Problem 47 - Algebra 1

Question

COMPUTER MODEMS In Exercises \(46-48\), use the following data which list the prices of several computer modems.\$ 230, \$ 220, \$ 170, \$ 215, \$ 190, \$ 200, \$ 200, \$ 150, \$ 170 Use a stem-and-leaf plot to order the data from least to greatest.

Step-by-Step Solution

Answer

A box-and-whisker plot for the data provides a visual representation of the data distribution, showing the variability and skewness effectively. The plot for this data would show the lowest point at \$150, the Q1 at \$170, the median at \$200, the Q3 at \$215, and the highest point at \$230.

Step 1: Preparation of the Data

First of all, the data should be organized in ascending order. As a result, it becomes: \$150, \$170, \$170, \$190, \$200, \$200, \$215, \$220, \$230.

Step 2: Identify the Five Components of the Box-and-Whisker Plot

It's necessary to define the minimum, lower quartile(Q1), median(Q2), upper quartile(Q3), and maximum. Minimum = \$150, Maximum = \$230. The median (Q2) separates the data into two halves, i.e., \$200 here (as there is an odd number of data). Further partition the data sets obtained into two more halves, and obtain Q1= \$170 and Q3= \$215.

Step 3: Creating the Box-and-Whisker Plot

Begin by drawing a number line that includes the range of the data. Then, upon this line, draw a rectangle to represent Q1 and Q3, and within that box, draw another line denoting the median (Q2). Finally, draw lines (the whiskers) from the box outward to the minimum and maximum.

Key Concepts

Data Organization

Data organization is a fundamental step in statistical analysis, and it forms the basis for creating effective visual representations such as box-and-whisker plots. Organizing data allows students and researchers to get an initial overview and easily identify patterns, trends, or outliers.

For instance, in our modem pricing example, organizing the prices in ascending order is the crucial first step. This step not merely simplifies the visual apprehension of the range of values but also sets the stage for the calculation of various statistics, including quartiles. Organized data paves the way for efficient computation and a clearer understanding of the dataset's distribution.

Benefits of Data Organization

• Accessibility: When data is neatly organized, it becomes easier to access and manipulate for different purposes.
• Error Minimization: A well-organized dataset reduces the chance of errors during analysis.
• Time Efficiency: It saves time during the analysis as the structure is predesigned for ease of computation.

Applying this approach to our modem pricing exercise, once the data is organized, the next steps, like computing quartiles and creating statistical plots, become much more manageable.

Quartiles

Quartiles are values that divide a data set into quarters when it's organized in ascending order. They are critical indicators of data dispersion and provide deeper insights into the dataset's composition by showing how data points are spread around the median.

In our modem pricing example, identifying the quartiles helps in understanding the distribution of modem prices across the sampled range. There are three quartiles usually denoted as Q1 (the first quartile), Q2 (the second quartile or the median), and Q3 (the third quartile).

Significance of Each Quartile

• Q1: Represents the median of the lower half of the data set, indicating that 25% of the prices are below this value.
• Q2/Median: Divides the dataset in half and shows the central tendency of the dataset.
• Q3: Represents the median of the upper half of the data set, meaning that 75% of the prices are below this value.

These quartiles are used to construct the box part of the box-and-whisker plot, which provides a visual summary of the dataset's distribution and helps in identifying the concentration of data points and potential outliers.

Statistical Plots

Statistical plots are visual representations of data that help to understand and interpret the data in a more intuitive way. Among various types of statistical plots, the box-and-whisker plot, or box plot, is particularly effective in displaying the distribution of data based on a five-number summary: minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum.

In the exercise involving modem prices, the box-and-whisker plot enables a clear visual comparison of different quartiles and the range of data values. It highlights where most of the data lies and quickly reveals outliers. To create this plot, a rectangle (the 'box') is drawn from Q1 to Q3 with a line inside indicating the median (Q2). 'Whiskers' extend from the box to the minimum and maximum values, thus encapsulating the total spread of the data.

Advantages of Using Box-and-Whisker Plots

• Summarization: They summarize a large amount of data in a visual snapshot.
• Comparison: Multiple box plots can be used side by side to compare different data sets.
• Outlier Spotting: They are efficient in displaying outliers and understanding data variability.

When students learn to interpret these plots properly, they become powerful tools in both descriptive statistics and exploratory data analysis.