Problem 15 - Algebra 2 and Trigonometry

Question

A student received the following grades on six tests: \(90,92,92,95,95, x\) a. For what value(s) of \(x\) will the set of grades have no mode? b. For what value(s) of \(x\) will the set of grades have only one mode? c. For what value (s) of \(x\) will the set of grades be bimodal?

Step-by-Step Solution

Answer

a. Any x not equal to 90, 92, 95. b. x=92 or x=95. c. x=90.

Step 1: Determine No Mode Value

The mode of a set of numbers is the number that appears most frequently. If a set of numbers has no mode, it means that each number appears the same number of times. In this problem, we have grades: 90, 92, 92, 95, 95, and x. For there to be no mode, the number of each grade must occur the same number of times. That means all three grades (90, 92, 95) and the unknown x must appear exactly once or another number should align them, which would only happen if all are unique. Thus, the value of x must be different from 90, 92, and 95. Any value not equal to 90, 92, or 95 will satisfy this condition.

Step 2: Determine One Mode Value

For the set to have one mode, one grade should appear more times than any other grade. Currently, 92 and 95 each appear twice. For either of these to be the mode alone, the value of x must duplicate one of them while not increasing the count of the other two. Therefore, x should be either 92 or 95 to produce a single mode. If x = 92, then 92 appears 3 times, and similarly, if x = 95, then 95 appears 3 times.

Step 3: Determine Bimodal Value

A bimodal set has two modes, which are two grades that appear more frequently than any other. In the current set, both 92 and 95 appear twice. To keep both as the modes, x must not increase the count of any number beyond two occurrences. Hence, x should be different from 92 and 95 to retain the two numbers as equally the most frequent. Since 90 appears once, choosing x = 90 will not affect the frequency of 92 and 95, maintaining the bimodal condition.

Key Concepts

Modes in Statistics

In statistics, the mode is a measure of central tendency that identifies the most frequently occurring value within a dataset. Calculating the mode is simple: just look for the value that appears the most. If all values appear once, the set is said to have no mode.
Unlike mean and median, the mode can vary greatly within different datasets because there can be more than one mode, or none at all.

• If each number appears the same number of times, there is no mode.
• If one value appears more frequently than others, that value is the mode.

The idea of mode is very useful for categorical data where numbers are less meaningful than the frequency of their occurrence.

Bimodal Distribution

A bimodal distribution refers to a situation in which a dataset has two values that occur with the highest frequency. In simpler terms, it's like a tie for the most popular value or grade.
Here's when bimodality matters:

• Observations show that having two modes can indicate that the dataset actually represents two different groups or phenomena.
• When both modes are equally common, it means that two separate categories are competing for frequency.

For instance, in a set of grades, if both 92 and 95 occur twice, the distribution is bimodal, meaning two scores share the title of most frequent.

Number Sets

A number set is a collection of numbers with which you are working. They can include integers, whole numbers, or any numbers like grades or scores.
Studying a number set involves looking at:

• How numbers relate to each other, such as frequency or averages.
• Variations within the numbers, like lowest and highest values and how often each occurs (frequency).

In statistical terms, understanding the composition of a number set is crucial because choosing or identifying a specific value, like "x" in our example, can determine the overall statistical properties, such as having no mode, one mode, or being bimodal.