Answer the questions below.
(a) The 10 members of a fraternity have the following test scores: 118, 120, 121, 123, 124, 125, 127, 128, 130, 131. Which measure should be used to summarize the data?
A. Mean
B. Median
C. Mode
(b) On a test, each student is given a grade of A, B, C, D, or F. Which measure tells the grade given most often?
A. Mean
B. Median
C. Mode
Answer (2)
For test scores: Since the mean and median are close and there are no outliers, the mean is the best measure.
For grades: The mode is the best measure because it represents the most frequent grade.
For data from the past 9 days: Assuming we want the most frequent value, the mode is the best measure.
Explanation
Understand the problem and provided data We are asked to determine the best measure to describe data in three different scenarios. (a) We have a set of 10 test scores: 118, 120, 121, 123, 124, 125, 127, 128, 130, 131. (b) Grades of students on a test are A, B, C, D, or F. (c) We are considering data from the past 9 days.
(a) Calculate mean and median (a) To determine whether the mean or median is a better measure for the given test scores, we need to consider the distribution of the data. If the data is symmetric, the mean is appropriate. If there are outliers or the data is skewed, the median is more appropriate. Let's calculate the mean and median for the given data.
First, let's sort the data: 118, 120, 121, 123, 124, 125, 127, 128, 130, 131.
The mean is calculated as the sum of the data divided by the number of data points: M e an = 10 118 + 120 + 121 + 123 + 124 + 125 + 127 + 128 + 130 + 131 = 10 1247 = 124.7
The median is the middle value of the sorted data. Since there are 10 data points (an even number), the median is the average of the two middle values, which are the 5th and 6th values (124 and 125). M e d ian = 2 124 + 125 = 2 249 = 124.5
Since the mean (124.7) and median (124.5) are very close, and there are no obvious outliers, the mean is an appropriate measure to summarize the data.
(b) Determine the appropriate measure for categorical data (b) To determine which measure (mean, median, or mode) tells the grade given most often, we need to consider the type of data. Since the data is categorical (grades A, B, C, D, or F), we cannot calculate the mean or median. The mode is the appropriate measure for categorical data, as it represents the most frequent category.
(c) Determine the appropriate measure for incomplete data (c) To determine which measure is appropriate for the data from the past 9 days, we need more information about the data. The question is incomplete, so I will assume that we are looking for the most frequent value. In this case, the mode would be the most appropriate measure.
Examples
Understanding which measure to use is crucial in many real-world scenarios. For example, when analyzing income data, the median is often preferred over the mean because it is less affected by extremely high incomes. In retail, the mode helps identify the most popular product, guiding inventory decisions. In education, understanding the distribution of test scores helps teachers tailor their instruction to meet the needs of their students.
For the test scores, the best measure to summarize the data is the Mean (A). For the grades given, the Mode (C) is the most appropriate measure as it indicates the most frequently assigned grade.
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