


Statistical Analysis
Statistical analysis sounds like dental surgery. Scientific
and sticky and gross. But SAT statistical analysis is actually not
so bad. On these questions, the SAT gives you a data set—a collection
of measurements or quantities. An example of a data set is the set
of math test scores for the 20 students in Ms. Mathew’s
fourthgrade class:
71, 83, 57, 66, 95, 96, 68, 71, 84, 85, 87,
90, 88, 90, 84, 90, 90, 93, 97, 99
You are then asked to find one or more of the following
values:
 Arithmetic Mean
 Median
 Mode
 Range
Arithmetic Mean (a.k.a. Average)
Arithmetic mean means the same thing as average. It’s
also the most commonly tested concept of statistical analysis on
the SAT. The basic rule of finding an average isn’t complicated:
It’s the value of the sum of the elements contained in a data set divided
by the number of elements in the set.
Take another look at the test scores of the 20 students
in Ms. Mathew’s class. We’ve sorted the scores in her class from
lowest to highest:
57, 66, 68, 71, 71, 83, 84, 84, 85, 87, 88,
90, 90, 90, 90, 93, 95, 96, 97, 99
To find the arithmetic mean of this data set, sum the
scores and then divide by 20, since there are 20 students
in her class:
But the SAT is Sneaky When It’s Mean
But that’s not the way that the SAT usually tests mean.
It likes to be more complicated and conniving. For example,

Here’s the key: If you know the average of a group and
also know how many numbers are in the group, you can calculate the
sum of the numbers in the group. The question above tells you that
the average of the numbers is 22 and that there are
four numbers in the group. If the average of four numbers is 22,
then the four numbers, when added together, must equal 4 × 22 = 88. Since you know three of the four numbers
in the set, and since you now know the total value of the set, you
can write
Solving for the unknown number is easy. All you have to
do is subtract the sum of 7, 11, and 18 from 88: x = 88
– (7 + 11 + 18) = 88 – 36 = 52.
As long as you realize that you can use an average to
find the sum of all the values in a set, you can solve pretty much
every question about arithmetic mean on the SAT:

This question seems really tough, since it keeps splitting
apart its set of seven mysterious numbers. Students often freak
out when SAT questions ask them for numbers that seem impossible
to determine. Chill out. You don’t have to know the exact numbers
in the set to answer this problem. All you have to know is how averages
work.
There are seven numbers in the entire set,
and the average of those numbers is 54. The sum of
the seven numbers in the set is 7 × 54 = 378. And, as the problem states, three
particular numbers from the set have an average of 38.
Since the sum of three items is equal to the average of those three
numbers multiplied by three, the sum of the three numbers in the
problem is 3 × 38 = 114. Once you’ve got
that, you can calculate the sum of the four remaining numbers, since
that value must be the total sum of the seven numbers minus the
sum of the miniset of three: 378 – 114 = 264. Now,
since you know the total sum of the four numbers, you can get the
average by dividing by 4: 264 ÷ 4 = 66.
And here’s yet another type of question the SAT likes
to ask about mean: the dreaded “changing mean” question.

Actually, you shouldn’t dread “changing mean” questions
at all. They’re as simple as other mean questions. Watch. Here’s
what you know from the question: the original number of members, 14,
and the original average age, 34. And you can use this
information to calculate the sum of the ages of the members of the
original group by multiplying 14 × 34 = 476. From the question, you also know
the total members of the group after the new member joined, 14
+ 1 = 15, and you know the new average age of the group, 37.
So, you can find the sum of the ages of the new group as well: 15 × 37 = 525. The age of the new member is just
the sum of the age of the new group minus the sum of the age of
the old group: 555 – 476 = 79. That is one ancient scuba
diver.
Median
The median is the number whose value is exactly in the
middle of all the numbers in a particular set. Take the set {6, 19, 3, 11, 7}.
If the numbers are arranged in order of value, you get
{3, 6, 7, 11, 19}
It’s clear that the middle number in this group is 7,
so 7 is the median.
If a set has an even number of items, it’s impossible
to isolate a single number as the median. Here’s the last set, but
with one more number added:
{3, 6, 7, 11, 19, 20}
In this case, the median equals the average of the two
middle numbers. The two middle numbers in this set are 7 and 11,
so the median of the set is ^{(7+11)}
/_{2} = 9.
Mode
The mode is the number within a set that appears most
frequently. In the set {10, 11, 13, 11, 20},
the mode is 11, since it appears twice and all the
others appear once. In a set where more than one number appears
at the same highest frequency, there can be more than one mode:
The set {2, 2, 3, 4, 4} has
modes of 2 and 4. In the set {1, 2, 3, 4, 5}, where
all of the numbers appear an equal number of times, there is no mode.
Range
The range measures the spread of a data set, or the difference
between the smallest element and the largest. For the set of test
scores in Ms. Mathew’s class, {57,
66, 68, 71, 71, 83, 84, 84, 85, 87, 88, 90, 90, 90, 90, 93, 95,
96, 97, 99}, the range is 99 – 57 = 42.
