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Chapter 2: Problem 53

Use the following information to answer the next three exercises: State whether the data are symmetrical, skewed to the left, or skewed to the right. When the data are symmetrical, what is the typical relationship between the mean and median?

Short Answer

Expert verified
In symmetrical data, the mean and median are equal.

Step by step solution

01

Understanding Symmetrical Data

Symmetrical data occurs when the left and right sides of the distribution are mirror images of one another. In such cases, the data is evenly distributed around a central point.
02

Compare Mean and Median

In a symmetrical distribution, the mean and median are typically equal because the data values are evenly distributed around a central point, which means there is no skewness to pull the mean away from the median.

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Key Concepts

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

Symmetrical Distribution
A symmetrical distribution is quite like looking at a mirror reflection from the center point of your data. When the data is symmetrical, it means that the shape of the distribution on one side is a perfect reflection of the other side. This balance gives an impression of a 'mountain peak' that is identical no matter which side you approach it from. Some important points to remember about symmetrical distributions:
  • The highest point, or peak, of a symmetrical distribution often represents the mean, median, and mode, which all align with each other in such cases.
  • Common examples of symmetrical distributions are those that resemble a bell shape, like the Normal Distribution.
  • Because of the equal distribution of data on both sides, such distributions are also known to portray fairness in data representation.
Recognizing a symmetrical distribution can help you quickly identify vital statistical characteristics, such as central tendency and variability, intuitively.
Mean and Median Relationship
The mean and median are both measures of central tendency but they tell us slightly different things about our data. When it comes to symmetrical distributions, these two measures behave in remarkable harmony. Let's uncover why. In a symmetrical distribution:
  • The mean, which is the average of all data points, will generally equal the median, the middle value when all data points are ordered from smallest to largest.
  • This equality occurs because there's a perfect balance - the data is distributed equally around a central point. Thus, any values that might pull the mean in one direction or the other are perfectly countered by equal values pulling back in the opposite direction.
  • This relationship implies that in symmetrical data, the differences at both extremes are balanced, ensuring both mean and median stay at the distribution's center.
Understanding the mean and median's relationship in various distributions helps analyze data more accurately and efficiently.
Skewness in Data
Skewness gives insight into the direction and magnitude of asymmetry in data, which is crucial for identifying if data is not symmetrical. Here's what to know about skewness:
  • When data is skewed to the right (or positively skewed), it means that the tail on the right side is longer or fatter than on the left side. In this case, the mean is typically greater than the median as higher values have pulled the mean to the right.
  • On the other hand, if data is skewed to the left (or negatively skewed), the tail on the left side is longer. Here, the mean tends to be less than the median since lower values pull it leftward.
  • Skewness can impact many statistical measures and analyses, such as testing hypotheses or creating regression models.
Recognizing skewness in your data is essential for choosing the correct statistical methods to apply and for accurately interpreting your results.

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Most popular questions from this chapter

The University of California has two criteria used to set admission standards for freshman to be admitted to a college in the UC system: a. Students' GPAs and scores on standardized tests (SATs and ACTs) are entered into a formula that calculates an "admissions index" score. The admissions index score is used to set eligibility standards intended to meet the goal of admitting the top 12% of high school students in the state. In this context, what percentile does the top 12% represent? b. Students whose GPAs are at or above the 96th percentile of all students at their high school are eligible (called eligible in the local context), even if they are not in the top 12% of all students in the state. What percentage of students from each high school are "eligible in the local context"?

Construct a frequency polygon for the following: $$ a.\begin{array}{|c|c|}\hline \text { Pulse Rates for Women } & {\text { Frequency }} \\ \hline 60-69 & {12} \\ \hline 70-79 & {14} \\ \hline 80-79 & {11} \\ \hline 90-99 & {1} \\ \hline 100-109 & {1} \\ \hline 100-119 & {1} \\ \hline 120-129 & {1} \\ \hline\end{array}$$ $$b.\begin{array}{|c|c|}\hline \text { Actual Speed in a 30 MPH zone } & {\text { Frequency }} \\ \hline 42-45 & {25} \\ \hline 46-49 & {14} \\\ \hline 46-49 & {14} \\ \hline 50-53 & {7} \\ \hline 54-57 & {3} \\ \hline 58-61 & {1} \\ \hline\end{array}$$ $$c.\begin{array}{|c|c|}\hline \text { Tar (mg) in Nonfiltered cigarettes } & {\text { Frequency }} \\ \hline 10-13 & {1} \\ \hline 14-17 & {0} \\ \hline 14-21 & {15} \\ \hline 22-25 & {7} \\ \hline 26-29 & {7} \\\ \hline\end{array}$$

Use the following information to answer the next three exercises: The following data show the lengths of boats moored in a marina. The data are ordered from smallest to largest: 16; 17; 19; 20; 20; 21; 23; 24; 25; 25; 25; 26; 26; 27; 27; 27; 28; 29; 30; 32; 33; 33; 34; 35; 37; 39; 40 Identify the mode.

Construct a frequency polygon from the frequency distribution for the 50 highest ranked countries for depth of hunger. $$ \begin{array}{|c|c|}\hline \text { Depth of Hunger } & {\text { Frequency }} \\\ \hline 230-259 & {21} \\ \hline 260-289 & {13} \\ \hline 260-389 & {5} \\\ \hline 390-349 & {7} \\ \hline 320-349 & {1} \\ \hline 380-409 & {1} \\\ \hline 410-439 & {1} \\ \hline\end{array} $$

Use the following information to answer the next nine exercises: The population parameters below describe the full-time equivalent number of students (FTES) each year at Lake Tahoe Community College from 1976鈥�1977 through 2004鈥�2005. \(\bullet \mu=1000 \mathrm{FTES}\) \(\bullet\) median \(=1,014 \mathrm{FTES}\) \(\bullet \quad \sigma=474 \mathrm{FTES}\) \(\cdot\) first quartile \(=528.5\) FTES \(\cdot\) third quartile \(=1,447.5\) FTES \(\cdot n=29\) years How many standard deviations away from the mean is the median?
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