Statistics

Data: 15, 17, 10, 13, 7, 18, 9, 6, 14, 11. Sum = 120. Mean = 12. Deviations: |15-12|=3, |17-12|=5, |10-12|=2, |13-12|=1, |7-12|=5, |18-12|=6, |9-12|=3, |6-12|=6, |14-12|=2, |11-12|=1. Sum of deviations = 3+5+2+1+5+6+3+6+2+1 = 34. Mean deviation = 34/10 = 3.4.

First, find the median of the frequency distribution. The total frequency is 54, so the median is the value corresponding to the 27th and 28th observation, which is 7. Calculate the mean deviation about 7 using the formula sum(f_i * |x_i - median|) / sum(f_i).

Mean = sum(f*x)/sum(f) = (18+40+105+225+88+52)/66 = 528/66 = 8. Mean deviation = sum(f*|x-mean|)/sum(f) = (6*|3-8| + 8*|5-8| + 15*|7-8| + 25*|9-8| + 8*|11-8| + 4*|13-8|)/66 = (30+24+15+25+24+20)/66 = 138/66 = 2.0909.

Mean = (3+4+5+6+7)/5 = 25/5 = 5. Mean deviation = (|3-5| + |4-5| + |5-5| + |6-5| + |7-5|) / 5 = (2+1+0+1+2) / 5 = 6/5 = 1.2.

Standard deviation is invariant to constant shifts (the '2' in 2-3x). For a linear transformation y = ax + b, SD(y) = |a| * SD(x). SD(2-3x) = |-3| * 3 = 9.

Data: 39, 40, 40, 41, 41, 42, 42, 43, 43, 44, 44, 45. Median is (42+42)/2 = 42. Deviations from 42: |-3|, |-2|, |-2|, |-1|, |-1|, |0|, |0|, |1|, |1|, |2|, |2|, |3|. Sum = 3+2+2+1+1+0+0+1+1+2+2+3 = 18. Mean deviation = 18 / 12 = 1.5.

Sum of observations = 12, n = 10. Mean = 1.2. Sum of squares = 18. Variance = (Sum of squares / n) - (Mean)^2 = 18/10 - (1.2)^2 = 1.8 - 1.44 = 0.36. Standard deviation = sqrt(0.36) = 0.6 = 3/5.

For w = lx + k, the standard deviation becomes |l| times the original standard deviation. Thus |l| = 15/12 = 1.25, and using the means gives 55 = 1.25 × 48 + k, so k = -5. The positive value l = 1.25 gives option A.