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Chapter 1: Problem 12

The accompanying specific gravity values for various wood types used in construction appeared in the article "Bolted Connection Design Values Based on European Yield Model" (J. Struct. Engrg., 1993: 2169-2186): \(\begin{array}{lllllllll}.31 & .35 & .36 & .36 & .37 & .38 & .40 & .40 & .40 \\\ .41 & .41 & .42 & .42 & .42 & .42 & .42 & .43 & .44 \\ .45 & .46 & .46 & .47 & .48 & .48 & .48 & .51 & .54 \\ .54 & .55 & .58 & .62 & .66 & .66 & .67 & .68 & .75\end{array}\) Construct a stem-and-leaf display using repeated stems (see the previous exercise), and comment on any interesting features of the display.

Short Answer

Expert verified
Most values cluster around 0.42, with fewer extremes and a slight right skew.

Step by step solution

01

Organize the Data

Begin by listing the specific gravity values in ascending order: 0.31, 0.35, 0.36, 0.36, 0.37, 0.38, 0.40, 0.40, 0.40, 0.41, 0.41, 0.42, 0.42, 0.42, 0.42, 0.42, 0.43, 0.44, 0.45, 0.46, 0.46, 0.47, 0.48, 0.48, 0.48, 0.51, 0.54, 0.54, 0.55, 0.58, 0.62, 0.66, 0.66, 0.67, 0.68, 0.75. Notice that the values range from 0.31 to 0.75.
02

Choose an Appropriate Stem

For the data given, use stems that represent the tens place (0.3, 0.4, etc.). In this particular case, to handle data points that might cluster around certain stems, you can use repeated stems to better visualize the distribution. For instance, use two stems for each whole number interval (e.g., 3, 3* for 0.3x values).
03

Assign Leaves to Each Stem

Assign each data point to a stem. For example, the number 0.31 will belong to the '3' stem and will have a leaf of '1'. Using repeated stems: - '3': 1, 5, 6, 6, 7, 8 - '3*': - '4': 0, 0, 0, 1, 1 - '4*': 2, 2, 2, 2, 2, 3, 4, 5, 6, 6, 7, 8, 8, 8 - '5': 1, 4, 4, 5 - '5*': 8 - '6': 2 - '6*': 6, 6, 7, 8 - '7': 5 - '7*':
04

Construct the Stem-and-Leaf Display

Construct the stem-and-leaf plot for better visualization: 3 | 1 5 6 6 7 8 3* | 4 | 0 0 0 1 1 4* | 2 2 2 2 2 3 4 5 6 6 7 8 8 8 5 | 1 4 4 5 5* | 8 6 | 2 6* | 6 6 7 8 7 | 5 7* | Each row represents a range of specific gravity values, with '3' and '3*' representing all values between 0.3 and just under 0.4, and so on.
05

Analyze the Display

Examine the stem-and-leaf plot for patterns or interesting features: - Most values are clustered around 0.42, showing a high frequency of occurrences between 0.40 and 0.49. - There are fewer data points at the extreme ends, both lower (0.30-0.39) and higher (above 0.60). - The data is slightly right-skewed, indicating a majority of the wood has a lower specific gravity.

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Key Concepts

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

Specific Gravity
Specific gravity is a measure that compares the density of a substance to the density of a reference substance, usually water. It is a dimensionless quantity, which means it doesn't have any units. In the context of wood used in construction, specific gravity helps determine how dense and heavy a type of wood is in comparison to water.
Understanding specific gravity is crucial for several reasons:
  • **Material Selection:** It informs engineers and architects about the heaviness and strength of wood, influencing design decisions.
  • **Performance Prediction:** A higher specific gravity implies a denser wood, usually leading to better strength and durability.
  • **Cost Efficiency:** Choosing wood with the appropriate specific gravity can help balance material costs and performance needs.
Overall, specific gravity is a vital characteristic that affects the functionality and cost-effectiveness of construction materials.
Data Visualization
Data visualization refers to the graphical representation of data and information. A stem-and-leaf plot is a popular data visualization tool that organizes data to show its shape and distribution effectively.
Here鈥檚 why data visualization, like a stem-and-leaf plot, is beneficial:
  • **Simplification of Data:** It breaks down complex data sets to highlight key patterns and trends.
  • **Immediate Insights:** This type of plot displays detailed information in a compact form, making it easier to spot frequency distribution.
  • **Comparative Analysis:** It allows for quick comparisons within datasets, such as identifying clusters or gaps in specific gravity values as shown.
  • **Data Integrity:** Stem-and-leaf plots maintain original data values for more detailed analysis.
Through effective visualization, like with a stem-and-leaf plot, you can gain intuitive understanding of large volumes of data.
Statistical Analysis
Statistical analysis involves collecting, reviewing, and summarizing data to discover patterns or trends. The stem-and-leaf plot is a form of statistical analysis that offers insights into the distribution and central tendencies of data.
Here鈥檚 how statistical analysis using a stem-and-leaf plot is advantageous:
  • **Identifying Patterns:** It helps identify trends, such as clustering around a certain range or the presence of outliers.
  • **Frequency Estimation:** You can easily see how often each specific gravity value occurs, as seen with the cluster at 0.42.
  • **Shape Recognition:** The plot shows the data鈥檚 shape, highlighting aspects such as skewness, with our example slightly right-skewed.
  • **Central Tendency:** It becomes easy to estimate the median and mode directly from the plot.
Ultimately, statistical analysis helps convert data into meaningful information that can guide decision-making.
Data Interpretation
Data interpretation is the process of making sense out of data, using visualizations and analyses to extract meaningful insights. A stem-and-leaf plot provides a strong basis for interpretation in statistical contexts.
To interpret data effectively:
  • **Observe Clusters:** Recognize regions where data points group together, such as values between 0.40 to 0.49, indicating a common specific gravity range for the wood types.
  • **Identify Trends and Anomalies:** Spot irregular or rare values at the beginning or end of a dataset (e.g., fewer points above 0.60).
  • **Recognize Skewness:** Determine the data distribution鈥檚 skewness, which can indicate a tendency towards lighter or denser woods.
  • **Harness Contextual Information:** Combine your insights with external knowledge (like construction standards) for better understanding.
Data interpretation harnesses statistical results and turns them into actionable insights, especially in choosing the right materials for construction.

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Most popular questions from this chapter

The first four deviations from the mean in a sample of \(n=5\) reaction times were \(.3, .9,1.0\), and 1.3. What is the fifth deviation from the mean? Give a sample for which these are the five deviations from the mean.

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Many people who believe they may be suffering from the flu visit emergency rooms, where they are subjected to long waits and may expose others or themselves be exposed to various diseases. The article "Drive-Through Medicine: A Novel Proposal for the Rapid Evaluation of Patients During an Influenza Pandemic" (Ann. Emerg. Med., 2010: 268-273 described an experiment to see whether patients could be evaluated while remaining in their vehicles. The following total processing times (min) for a sample of 38 individuals were read from a graph that appeared in the cited article: \(\begin{array}{llllllll}9 & 16 & 16 & 17 & 19 & 20 & 20 & 20 \\ 23 & 23 & 23 & 23 & 24 & 24 & 24 & 24 \\ 25 & 25 & 26 & 26 & 27 & 27 & 28 & 28 \\ 29 & 29 & 29 & 30 & 32 & 33 & 33 & 34 \\ 37 & 43 & 44 & 46 & 48 & 53 & & \end{array}\) a. Calculate several different measures of center and compare them. b. Are there any outliers in this sample? Any extreme outliers? c. Construct a boxplot and comment on any interesting features.
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