


Statistical Analysis
On the Math IC, you will occasionally be presented with
a data set—a collection of measurements or quantities. For example,
the set of test scores for the 20 students in Ms. McCarthy’s
math class is a data set:
71, 83, 57, 66, 95, 96, 68, 71, 84, 85, 87,
90, 88, 90, 84, 90, 90, 93, 97, 99
The Math IC will test your ability to use the basic tools
of statistics. From a given data set, you should be able to derive
the following four values:
 Arithmetic Mean
 Median
 Mode
 Range
Arithmetic Mean
The arithmetic mean is the value of the sum
of the elements contained in a data set divided by the number of
elements in the set.
On the Math IC and in many high school math classes, the
arithmetic mean is often called an “average” or “mean.”
Let’s take another look at the test scores of the 20 students
in Ms. McCarthy’s math class. We’ve sorted the scores in order 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, we must
sum the scores and then divide by 20—the number of scores in the
set. The mean of the math test scores in Ms. McCarthy’s class is:
While some Math IC questions might cover arithmetic mean
in the straightforward manner shown in this example, it is more
likely the test will cover mean in a more complicated way.
The Math IC might give you n – 1 numbers
of an nnumber set and the average of that set,
and ask you to find the last number:

Remember that the mean of a set of numbers is intimately
related to the number of terms in the set and the sum of those terms.
In the question above, you know that the average of the 4 numbers
is 22. This means that the four numbers, when added together, must
equal 4 22, or 88. Based on the sum of the
three terms you are given, you can easily determine the fourth number
by subtraction:
Solving for the unknown number is easy: all you need to
do is subtract 7, 11, and 18 from 88 to get 52, which is the answer.
The test might also present you with what we call an “adjusted
mean” question. For example:

Here you know the original number of students in the class
and the original mean of the students’ ages, and you are asked to
determine the mean after an additional term is introduced. To figure
out the age of the new student, you simply need to find the sum
of the ages in the adjusted class (with one extra student) and subtract
from that the sum of the ages of the original class. To calculate
the sum of the ages of the adjusted class:
By the same calculations, the sum of the students’ ages
in the original class is 14 34 = 476. So the new student added
an age of 525 – 476 = 49 years.
Median
The Math IC might also ask you about the median of a set
of numbers. The median is the number whose value is
in the middle of the numbers in a particular set. Take the set:
6, 19, 3, 11, 7. Arranging the numbers in order of value results
in the list below:
3, 6, 7, 11, 19
Once the numbers are listed in this ordered way, it becomes
clear that the middle number in this group is 7, making 7 the median.
The set 3, 6, 7, 11, 19 contains an odd number
of items; in a set with an even number of items, it’s impossible
to isolate a single number as the median, so calculating the median
requires an extra step. Let’s add one number to the set from the
previous example:
3, 6, 7, 11, 15, 19
When the set contains an even number of elements, the
median is found by taking the mean 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 element of a set that appears
most frequently. In the set 10, 11, 13, 11, 20, the mode is 11 since
it appears twice and each of the other numbers appears only once.
In a set in which more than one number appears with the same highest
frequency, there is more than one mode: the set 2, 2, 3, 4, 4 has
modes of 2 and 4. In a set in which each of the elements appears
an equal number of times, there is no mode.
Mode questions are easy; unfortunately they don’t often
appear on the Math IC.
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. McCarthy’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.
