91Ó°ÊÓ

Consider numerical observations \(x_{1}, \ldots, x_{n^{-}}\)- It is frequently of interest to know whether the \(x_{i}\) s are (at least approximately) symmetrically distributed about some value. If \(n\) is at least moderately large, the extent of symmetry can be assessed from a stem-and-leaf display or histogram. However, if \(n\) is not very large, such pictures are not particularly informative. Consider the following alternative. Let \(y_{1}\) denote the smallest \(x_{p}, y_{2}\) the second smallest \(x_{i}\), and so on. Then plot the following pairs as points on a two-dimensional coordinate system: \(\left(y_{n}-\tilde{x}, \tilde{x}-y_{1}\right)\), \(\left(y_{n-1}-\tilde{x}, \tilde{x}-y_{2}\right),\left(y_{n-2}-\tilde{x}, \tilde{x}-y_{3}\right), \ldots\) There are \(n / 2\) points when \(n\) is even and \((n-1) / 2\) when \(n\) is odd. a. What does this plot look like when there is perfect symmetry in the data? What does it look like when observations stretch out more above the median than below it (a long upper tail)? 83. Consider numerical observations \(x_{1}, \ldots, x_{n^{-}}\)- It is frequently of interest to know whether the \(x_{i}\) s are (at least approximately) symmetrically distributed about some value. If \(n\) is at least moderately large, the extent of symmetry can be assessed from a stem-and-leaf display or histogram. However, if \(n\) is not very large, such pictures are not particularly informative. Consider the following alternative. Let \(y_{1}\) denote the smallest \(x_{p}, y_{2}\) the second smallest \(x_{i}\), and so on. Then plot the following pairs as points on a two-dimensional coordinate system: \(\left(y_{n}-\tilde{x}, \tilde{x}-y_{1}\right)\), \(\left(y_{n-1}-\tilde{x}, \tilde{x}-y_{2}\right),\left(y_{n-2}-\tilde{x}, \tilde{x}-y_{3}\right), \ldots\) There are \(n / 2\) points when \(n\) is even and \((n-1) / 2\) when \(n\) is odd. a. What does this plot look like when there is perfect symmetry in the data? What does it look like when observations stretch out more above the median than below it (a long upper tail)?

Step by step solution

01

Understand the Pair Plot Concept

To assess symmetry, we plot pairs \((y_{n}-\tilde{x}, \tilde{x}-y_{1})\), \((y_{n-1}-\tilde{x}, \tilde{x}-y_{2})\), etc., where \((y_{i})\) are the ordered observations and \((\tilde{x})\) is the median. The number of points depends on whether \(n\) is odd or even.

02

Symmetry in the Plot

When the observations are perfectly symmetric about the median, each negative of the distance above the median is exactly counteracted by the distance below the median. This results in the plot being a diagonal line, specifically \(y = x\). Each point \((y_{n-i}-\tilde{x}, \tilde{x}-y_{i+1})\) would fall on this line.

03

Asymmetry with a Long Upper Tail

If the observations have a long upper tail, points would plot above the line \(y = x\) because the distances above the median (for larger \(y\)'s) are greater than those below it (for smaller \(y\)'s). This shifts the plot upward, deviating from the diagonal.

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Key Concepts

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

Stem-and-Leaf Display

Visualizing data is important for understanding its distribution, particularly symmetry. One way to do this is by using a **stem-and-leaf display**. This method organizes data into an easily readable format by splitting each data point into a "stem" and a "leaf." The stem represents the largest common shared element, often the initial digits, while the leaf represents final digits. For example, in a list of numbers like 54, 57, and 59, the stem would be 5 and the leaves would be 4, 7, and 9. This gives us a simple visual to spot the shape of the data distribution. - **Benefits**: - Compact and organized - Maintains original data values When assessing statistical symmetry, a symmetrical stem-and-leaf display will show a balanced spread of numbers around a central stem, indicating that values are evenly distributed above and below a central point, typically the median.

Histogram

Another tool for visualizing data distribution is the **histogram**. This graphical representation is created by dividing data ranges into intervals, or "bins," and taller bars represent more data points falling within that range. Histograms are particularly useful when assessing symmetry in large datasets. If the data is symmetrically distributed, the histogram will show a bell-shaped curve with equal tails on both sides. This method helps in visually determining whether data points are skewed, either with a long tail to the left or right. - **Key Features**: - Uses bars to represent frequency of data ranges - Highlights any skewness in data A symmetrical histogram not only indicates balance around the median but also helps in identifying any outliers that may cause asymmetry. A longer tail on one side, for instance, suggests that data stretches more on that side, impacting the distribution around the median.

Median

The **median** is a central value that divides a dataset into two equal halves, and it's crucial for assessing symmetry. When data is ordered, the median represents a boundary where 50% of data points fall below and 50% above. Finding the median is straightforward: - For an odd number of data points, the median is the middle value. - For an even number, it's the average of the two middle values. The median is especially pivotal when visualizing symmetry, as it serves as a reference line. If data is symmetrically distributed, we expect variations above and below this line to be mirrored. When combined with a stem-and-leaf display or histogram, analyzing data symmetry becomes clearer, highlighting how well data clusters around the median or if there's any skew, suggesting a long tail dominates one side.