Final answer:
The range, population variance, population standard deviation, sample variance, and sample standard deviation of the data set [0, 5, 10, 10, 15, 20] are 20, 41.67, 6.45, 50, and 7.07 respectively. The sample standard deviation tends to be higher than the population standard deviation as it compensates for estimation from a sample.
Step-by-step explanation:
The given data set is [0, 5, 10, 10, 15, 20]. We will calculate the range, population variance (σ²), population standard deviation (σ), sample variance (s²), and sample standard deviation (s).
The range is the difference between the maximum and minimum values in the data set: 20 - 0 = 20.
To calculate the mean (μ) for the population and the sample mean (σ), we sum all the values and divide by the number of values. Mean = (0 + 5 + 10 + 10 + 15 + 20) / 6 = 60 / 6 = 10.
Population variance (σ²) = Σ(x-μ)² / N = ((0-10)² + (5-10)² + (10-10)² + (10-10)² + (15-10)² + (20-10)²) / 6 = (100 + 25 + 0 + 0 + 25 + 100) / 6 = 250 / 6 ≈ 41.67.
Population standard deviation (σ) is the square root of population variance: σ = √σ² ≈ √41.67 ≈ 6.45.
Sample variance (s²) differs from population variance as it divides by the number of values minus one (N-1). Hence, s² = Σ(x-μ)² / (N-1) = 250 / 5 = 50.
Sample standard deviation (s) is the square root of sample variance: s = √s² = √50 ≈ 7.07.
Comparing the sample and population standard deviations, we find that the sample standard deviation is usually higher as it compensates for the reduced degree of freedom when estimating the population standard deviation from a sample.