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Chapter 2: Problem 137

The mean GPA for all students at a community college in the fall semester was 2.77. A student with a GPA of 2.0 wants to know her relative standing in relation to the mean GPA. A numerical summary that would be useful for this purpose is the a. standard deviation b. median c. interquartile range d. number of students at the community college

Short Answer

Expert verified
The standard deviation would be the most useful numerical summary.

Step by step solution

01

Understand the Problem

The student wants to understand how her 2.0 GPA compares to the mean GPA of 2.77 for the semester. To determine her relative standing, a measure of the dispersion of GPAs around the mean is needed.
02

Evaluate Possible Options

Consider each of the provided options to determine which would best describe the dispersion of GPAs. The standard deviation measures how spread out numbers are around the mean, the median provides a middle value, the interquartile range measures the middle spread around a median, and the number of students does not directly relate to GPA dispersion.
03

Determine the Best Option

Since the problem requires understanding relative standing in relation to the mean GPA, the most appropriate numerical summary is the standard deviation. It indicates how much individual GPAs, like that of the student with 2.0, deviate from the average GPA.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Standard Deviation
The concept of standard deviation plays a crucial role in understanding the spread of data. In statistics, it provides insight into how much individual data points, such as a student's GPA, vary from the average or mean value. The formula for standard deviation is calculated as the square root of the variance, which is the average of the squared differences from the mean.
To put it in simpler terms, if you have a set of GPAs, the standard deviation tells you how much each student's GPA typically differs from the average GPA. When GPAs are close to the mean, the standard deviation is small. Conversely, if they vary widely, the standard deviation is large.
  • If a student has a GPA of 2.0 and the mean GPA is 2.77, understanding the standard deviation helps determine how typical or atypical that GPA is compared to the rest.
  • A higher standard deviation implies more variation, while a lower one suggests most students have similar GPAs.
Mean
The mean, or average, is fundamental in statistics, acting as a basic measure of central tendency. It's calculated by summing up all individual values and dividing by their count. For example, to find the mean GPA of all students at a community college, you add all their GPAs and divide by the number of students.
The mean provides a single value that represents the entire dataset. It's particularly useful when comparing individual data points, such as a student's GPA, against this overall average.
  • For example, a student with a GPA lower than the mean, like a 2.0 GPA compared to a 2.77 mean, can see they are below average.
  • This helps students understand where they fall within the context of their peers, providing perspective on their academic performance.
GPA Analysis
GPA analysis allows students and educators to gain insights into academic performance. Understanding how one's GPA compares to the average can highlight areas needing improvement or confirm solid performance.
GPA analysis typically involves:
  • Comparing individual GPAs to the mean to assess academic standing.
  • Using measures of dispersion, such as the standard deviation, to understand variability and relative performance.
  • Incorporating other statistical measures, like the median or interquartile range, for comprehensive analysis.
For a student with a low GPA, analyzing their position relative to the mean and understanding standard deviation can guide them in identifying areas for focus and growth.
Dispersion Measurement
Dispersion measurement is essential for understanding how data is spread out, providing a picture of variability within a dataset. In the context of GPAs, it helps illustrate how much individual scores differ from each other and the mean.
Common measures of dispersion include:
  • Standard Deviation: This measures how close data is to the mean and is used to assess overall variability.
  • Interquartile Range (IQR): This represents the range between the middle 50% of data and is useful in understanding the spread without the influence of outliers.
By evaluating dispersion, students can better understand their GPA in the context of the entire student body, providing insights into whether a GPA is an outlier or typical compared to peers.

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Most popular questions from this chapter

Spring break hotel prices For a trip to Miami, Florida, over spring break in \(2014,\) the data below (obtained from travelocity.com) show the price per night (in U.S. dollars) for various hotel rooms. $$ \begin{array}{lllllllll} 239 & 237 & 245 & 310 & 218 & 175 & 330 & 196 & 178 \\ 245 & 255 & 190 & 330 & 124 & 162 & 190 & 386 & 145 \end{array} $$ a. Construct a stem-and-leaf plot. Truncate the data to the first two digits for purposes of constructing the plot. For example, 239 becomes \(23 .\) b. Reconstruct the stem-and-leaf plot in part a by using split stems; that is, two stems of \(1,\) two stems of \(2,\) and so on. Compare the two stem-and-leaf plots. Explain how one may be more informative than the other. c. Sketch a histogram by hand (or use software), using 6 intervals of length 50 , starting at 100 and ending at 400 . What does the plot tell about the distribution of hotel prices? (Mention where most prices tend to fall and comment about the shape of the distribution.)

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