Measures of central tendency (mean, median, and mode) and dispersion (variance and standard deviation) are indicators that express the characteristics of data numerically.
Variance and Standard Deviation
Variance s² = Σ(xi - x̄)² / n
Standard deviation s = √(variance)
（x̄ is the mean, and n is the number of data values）
Example ①
Find the variance and standard deviation of the data: 2, 4, 6, 8.
Answer: Mean = 5. Variance = {(2-5)² + (4-5)² + (6-5)² + (8-5)²}/4 = (9+1+1+9)/4 = 5.
Standard deviation = √5.
Find the variance and standard deviation of the data: 2, 4, 6, 8.
Answer: Mean = 5. Variance = {(2-5)² + (4-5)² + (6-5)² + (8-5)²}/4 = (9+1+1+9)/4 = 5.
Standard deviation = √5.
Key Point
- Variance = the mean of the squared deviations (xi - x̄)
- Standard deviation = the positive square root of the variance
- The greater the spread, the greater the variance and standard deviation
Practice Problems
- Find the mean, variance, and standard deviation of the data: 1, 3, 5, 7, 9.
Answers and Explanations
- Answer: Mean 5, variance 8, standard deviation 2√2. Deviations: -4,-2,0,2,4. Variance=(16+4+0+4+16)/5=8.