GMAT Math Strategies Solving Not-so-simple Statistics ProblemsThis tutorial is one of two focusing on GMAT math strategies. In this Q&A, you'll gain insight intodescriptive statisticsproblems specifically, those involving the concepts of arithmetic mean (simple average), median and range. You'll discover that these seemingly simple concepts can make for surprisingly difficult GMAT questions, unless you're truly prepared for them.

Q:Can you briefly define the termdescriptive statistics, and describe what aspects of descriptive statistics GMAT test takers are likely to encounter?A:The termdescriptive statisticsembraces such concepts as arithmetic mean (simple average), median, range, and standard deviation. All but the last of these concepts are easily understood. Just for the record, here's a definition of each one:For any set of numerical terms: Thearithmetic meanis the sum of the terms, divided by the number of terms in the set. Themedianis the middle term in value if the set contains an odd number of terms, or the arithmetic mean (average) of the two middle terms if the set contains an even number of terms. Therangeis the difference on the real-number line between the term with the greatest value and the term with the least value.You should understand each of these terms, because the test makers won't provide you with their definitions during the test. By the way, the same goes for standard deviation a more advanced statistical concept that inherently makes for a relatively challenging GMAT question.

Q:The three concepts you just defined seem very straightforward. How can the test makers design challenging questions or even moderately difficult ones involving these concepts?A:To increase the difficulty of questions that focus on these concepts, the test designers often usevariables, which add an algebraic dimension to these questions. Consider, for instance, the following Problem-Solving question (answer choices are omitted here):Which of the following expressions represents the arithmetic mean (average) of the five termsp,q,p+q,p 1, andq+ 1 ?Solving the problem requires not only application of the arithmetic-mean concept, but also a bit of algebraic manipulation, rendering the question a bit more complex than simply adding together five numbers and dividing by five. Here are the algebraic steps, plugging the variable expressions into a general equation for arithmetic mean (AM):

Still not to difficult, is it? So to further increase the difficulty level of an arithmetic-mean question, the test makers might provide the arithmetic mean and ask instead for the value of an unknown term in the set. Consider this variation on the problem just solved (again, omitting answer choices):Which of following expressions represents the fifth term of a set that also includes the termsp,q,p+q, andp 1, ifrepresents the arithmetic mean (average) of the five terms?This one's a bit more challenging, isn't it? It's more difficult to understand and to determine how to approach and solve. Moreover, although you apply the same arithmetic-mean formula to solve this problem as for the previous one, you need to perform more algebraic steps along the way:

Q:What about the concepts of median and range? How might the test makers design a challenging GMAT question involving either of these simple concepts?A:Again, the use of variables, instead of or in addition to numbers, adds complexity to the question. Also, the concepts of median and range are typically incorporated into an arithmetic-mean problem. Here's a Problem-Solving question that employs both devices (once again, answer choices are omitted here):If 0 q, whether (p+q) is greater or less thanpandqdepends on the sign ofq. Ifqis positive, then (p+q) >p>q. But ifqis negative, thenp> (p+q) >q. Even if you assumepandqare both positive, the median value might be either (p 1) orq, depending on the difference betweenpandq. If the difference is less than 1, then the median isp, whereas if the difference is greater than 1, then the median is (p 1):Ifpq< 1, then (p+q) > (q+ 1) >p>q>p 1.Ifpq> 1, then (p+q) >p> (p 1) > (q+ 1) >q.These sorts of dynamics between variable expressions is great fodder for Data-Sufficiency questions, because whether you can determine the relationships between the expressions depends on how much and what type of information you're provided about them. For example, here's the scenario we just looked at, transformed into a Data-Sufficiency question:Among the termsp,q,p+q,p 1, andq+ 1, which represents the median value?(1)p>q(2)pq< 1The correct answer is (E). Even considering both statements (1) and (2) together, the median value depends on the signs ofpandq.

Q:In your last example, whether the question was answerable depended on the sign and relative values of the variable expressions. Is this typical of GMAT Data-Sufficiency questions? If so, is there a systematic process for ensuring that your analysis accounts for all possible values of the variable expressions?A:Yes, it's very typical. In fact, identifying possible value ranges for variables is at the heart of many Data-Sufficiency questions. Whenever you encounter a Data-Sufficiency question involving variable expressions as opposed to numbers check to see whether the question itself asks: Which of two variable expressions is greater in value Whether two variable expressions are equal in value Whether the value of a variable expression is positive or negativeThe question might look something like one of the following:If..., isx>y?If..., doesx=y?If..., isx> 0 ?Your immediate reaction to this sort of question should be to consider the following value ranges along the real-number line: Values greater than 1 Fractional values between 0 and 1 Fractional values between 1 and 0 Values less than 1Why these four ranges? Well, when you perform certain operations with variables, the result depends on what range the variable falls into. For instance, when you square a number or take its cube root, whether you end up with a positive number, negative number, a smaller number, or a larger number, depends on which of the five ranges the original number falls into:Ifx> 1, then 1