91Ó°ÊÓ

Every score in the following batch of exam scores is in the \(60 \mathrm{~s}, 70 \mathrm{~s}, 80 \mathrm{~s}\), or \(90 \mathrm{~s}\). 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 \(6 \mathrm{~L}\) 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? \(\begin{array}{lllllllllllll}74 & 89 & 80 & 93 & 64 & 67 & 72 & 70 & 66 & 85 & 89 & 81 & 81 \\ 71 & 74 & 82 & 85 & 63 & 72 & 81 & 81 & 95 & 84 & 81 & 80 & 70 \\ 69 & 66 & 60 & 83 & 85 & 98 & 84 & 68 & 90 & 82 & 69 & 72 & 87\end{array}\)

Step by step solution

01

Understand Stem-and-Leaf Display

A stem-and-leaf display is a method of displaying quantitative data in a graphical format. It involves splitting each value into two parts: the 'stem', which is a leading digit or digits, and the 'leaf', which is the trailing digit or digits. In this exercise, we're using repeated stems for better detail.

02

Create Repeated Stem Groups

Separate the stems into two groups based on the tens digit, creating a 'low' (L) and 'high' (H) version of each stem. For example, split the 60s into '6L' for scores 60-64 and '6H' for scores 65-69. Do similarly for 70s, 80s, and 90s.

03

Assign Scores to Repeated Stems

Take each score from the data set and assign it to the appropriate stem and leaf. For instance, the score 64 goes under '6L' with the leaf 4, and the score 67 goes under '6H' with the leaf 7.

04

Construct the Stem-and-Leaf Display

List each stem group in ascending order, then append the corresponding leaves (the trailing digits of the scores) in ascending order. The stems will be [6L, 6H, 7L, 7H, 8L, 8H, 9L, 9H], and the leaves are arranged accordingly.

05

Analyze the Display

Look at the stem-and-leaf display for patterns, trends, or concentrations of scores. This can highlight clusters of exam scores and see in which ranges they predominantly fall.

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Key Concepts

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

Quantitative data display

Displaying quantitative data in a visual format makes it easier to identify patterns and trends. One effective method is using a stem-and-leaf plot, which organizes data based on place value. This type of display highlights the overall distribution of the data set, providing a clear picture of where most data points lie.
In a stem-and-leaf plot, each number is split into a "stem" and a "leaf." Often, the stem represents the higher value digits, while the leaf is the lower value digit. This strategy ensures every individual data point is visible, unlike histograms or bar graphs which might conceal exact data values.
For exam scores ranging from the 60s to 90s, employing repeated stems (e.g., "6L" and "6H") further enhances the clarity of the distribution. By refining the sorting into smaller intervals, we achieve a more detailed picture, making it easier to discern where scores cluster or sparse out.

Data distribution

Understanding data distribution is essential in statistical analysis, especially in identifying how values compare across a dataset. With the stem-and-leaf plot, we visualize data to observe natural groupings or clusters. This is crucial in spotting any anomalies or significant trends.
In the given exercise, using repeated stems within the stem-and-leaf plot allows us to see finer details of score distribution. Instead of simply noting scores are within the 60s or 70s range, we break it down to indicate if they're in the lower or higher end of these ranges. This refinement can highlight whether more students scored in the lower or upper halves of these segments.
Such insights help identify the median and mode more effectively, as you can quickly scan which stems carry the heaviest distribution of leaves. Thus, the stem-and-leaf plot serves as a unique representation reflective of the data's spread and central tendency.

Statistical data analysis

Statistical data analysis involves carefully evaluating datasets to infer meaningful conclusions, and the stem-and-leaf plot is a handy tool in this process. First, it enables a visual summary of the data, showing distributions and facilitating immediate comprehension of the dataset’s outline.
Furthermore, it helps in calculating various statistical measures. For instance, by observing the plot, one can quantify mean, median, and mode with greater accuracy. Since the stem-and-leaf plot preserves the original data points, recalculating statistics such as the range or three-quarter spread becomes more straightforward.
This graphical approach also aids in examining the shape of data distribution; whether it's symmetric, skewed, or shows any outliers. Detecting outliers can prompt further investigation into their causes. Overall, stem-and-leaf plots are a beneficial element in statistical data analysis, offering unobstructed insights into complex datasets.