The data shows the number of seconds it took two randomly selected groups to thread a needle. Compare the measures of center and variability of these data sets.
Student Group A: 56, 14, 26, 28, 29, 45, 32, 80, 3, 10
Student Group B: 27, 24, 18, 19, 5, 16, 22, 10, 6, 20
To compare the measures of center and variability of these data sets, we will calculate the mean and median for each group, as well as the range and standard deviation.
For Student Group A:
Mean: (56 + 14 + 26 + 28 + 29 + 45 + 32 + 80 + 3 + 10) / 10 = 323 / 10 = 32.3
Median: In ascending order, the data set is 3, 10, 14, 26, 28, 29, 32, 45, 56, 80. The median is the middle value, so the median is 29.
Range: The range is the difference between the largest and smallest values, so the range is 80 - 3 = 77.
Standard Deviation: The standard deviation measures the spread or variability of the data. To calculate the standard deviation, we need to find the difference between each data point and the mean, square each difference, find the average of the squared differences, and then take the square root. However, since this is a sample data set and not a population, we need to use the formula for sample standard deviation, which divides the sum of squared differences by n-1 instead of n (where n is the sample size).
Using the sample standard deviation formula, we have:
Standard Deviation = √([Σ(x - mean)²] / (n-1))
= √([(56-32.3)² + (14-32.3)² + (26-32.3)² + (28-32.3)² + (29-32.3)² + (45-32.3)² + (32-32.3)² + (80-32.3)² + (3-32.3)² + (10-32.3)²] / (10-1))
= √([529.29 + 387.69 + 40.96 + 21.16 + 9 + 157.29 + 0.09 + 1796.74 + 865.69 + 532.09] / 9)
= √([4340.5] / 9)
= √482.28
≈ 21.97
For Student Group B:
Mean: (27 + 24 + 18 + 19 + 5 + 16 + 22 + 10 + 6 + 20) / 10 = 167 / 10 = 16.7
Median: In ascending order, the data set is 5, 6, 10, 16, 18, 19, 20, 22, 24, 27. The median is the middle value, so the median is 19.
Range: The range is the difference between the largest and smallest values, so the range is 27 - 5 = 22.
Standard Deviation: Using the same formula as we did for Group A, we have:
Standard Deviation = √([Σ(x - mean)²] / (n-1))
= √([(27-16.7)² + (24-16.7)² + (18-16.7)² + (19-16.7)² + (5-16.7)² + (16-16.7)² + (22-16.7)² + (10-16.7)² + (6-16.7)² + (20-16.7)²] / (10-1))
= √([120.49 + 53.29 + 2.89 + 4.49 + 141.61 + 0.09 + 28.09 + 42.49 + 136.89 + 10.89] / 9)
= √([540.32] / 9)
= √60.03
≈ 7.75
In summary, for Student Group A, the mean is 32.3, the median is 29, the range is 77, and the standard deviation is approximately 21.97. For Student Group B, the mean is 16.7, the median is 19, the range is 22, and the standard deviation is approximately 7.75.