Statistical Analysis
Statistical Analysis
On the Math IIC 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
From a given a data set, you should be able to derive the four following 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 found in the set.
On the Math IIC and in many high school math classes, the arithmetic mean is often called an “average” or is simply referred to as the mean.
Let’s take another look at the test scores of Ms. McCarthy’s math class. We’ve sorted the scores in her class 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, since that is the number of scores in the set. The mean of the math test scores in Ms. McCarthy’s class is:
While some Math IIC 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 IIC 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. For example:
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 four 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 have 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?
This question is really just a rephrasing of the previous 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 IIC 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.
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. For our purposes, let’s add an extra number to the previous example to produce an even set:
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 all the other numbers appear just once. In a set where 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 where all of the elements appear an equal number of times, there is no mode.
The good news is that mode questions are easy. The bad news is that mode questions don’t appear all that much on the Math IIC.
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.
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