Statistics

The sum of deviations from 15 is 7, so the sum of the observations is 10 × 15 + 7 = 157. Dividing by 10 gives a mean of 15.7.

Calculate midpoints of classes, multiply by frequency, and use standard deviation formula. The values provided in option A match the statistical calculation for the given frequency distribution.

The sorted data is 34, 38, 42, 44, 46, 48, 54, 55, 63, 70. The median is (46+48)/2 = 47. The absolute deviations from 47 are 13, 9, 5, 3, 1, 1, 7, 8, 16, 23. The sum is 86. The mean deviation is 86/10 = 8.6.

Mean = (8+12+13+15+22)/5 = 70/5 = 14. Variance = [(8-14)^2 + (12-14)^2 + (13-14)^2 + (15-14)^2 + (22-14)^2] / 5 = (36 + 4 + 1 + 1 + 64) / 5 = 106 / 5 = 21.2.

Coefficient of variation is (standard deviation / mean) * 100%. Here, (5 / 25) * 100% = 0.2 * 100% = 20%.

Mean = (6+8+12+15+10+9)/6 = 60/6 = 10. Mean deviation = (|6-10| + |8-10| + |12-10| + |15-10| + |10-10| + |9-10|) / 6 = (4 + 2 + 2 + 5 + 0 + 1) / 6 = 14/6 = 2.33.

The mean deviation from the mean is the average of the absolute differences between each data point and the mean. With mean 10, the differences are |6-10|=4, |7-10|=3, |10-10|=0, |12-10|=2, |13-10|=3, |a-10|, |12-10|=2, |16-10|=6. Summing these and dividing by 8 gives 3.25, which implies |a-10|=2, meaning a=12 or 8.

The empirical relationship between mean, median, and mode is: Mode = 3 * Median - 2 * Mean. Plugging in the values: 23.3 = 3 * Median - 2 * 22.2. 23.3 = 3 * Median - 44.4. 3 * Median = 67.7. Median = 22.566, which rounds to 22.6.

For a set of 2n observations with n equal to alpha and n equal to -alpha, the mean is 0. Variance = [n(alpha^2) + n(-alpha^2)] / 2n = (2n * alpha^2) / 2n = alpha^2. Standard deviation = sqrt(variance) = |alpha|. Given SD = 2, |alpha| = 2.

Data: 90, 100, 110, 115, 125. Median = 110. Deviations: |90-110|=20, |100-110|=10, |110-110|=0, |115-110|=5, |125-110|=15. Sum = 20+10+0+5+15 = 50. Mean deviation = 50/5 = 10.