91Ó°ÊÓ

The article "Determination of Most Representative Subdivision" (J. of Energy Engr., 1993: 43-55) gave data on various characteristics of subdivisions that could be used in deciding whether to provide electrical power using overhead lines or underground lines. Here are the values of the variable \(x=\) total length of streets within a subdivision: \(\begin{array}{rrrrrrr}1280 & 5320 & 4390 & 2100 & 1240 & 3060 & 4770 \\ 1050 & 360 & 3330 & 3380 & 340 & 1000 & 960 \\ 1320 & 530 & 3350 & 540 & 3870 & 1250 & 2400 \\ 960 & 1120 & 2120 & 450 & 2250 & 2320 & 2400 \\ 3150 & 5700 & 5220 & 500 & 1850 & 2460 & 5850 \\ 2700 & 2730 & 1670 & 100 & 5770 & 3150 & 1890 \\ 510 & 240 & 396 & 1419 & 2109 & & \end{array}\) a. Construct a stem-and-leaf display using the thousands digit as the stem and the hundreds digit as the leaf, and comment on the various features of the display. b. Construct a histogram using class boundaries 0,1000 , \(2000,3000,4000,5000\), and 6000 . What proportion of subdivisions have total length less than 2000 ? Between 2000 and 4000 ? How would you describe the shape of the histogram?

Step by step solution

01

Stem-and-Leaf Display Construction

To construct a stem-and-leaf display, we first identify the stem as the thousands digit and the leaf as the hundreds digit of each value. For example, the number 1280 would have a stem of 1 and a leaf of 2. After listing all values, group them based on their stems and record the corresponding leaves in each group.

02

Note Features of the Stem-and-Leaf Display

Examine the stem-and-leaf display for data patterns such as clustering of values, gaps, and outliers. Note any interesting features such as whether most values are concentrated within certain stems or spread uniformly.

03

Histogram Construction

Create a histogram with bins defined by the class boundaries 0-1000, 1000-2000, 2000-3000, 3000-4000, 4000-5000, and 5000-6000. Count the number of values (frequency) in each bin by checking which total street lengths fall within the defined boundaries.

04

Proportion Calculation

Calculate the proportion of subdivisions in specified ranges. For less than 2000, add the number of values in the first two bins (less than 2000) and divide by the total number of subdivisions. For 2000 to 4000, count the frequency in the bins 2000-3000 and 3000-4000 and divide by the total number of subdivisions.

05

Histogram Shape Analysis

Analyze the shape of the histogram by looking for patterns. Determine whether the histogram is skewed left, skewed right, or symmetric. Count the peaks to check if the distribution is unimodal or multimodal.

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Key Concepts

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

Histogram Construction

Creating a histogram is a practical way to visually represent the distribution of numerical data. To construct a histogram, you need to divide the data into several categories known as bins or class intervals. Each bin represents a range of data values, and the height of the bar indicates how many values fall into each range.

For example, if we have class boundaries such as 0-1000, 1000-2000, and so on, we count how many data points, like the total street lengths, fall into each interval. As a result, we get a graphical display where:

• Each bar represents the frequency of data points within the particular class interval.
• The width of each bar is consistent (each represents the same range of values).
• The height of the bar corresponds to the number of entries within that range.

Building a histogram allows you to get a quick visual overview of where most of the data values fall and helps identify the spread and shape of the data distribution.

Data Visualization in Statistics

Data visualization in statistics is essential to understand the underlying patterns within datasets. It involves using graphical representations such as histograms, pie charts, line graphs, and stem-and-leaf displays to convey complex data insights in an easily digestible format.

The goal is to make data intuitive, allowing viewers to grasp trends, outliers, and distributions at a glance. A well-constructed visual can help:

• Identify clusters, gaps, and potential anomalies in the data.
• Provide insights into the data's shape and variability.
• Facilitate decision-making based on visual trends rather than just numerical data.

When you see a histogram, for example, you can quickly determine which ranges have more data points. This instant recognition makes data visualization a powerful tool in statistics and analysis. Moreover, examining visuals can lead to better communication of results and findings with others who might not have a statistical background.

Proportion Calculation

Proportion calculation is a fundamental concept in statistics that measures the relative size of a part compared to the whole. To calculate a proportion, you divide the part by the total. This method is particularly useful when analyzing categories or ranges within data.

For example, if we want to calculate the proportion of subdivisions with street lengths less than 2000, we would first count how many data points fall into this category. Suppose there are 15 subdivisions with such lengths, and the total number of subdivisions is 50. The proportion is calculated as follows:\[\text{Proportion} = \frac{15}{50} = 0.3\]This result shows that 30% of the subdivisions have a street length of less than 2000.

Proportion gives a clearer perspective than just frequencies, particularly when datasets are large or when comparisons are needed between different ranges or categories.