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Every score in the following batch of exam scores is in the 60 's, 70 's, 80 's, or 90 's. A stem-and-leaf display with only the four stems \(6,7,8\), and 9 would not give a very detailed description of the distribution of scores. In such situations, it is desirable to use repeated stems. Here we could repeat the stem 6 twice, using 6L for scores in the low 60's (leaves 0, 1,2, 3 , and 4) and \(6 \mathrm{H}\) for scores in the high 60 's (leaves \(5,6,7,8\), and 9). Similarly, the other stems can be repeated twice to obtain a display consisting of eight rows. Construct such a display for the given scores. What feature of the data is highlighted by this display? \(\begin{array}{lllllllllllll}74 & 89 & 80 & 93 & 64 & 67 & 72 & 70 & 66 & 85 & 89 & 81 & 81 \\ 71 & 74 & 82 & 85 & 63 & 72 & 81 & 81 & 95 & 84 & 81 & 80 & 70 \\ 69 & 66 & 60 & 83 & 85 & 98 & 84 & 68 & 90 & 82 & 69 & 72 & 87 \\ 88 & & & & & & & & & & & & \end{array}\)

Step by step solution

01

Classify Scores for Each Stem

First, we need to distribute the scores based on the repeated stem categories: 6L, 6H, 7L, 7H, 8L, 8H, 9L, and 9H. - **6L (60-64)**: 60, 63, 64 - **6H (65-69)**: 66, 66, 67, 68, 69, 69 - **7L (70-74)**: 70, 70, 71, 72, 72, 72, 74, 74 - **7H (75-79)**: None - **8L (80-84)**: 80, 80, 81, 81, 81, 81, 82, 82, 83, 84, 84 - **8H (85-89)**: 85, 85, 85, 87, 88, 89, 89 - **9L (90-94)**: 90, 93 - **9H (95-99)**: 95, 98

02

Create the Stem-and-Leaf Display

Now, create the stem-and-leaf display using the above classification for each repeated stem. - **6L |** 0, 3, 4 - **6H |** 6, 6, 7, 8, 9, 9 - **7L |** 0, 0, 1, 2, 2, 2, 4, 4 - **7H |** (No Scores) - **8L |** 0, 0, 1, 1, 1, 1, 2, 2, 3, 4, 4 - **8H |** 5, 5, 5, 7, 8, 9, 9 - **9L |** 0, 3 - **9H |** 5, 8

03

Highlight Features of the Data

This display highlights the clustering of scores, shows specific distributions within the score ranges (like the absence of scores in 75-79), and provides more detail in areas with high frequency, such as scores around the low 80s.

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Key Concepts

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

Data Visualization

Data visualization is a powerful tool that helps us understand numerical data by presenting it in an intuitive and visual way. The stem-and-leaf display is one such method used to organize and present data. It involves splitting each data point into a "stem" and a "leaf," where the stem consists of the leading digit(s) and the leaf represents the trailing digit. This simple yet effective display method allows us to quickly see patterns, trends, and outliers in the data set.

In the context of the exercise, we see the utilization of a stem-and-leaf plot with repeated stems to better capture the nuances in the score distribution. By visualizing this data in a structured format, we turn abstract numbers into something more tangible and accessible. It helps to

• Identify clusters of data points.
• Reveal gaps or missing ranges, as seen with missing scores from 75 to 79.
• Highlight frequency and distribution patterns distinctly.

Using repeated stems enhances this visualization by providing more detail within each score range, improving the clarity and interpretability of the data set.

Score Distribution

Score distribution refers to how scores are spread or clustered across different intervals in a data set. In the given exercise, scores are located within the ranges of 60s, 70s, 80s, or 90s. Through a stem-and-leaf display, repeated stems like 6L and 6H depict the detailed distribution of scores within those larger ranges.

Such a granular approach in displaying score distribution allows for:

• Better resolution of data clusters, where multiple scores fall under specific sub-ranges.
• Identifying peaks or parts with no scores, adding further insights into the overall score pattern.
• Easy comparison of score frequencies between these sub-ranges.

This method breaks down broad score ranges into smaller, more detailed parts, making it easier for analysts to assess data characteristics and draw conclusions.

Statistical Analysis

Statistical analysis involves interpreting data to uncover underlying trends, patterns, and insights. A stem-and-leaf display aids in the preliminary stages of statistical analysis by clearly outlining the data's distribution and helping identify any prominent features, like clustering and gaps.

This insight can guide further steps in statistical analyses, such as:

• Determining the central tendency of the data.
• Assessing data variability through visible spread and outlier inspection.
• Evaluating symmetry or skewness in the data distribution.

With the repeated stems, the given problem allows a detailed preliminary analysis, facilitating a deeper exploration of data characteristics before applying more advanced statistical methods.

Repeated Stems

Repeated stems are a technique used in stem-and-leaf displays to provide a more detailed view of the data. By dividing each stem into two subcategories ("L" for lower and "H" for higher), we increase the resolution of the visualization. This allows us to see variations within broader data categories, a crucial feature when dealing with dense data sets.

In the context of the exercise:

• The stem "6" is divided into 6L and 6H, capturing scores from 60-64 and 65-69, respectively.
• This repetition is mirrored for the 70s, 80s, and 90s, increasing the granularity in each range.
• This approach prevents loss of detail, making it easier to interpret dense clusters of data.

Using repeated stems enhances data readability and ensures that subtle trends, clusters, or gaps are more adequately captured, leading to more informative and actionable data insights.