Problem 33 - Algebra 1: Concepts and Skills

Question

The table shows the wing length (in millimeters) and the wing-beat rate (in beats per second) for five birds. $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { Bird } & {\text { Flamingo }} & {\text { Shellduck }} & {\text { Velvet Scoter }} & {\text { Fulmar }} & {\text { Great Egret }} \\ \hline \text { Wing length } & {400} & {375} & {281} & {321} & {437} \\ \hline \text { Wing-beat rate } & {24} & {3.0} & {43} & {3.6} & {21} \\ \hline \end{array} $$ Make a scatter plot that shows the wing-beat rates and wing lengths for the five birds. Use the horizontal axis to represent the wing-beat rate.

Step-by-Step Solution

Answer

The scatter plot will have the wing-beat rate on the horizontal axis, spanning from 0 to 50 (with each 10 units representing 10 beats per second), and the wing length on the vertical axis, spanning from 250 to 450 (with each 50 units representing 50 millimeters). Five different points representing the five birds - Flamingo, Shellduck, Velvet Scoter, Fulmar, and Great Egret - should be plotted on the graph, based on their corresponding wing-beat rates and wing lengths. Each point should be labelled with the corresponding bird's name.

Step 1: Determining the scale of the axes

First, identify the range of values for both wing-beat rate and wing length. For wing-beat rate, the values range from 3.0 to 43 (rounding up from 43), and for wing length, the values range from 281 to 437. Select a scale for both axes that will comfortably fit all data points. One possible choice could be to have the x-axis (wing-beat rate) go from 0 to 50, with 10 units representing 10 beats per second; and the y-axis (wing length) go from 250 to 450, with 50 units representing 50 millimeters.

Step 2: Plotting the points

For each bird, find the corresponding point on the graph. For example, the Flamingo has a wing-beat rate of 24 and a wing length of 400 mm. This would correspond to a point on the graph where x=24 and y=400.

Step 3: Labeling the points

Label each point on the graph with its corresponding bird's name. This would help in identifying which dot represents which bird on the scatter plot.

Step 4: Review the scatter plot

Review the scatterplot to ensure all points are correctly plotted and clearly labelled. Each bird should have its corresponding point on the graph, and no points should overlap.

Key Concepts

Data Visualization

Data visualization is a powerful method for interpreting and communicating data through graphic representation. In our example, we aim to present the relationship between the wing length and the wing-beat rate of five different birds. A scatter plot is an excellent choice for this task as it helps visualize the correlation between these two variables. By representing each bird as a unique point on the graph, we can easily compare their wing-beat rates and wing lengths visually.
To begin, it's essential to determine a scale that will allow all points to fit well within the axes. Choosing a clear and logical scale ensures that the data is represented accurately and legibly. The chosen scale should provide enough space for the display without distorting the data.

• A clear x-axis ranging from 0 to 50 will aptly show the wing-beat rates.
• The y-axis spanning from 250 to 450 provides a good range for wing lengths.

This visualization method offers a snapshot comparison of the varied physiological characteristics in birds, simplifying complex statistics into cleaner visual comparisons.

Coordinate Graphing

Coordinate graphing is the process of plotting data on a grid defined by two perpendicular axes. In a scatter plot, each piece of data is represented as a point on this graph according to its x (horizontal) and y (vertical) values. Our task is to plot the wing-beat rate on the horizontal axis and the wing length on the vertical axis.
To accurately graph the five birds:

• Locate the x-coordinate for each bird representing the wing-beat rate.
• Identify the y-coordinate for each bird indicating the wing length.

For example, plotting the Flamingo involves finding the point where x equals 24 and y equals 400. This places the Flamingo's point at the intersection of these coordinates. Repeating this process for each bird allows us to fill out the scatter plot. Once plotted, label each point with the corresponding bird name to track which data point belongs to which bird easily.

Statistical Representation

Statistical representation using a scatter plot provides a visual means to interpret the data. It highlights patterns, trends, and potential outliers which can inform further analysis or hypothesis. Our primary focus here with the bird's data is to observe possibly any correlation or lack thereof between wing-beat rate and wing length.
Key insights can quickly emerge from such a plot. For instance:

• Birds with longer wings might show a different wing-beat rate trend than those with shorter wings.
• Outliers such as the Shellduck, with a noticeably lower wing-beat rate, can prompt additional questions or investigations.

With all points labeled and correctly plotted, the scatter plot can serve as a basis for more advanced statistical analysis, such as linear regression, to determine any meaningful relationship or correlation between the tested features. Such analytical capacity is foundational for students and professionals in fields like ecology and biology.