### Story Problems: Make Them Real (Part 2)

Last time, we talked about how to make story problems real; if you haven’t read that article yet, go take a look before you continue with this one.

I’ve got another one for you that’s in that same vein: the math topic is different, but the “story” idea still hold in general. This one has something extra though: you need to know how a certain math topic (standard deviation) works in general. Otherwise, you won’t be able to think your way through the problem.

Try this GMATPrep® problem:

* ” During an experiment, some water was removed from each of 6 water tanks. If the standard deviation of the volumes of water in the tanks at the beginning of the experiment was 10 gallons, what was the standard deviation of the volumes of water in the tanks at the end of the experiment?

“(1) For each tank, 30 percent of the volume of water that was in the tank at the beginning of the experiment was removed during the experiment.

“(2) The average (arithmetic mean) volume of water in the tanks at the end of the experiment was 63 gallons.”

Standard deviation! Ugh. : )

Okay, it’s no accident that they’re using a DS-format problem for this one. It’s not possible to calculate a standard deviation in 2 minutes without a calculator (unless, perhaps, that standard deviation is zero!). They never expect us to calculate standard deviation on this test, but they do want to know whether we understand the concept in general.

(Why did I say “aloud”? Often, we tell ourselves that we can explain something, but not until we actually try do we realize that we need a refresher on the concept. Giving an explanation aloud forces you to prove that you really do know how to explain the concept. If you don’t, you’ll hear your uncertainty in your own explanation.)

Standard deviation is the measure of how spread apart a set of data points is. For example, let’s say you have the following 5 numbers in a set: {3, 3, 3, 3, 3}. The standard deviation is zero because the numbers are all exactly the same—there is no “spread” at all in the set.

Which of the following two sets has a larger standard deviation?

{1, 2, 3, 4, 5}

{1, 10, 20, 80, 2,000}

The second one! The numbers are much more spread apart than in the first set.

Right now, some of you are wondering: okay, but what’s the actual standard deviation of those two sets?

I don’t know. I could calculate it—I’m sure there are many online “standard deviation” calculators I could use. But I don’t care. The real test is never going to make me calculate this! (And that’s why I haven’t gotten into the actual calculation method here… nor will I.)

There are a few concepts that we should know, though, in terms of how changes to sets can affect the standard deviation.

First, if you add the same number to every element in the set (or subtract the same number from each), the standard deviation won’t change. For example, these two sets have the same standard deviation:

{1, 2, 3, 4, 5}

{6, 7, 8, 9, 10}

The second set is the result of adding 5 to every number in the first set.

If you multiply every number in the set by the same number, the standard deviation could increase, decrease, or stay the same. (We’ll ignore the case where we multiply by 0, which makes everything 0. We’ll also ignore the cases where we multiply by 1 or -1, which doesn’t change the spread of numbers, just potentially the sign.)

Here are the cases:

If the starting numbers were all the same (e.g., 3, 3, 3, 3, 3), such that the standard deviation is zero, then the new set will have the same standard deviation of zero—because the new set will still consist of the five of the same numbers. For example, if we multiply every number in the above set by 2, we get {6, 6, 6, 6, 6}. The standard deviation doesn’t change.

If the starting numbers are different, then two possibilities will occur:

(a) If we multiply by an integer greater than 1 or less than -1, then the numbers will become more spread apart. For example, multiply every number in the set {1, 2, 3, 4, 5} by 2. The new set is {2, 4, 6, 8, 10}. Here’s the interesting thing: whatever the standard deviation of the first set is, the new set’s standard deviation is also multiplied by 2. Basically, the numbers get twice as far apart, so the standard deviation also gets twice as big.

(b) If, on the other hand, we multiply by a fraction between 0 and 1 (or between 0 and -1), then the numbers will get closer together. For example, multiply every number in the set {1, 2, 3, 4, 5} by 0.5. the new set is {0.5, 1, 1.5, 2, 2.5}. Can you guess what happens this time? Whatever the standard deviation of the first set, the standard deviation of the second set is also multiplied by 0.5. The numbers are half as far apart, so the standard deviation also gets cut in half.

Okay, now that you know that, take a look at the problem again. Can you think your way through?

There are six water tanks, each with some amount of water in it; picture the tanks. The problem also says that the standard deviation before the experiment was 10 gallons. This tells us one important piece of info: the tanks did not all start out with the same amount of water. (If they had, the standard deviation would have been zero.) Picture (or draw!) 6 tanks with different levels of water.

Next, the problem asks for the standard deviation of the water volumes after the experiment.

Now, if they gave the water volume in each tank after the experiment, then I could calculate the standard deviation (though I would never actually do so, of course!). Chances are they’re not going to be that obvious though. So what else could help me to answer the question?

Bingo—that stuff I was talking about a minute ago. If, for example, the volume in each tank was halved, then the standard deviation (10) would also be halved.

Check it out!

“(1) For each tank, 30 percent of the volume of water that was in the tank at the beginning of the experiment was removed during the experiment.”

Look at your tanks. 30% of the initial water volume was removed from every tank. That’s equivalent to multiplying the volume in each by 0.7. If this is true for every single tank (which it is, according to statement 1!), then the standard deviation must also be 0.7 of the original standard deviation. In other words, 10(0.7) = 7 (though we don’t actually have to calculate this—it’s enough to know that the volume of each tank was reduced by the same factor). Statement 1 is sufficient. Cross off answers B, C, and E.

“(2) The average (arithmetic mean) volume of water in the tanks at the end of the experiment was 63 gallons.”

I’m going to tell you the answer right up front: this statement is not sufficient. Many people say that it is, though, so this is where I want you to try the “explain it aloud” technique. If you thought statement 2 was sufficient, argue with me! Try to explain it to me right now. (I apologize in advance for not responding.)

If the 6 tanks average 63 gallons… write down some possibilities. Each one could have 63 gallons, for a standard deviation of zero. Or maybe the 6 volumes are 61, 62, 63, 63, 64, 65, in which case the standard deviation is greater than zero. There are infinite possible combinations, so statement 2 is insufficient.

Key Takeaways for Advanced Math Topics + Story Problems

(1) You do still need to know some “textbook” math to answer the more advanced problems correctly. : )

(2) You still don’t have to calculate anything! As a general rule, if you’re looking at a GMAT problem and it looks like the solution is unbelievably calculation-heavy, then the next thought in your head should be, “Do I really want to spend my precious, limited time doing that math?”

(3) And the answer is no! There’s a good chance that there’s some easier way to do the problem. Try to learn as many of those approaches as possible before you get into the test, but know that you’re not going to know them all. When you do hit a problem for which you can only think of a process that seems way too involved for 2 minutes… guess and move on!

* GMATPrep® questions courtesy of the Graduate Management Admissions Council. Usage of this question does not imply endorsement by GMAC.

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