Statistics

Sort the observations: 3, 6, 10, 12, 16, 18. There are 6 observations (even number). The median is the average of the two middle terms (10 and 12). Median = (10+12)/2 = 11.

First, arrange the 10 observations in ascending order: 0, 5, 30, 30, 42, 64, 71, 80, 91, 117. Since there is an even number of observations, the median is the average of the 5th and 6th terms: (42 + 64) / 2 = 106 / 2 = 53.

For a grouped frequency distribution, the intersection point of the 'less than' and 'more than' ogives gives the median. The x-coordinate of this intersection point is the median.

The intersection point of the less than and more than ogives gives the median on the x-axis.

The median is represented by the x-coordinate of the point where the less-than and more-than ogives intersect. Since the intersection is at (20.5, 15), the median is 20.5.

The combined mean is (n1 * mean1 + n2 * mean2) / (n1 + n2). (25 * 80 + 75 * 60) / (25 + 75) = (2000 + 4500) / 100 = 6500 / 100 = 65.

Mean = (40+60+180+200+100)/20 = 580/20 = 29. Variance = sum(f(x-mean)^2)/N = (4(10-29)^2 + 3(20-29)^2 + 6(30-29)^2 + 5(40-29)^2 + 2(50-29)^2)/20 = (1444+243+6+605+882)/20 = 3180/20 = 159. SD = sqrt(159) approx 12.61.

The initial sum of squares of heights can be found using the variance formula: Variance = (Sum(x^2)/n) - (Mean)^2. With n=5, mean=150, and variance=18, Sum(x^2) = 5 * (18 + 150^2) = 112590. Adding a student with height 156, the new sum of squares is 112590 + 156^2 = 136926, and the new mean is (5*150 + 156)/6 = 151. The new variance is (136926/6) - 151^2 = 22821 - 22801 = 20.

The mean of the transformed values is the variance plus the square of the difference between the original mean and 4. Therefore it is 16^2 + (16 - 4)^2 = 256 + 144 = 400.

Combined mean = (Sum of all observations) / (Total number of observations). Sum 1 = 9 * 100 = 900. Sum 2 = 6 * 80 = 480. Total sum = 1380. Total count = 15. Mean = 1380 / 15 = 92.

Combined mean = (50*60 + 50*55) / 100 = 57.5. Combined variance = (n1(s1^2 + d1^2) + n2(s2^2 + d2^2)) / (n1+n2), where d1 = 60-57.5 = 2.5 and d2 = 55-57.5 = -2.5. Variance = (50(64 + 6.25) + 50(49 + 6.25)) / 100 = (70.25 + 55.25) / 2 = 62.75. SD = sqrt(62.75) approx 7.92.

To find variance, calculate the mean first using midpoints (5, 15, 25, 35, 45). Mean = (6*5 + 16*15 + 24*25 + 25*35 + 17*45) / 88 = 2520 / 88 = 28.636. Variance = sum(f * (x - mean)^2) / N. Calculations: 6*(5-28.6)^2 + 16*(15-28.6)^2 + 24*(25-28.6)^2 + 25*(35-28.6)^2 + 17*(45-28.6)^2 = 3335 + 2967 + 311 + 1024 + 4573 = 12210. Variance = 12210 / 88 = 138.75. Option C is the closest.

Mean = sum(f*x) / sum(f). Midpoints (x): 5, 15, 25, 35, 45. Frequencies: 12, 16, 6, 7, 9. Sum(f) = 50. Sum(f*x) = (12*5) + (16*15) + (6*25) + (7*35) + (9*45) = 60 + 240 + 150 + 245 + 405 = 1100. Mean = 1100 / 50 = 22.

Original sum = 4 * 28 = 112. The error was replacing 12 with 21 (an increase of 9). Correct sum = 112 - 9 = 103. Correct mean = 103 / 4 = 25.75.