The relationships between mean, median, standard deviation and range are quite complex. A few generalisations can be offered:
1. Mean and median are similar but not identical measures. Intuitively, both are a measure of the "middle" of the data. In particular, if the data is distributed symmetrically about the mean (as in for instance an arithmetic series), the mean and median coincide. When the median is, for example, lower than the mean, this suggests that the mean is being 'pulled up' by some very large values, and vice versa.
2. Standard deviation and range share some features but are not the same. Both reflect, intuitively, a measure of the "spread" of data, or how much variability it contains. Range is a more trivial measure of the difference between the outlying points (minimum and maximum). Standard deviation is a more practical measure which considers the spread of the data as a whole (not just the outlying points).
Now for the questions.
7. Here we could use the standard deviation formula and plug in each dataset in turn. More intuitively however, we can look at the datasets and see how widely they are spread around the mean. D and E have "gaps" of 2 between each number and so have the largest SD. A has a consistent "gap" of 1 and B and C are more tightly distributed. Hence A has the middle, or third highest, SD.
8. Interpreting the question, we're told that in A the median is greater than the mean, whilst in B the two measures are equal. Going through the statements:
I -- we don't know anything about the standard deviations of either dataset, so we can't know which is greater.
II -- for this to be true, the mean of the combined set should be higher than the mean of set B. This would have to be true if Y>M but we don't know that this is the case.
III -- this is more difficult. Plugging in a few example numbers appears to be the easiest approach. If we take 0, 2 and 3 for A (ensuring X>Y) and 5 and 6 for B, we get X=2, Y=5/3, L=5.5, M=5.5. The combined set is 0, 2, 3, 5, 6, so Q=3 and R=16/5. R is slightly higher than Q in this example so statement III is not necessarily true.
Answer: E
9. Here we need to establish what E could contain. Since the range is 4 we must (at least) have two odd numbers 4 apart, e.g. 1 and 5. The other two numbers in the set could each be 1, 3 or 5. To save time, we can see that being 5 is the same (in terms of impact on SD) as being 1. Hence the combinations we need to consider are:
1,1,1,5
1,1,3,5 or 1,3,1,5 (the same)
1,3,3,5
Note that the choice of outliers (1 and 5) does not affect SD (because the distance from the mean is what matters -- try it and see).
Answer: A
