Statistics

The empirical relationship between mean, median, and mode is: Mode = 3 * Median - 2 * Mean. Substituting the values: Mode = 3 * 18 - 2 * 20 = 54 - 40 = 14.

If each observation is multiplied by a constant k, the variance is multiplied by k^2. Adding a constant does not change the variance. New variance = 25 * 2^2 = 25 * 4 = 100.

Variance = (Sum of squares of deviations) / N. Variance = 500 / 10 = 50.

Standard deviation is a measure of dispersion. Subtracting a constant from every observation shifts the data but does not change the spread, so the standard deviation remains the same.

Total sum = 100 * 50 = 5000. Sum of first 60 = 60 * 45 = 2700. Remaining sum = 5000 - 2700 = 2300. Mean of remaining 40 = 2300 / 40 = 57.5.

For a moderately skewed distribution, the empirical relationship is Mode = 3*Median - 2*Mean. Substituting the values: 16 = 3(20) - 2*Mean. 16 = 60 - 2*Mean. 2*Mean = 44. Mean = 22.

Standard deviation is a measure of dispersion and is invariant to change of origin (adding a constant to all observations). Therefore, it remains 4.

The total sum of 100 observations increases by 100 * (50.5 - 50) = 50. Since one observation is replaced, the difference between the new value and the old value must equal this total increase of 50.

Total sum = 9 * 20 = 180. Sum of first 5 = 5 * 18 = 90. Sum of last 5 = 5 * 22 = 110. Sum of first 5 + last 5 = 200. The middle observation (5th) is counted twice. Middle = 200 - 180 = 20.

Sum of 18 students = 18 * 72 = 1296. Sum of 19 students = 19 * 74 = 1406. Score of the 19th student = 1406 - 1296 = 110.

Standard deviation is invariant under translation (subtracting 5) but scales with multiplication. New SD = |3| * Old SD = 3 * 6 = 18.

Total frequency = 18+x. Median is 24, which falls in 20-30. Median = L + ((N/2 - CF)/f) * h. 24 = 20 + ((9+x/2 - 10)/x) * 10. 4 = (x/2 - 1)/x * 10. 0.4x = 0.5x - 1 => 0.1x = 1 => x = 10.

Mean of R = (18+22+25+29+31+35)/6 = 160/6. Mean of S = (16+24+27+30+33+t)/6 = (130+t)/6. Since means are equal, 160 = 130 + t, so t = 30.

Initial sum = 30 * 40 = 1200. The error is 65 - 56 = 9. The correct sum is 1200 + 9 = 1209. The correct mean is 1209 / 30 = 40.3.