Comparing the SD of multiple sets?

[yo4561]

Happy Sunday!

I learned that you can look at how large the jumps are when comparing the SDs of multiple sets to see which set has the largest SD. For example, let's say I have this made up example:

Set 1: {3, 7, 8, 9, 1} ---> 3 to 7 is a jump of 4, 7 to 8 is a jump of 1, 8 to 9 is a jump of 1, and 9 to 1 is a jump of 8 --> 4+1+1+8=14 jumps
Set 2: {15, 19, 21, 30, 32} --> repeated process above= 17 jumps
Set 3: {1, 4, 9, 12, 17} =repeated process above= 16 jumps

Therefore set 2 has the largest SD. I realize that this method might not work when each set does not have the same amount of numbers, so what would be the best way to handle finding the SD when comparing sets?

Do you recommend counting the jumps and ALSO calculating the average of each set (to compare each jump relative to the average....but this is essentially calculating the SD then... which I thought that you are not supposed to do on the GMAT)? I realize you can eliminate a set(s) based on some reasoning without doing any math, but what would be your standard way of dealing with problems in this nature when you need to do some math?

Many thanks in advance

[esledge]

Do you recommend counting the jumps and ALSO calculating the average of each set (to compare each jump relative to the average....but this is essentially calculating the SD then... which I thought that you are not supposed to do on the GMAT)? I realize you can eliminate a set(s) based on some reasoning without doing any math, but what would be your standard way of dealing with problems in this nature when you need to do some math?

I do not recommend this "jump" method. For one thing, the reasoning can be flawed if the terms are out of order, as in this example:

Set 1: {3, 7, 8, 9, 1} ---> 3 to 7 is a jump of 4, 7 to 8 is a jump of 1, 8 to 9 is a jump of 1, and 9 to 1 is a jump of 8 --> 4+1+1+8=14 jumps

...where that last jump of 8 isn't real. Both 9 and 1 are closer to other terms (e.g. the jump to 1 should be made from 3, so it's only a jump of 2).

The second reason is what you mentioned about the limited utility of this for comparing sets with different number of terms.

Finally, notice that this method is just a more labor-intensive way to determine the range of a set: the sum of all these jumps (from lowest to highest terms) equals the Highest - Lowest difference. Like standard deviation, range is a measure of the "spread" of the data, but range is a more coarse measure of spread, whereas standard deviation discerns the "spread" more finely, and actually measures the average spread of all the individual terms, not just Lowest and Highest..

My experience on the GMAT is that you only really need to understand this conceptual distinction between range and standard deviation. For example, the following sets have the same range:

Set A: 1, 1, 93, 93, 93
Set B: 1, 24, 47, 70, 93
Set C: 1, 47, 47, 47, 93

...but I bet you can rank these from least to greatest standard deviation without doing any calculation. So, just know what the std. dev. formula is, if only to understand what makes it increase (more distance from the average, and/or more terms with more distance) and what makes it decrease (the opposite).