subtract mean from each score, square result, sum all squares, divide by number of scores.
- Example: Total sum of squares is 146; ( \sigma^2 = \frac{146}{10} = 14.6 ).
Standard Deviation
- Standard deviation is the square root of the variance, representing average distance from the mean.
- Formula: ( SD = \sqrt{\frac{\sum (x - \bar{x})^2}{N}} )
- Steps: Calculate variance, then take the square root.
- Example: ( SD = \sqrt{14.6} \approx 3.82 ).
Key Terms & Definitions
- Range — Difference between the highest and lowest scores in data.
- Average Deviation — Mean of the absolute differences between each score and the mean.
- Variance (( \sigma^2 )) — Mean of the squared differences from the mean.
- Standard Deviation (SD) — Square root of variance; measures average distance of scores from the mean.
Action Items / Next Steps
- Practice calculating range, average deviation, variance, and standard deviation with different data sets.
- Review and memorize all related formulas and their steps.