SAT Mc Graw Hill 2011

CHAPTER 9 / SPECIAL MATH PROBLEMS 337

Average (Arithmetic Mean) Problems

You probably know the procedure for finding an
average of a set of numbers: add them up and divide
by how many numbers you have. For instance, the av-
erage of 3, 7, and 8 is (3 + 7 +8)/3 =6. You can de-
scribe this procedure with the “average formula”:

Since this is an algebraic equation, you can manipu-
late it just like any other equation, and get two more
formulas:

Sum = average ×how many numbers

This is a great tool for setting up tough problems. To find
any one of the three quantities, you simply need to find
the other two, and then perform the operation between
them. For instance, if the problem says, “The average
(arithmetic mean) of five numbers is 30,” just write 30 in
the “average” place and 5 in the “how many” place. No-
tice that there is a multiplication sign between them, so
multiply 30 × 5 =150 to find the third quantity: their sum.

Medians

How many numbers =

sum

average

Average =

sum

how many numbers

Lesson 2: Mean/Median/Mode Problems

Just about every SAT will include at least one

question aboutaverages,otherwise known as

arithmetic means. These won’t be simplistic

questions like “What is the average of this set

of numbers?” You will have to really under-

stand the concept of averages beyond the basic

formula.

Occasionally the SAT may ask you about the

modeof a set of numbers. A mode is the num-

ber that appears the most frequently in a set.(Just

remember: MOde =MOst.) It’s easy to see that

not every set of numbers has a mode. For

instance, the mode of [−3, 4, 4, 1, 12] is 4, but

[4, 9, 14, 19, 24] doesn’t have a mode.

The average (arithmetic mean) and the me-

dian are not always equal, but they are equal

whenever the numbers are spaced symmetri-

cally around a single number.

it splits the highway exactly in half. The median

of a set of numbers, then, is the middle number

when they are listed in increasing order. For in-

stance, the median of {−3, 7, 65} is 7, because

the set has just as many numbers bigger than

7 as less than 7. If you have an even number of

numbers, like {2, 4, 7, 9}, then the set doesn’t

have one “middle” number, so the median is

the average of the two middle numbers. (So

the median of {2, 4, 7, 9} is (4+7)/2 =5.5.)

All three of these formulas can be summarized

in one handy little “pyramid”:

When you take standardized tests like the SAT, your

score report often gives your score as a percentile,

which shows the percentage of students whose scores

were lower than yours. If your percentile score is

50%, this means that you scored at the medianof all

the scores: just as many (50%) of the students scored

below your score as above your score.

Example:

Consider any set of numbers that is evenly spaced,

like 4, 9, 14, 19, and 24:

Notice that these numbers are spaced symmetrically

about the number 14. This implies that the mean

and the median both equal 14. This can be helpful to

know, because finding the median of a set is often

much easier than calculating the mean.

Modes

41492419

A medianis something that splits a set into two

equal parts. Just think of the median of a

highway:

average

how

many

sum

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