Statistical Analysis
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:
  1. Arithmetic Mean
  2. Median
  3. Mode
  4. 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 n-number set and the average of that set, and ask you to find the last number:
If the average of four numbers is 22, and three of the numbers are 7, 11, and 18, then what is the fourth 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:
The mean age of the 14 members of a ballroom dance class is 34. When a new student enrolled, the mean age increased to 35. How old is the new student?
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.
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+112 = 9.
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.
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.
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