Question

The two claws of the lobster (Homarus americanus) are identical in the juvenile stages. By adulthood, however, the two claws normally have differentiated into a stout claw called a "crusher" and a slender claw called a "cutter." In a study of the differentiation process, 26 juvenile animals were reared in smooth plastic trays and 18 were reared in trays containing oyster chips (which they could use to exercise their claws). Another 23 animals were reared in trays containing only one oyster chip. The claw configurations of all the animals as adults are summarized in the table. \({ }^{31}\) $$ \begin{array}{|lccc|} \hline&& {\text { Claw Configuration }} \\ \text { Treatment } & \begin{array}{c} \text { Right } \\ \text { crusher, } \\ \text { left cutter } \end{array} & \begin{array}{c} \text { Right } \\ \text { cutter, } \\ \text { left crusher } \end{array} & \begin{array}{c} \text { Right and } \\ \text { left cutter } \\ \text { (no crusher) } \end{array} \\ \hline \text { Oyster chips } & 8 & 9 & 1 \\ \text { Smooth plastic } & 2 & 4 & 20 \\ \text { One oyster chip } & 7 & 9 & 7 \\ \hline \end{array} $$ (a) Create a stacked frequency bar chart to display these data. (b) Create a stacked relative frequency bar chart to display these data. (c) Of the two charts you created in parts (a) and (b), which is more useful for comparing the claw configurations across the three treatments? Why?

Short answer

The stacked relative frequency bar chart is more useful for comparing claw configurations across treatments because it allows for direct comparisons of proportions rather than total numbers, which can be misleading if the sample sizes are different.

Step-by-step solution

Calculate Total Frequencies

Sum the frequency of each claw configuration for all treatments to obtain the total frequency for 'Right crusher, left cutter,' 'Right cutter, left crusher,' and 'Right and left cutter (no crusher).'

Create the Stacked Frequency Bar Chart

Draw a bar chart with 'Treatment' on the x-axis and 'Number of Lobsters' on the y-axis. Stack the frequencies of each claw configuration for each treatment on top of each other in different colors or shades to differentiate them. Label each section with the corresponding frequency.

Calculate Relative Frequencies

Divide the frequency of each configuration by the total number of lobsters in each treatment to obtain the relative frequency. Multiply by 100 if you want to convert it to a percentage.

Create the Stacked Relative Frequency Bar Chart

Draw a similar bar chart to the one in Step 2, but this time the y-axis will represent 'Relative Frequency (%)'. Stack the relative frequencies of each claw configuration for each treatment in the same way.

Compare the Charts

Examine both charts to determine which gives a clearer comparison of claw configurations across the three treatments. Consider how easily one can compare the proportion of each configuration within and between treatments in each chart type.

Key concepts

Stacked Frequency Bar Chart

When visualizing the differentiation of lobster claw configurations in a study, a stacked frequency bar chart is a beneficial tool. This type of chart is excellent for showcasing multiple data categories stacked upon one another within individual bars, allowing a cumulative total to be displayed. For instance, in the described lobster study involving different rearing conditions, one could create a bar for each treatment—oyster chips, smooth plastic, and one oyster chip—on the x-axis. The y-axis would indicate the number of lobsters. Within each bar, the frequencies of the observed claw configurations, such as 'Right crusher, left cutter', are stacked on top of each other. Different colors or shades represent the different configurations, making it easy to discern between them at a glance.

By using a stacked frequency bar chart, researchers and students can quickly understand the distribution and total counts of each claw configuration under each experimental condition. This chart type shows absolute numbers, allowing a straightforward assessment of the results within each category but does not directly allow for easy comparison of proportions between categories.

Relative Frequency Bar Chart

In comparison, a stacked relative frequency bar chart is highly useful when comparing proportions rather than absolute numbers. This chart type normalizes data by converting frequencies into relative terms, which is particularly useful for comparing categories of different sizes. In the lobster study, relative frequencies would be calculated by dividing the number of lobsters with each claw configuration by the total number of lobsters within each treatment. The results are often multiplied by 100 to get a percentage. These percentages would then be stacked in a bar chart similar to the stacked frequency bar chart.

With this approach, it becomes straightforward to compare the proportion of lobsters with each claw configuration across different treatments. A relative frequency bar chart answers questions like, 'What proportion of lobsters develop a right crusher when reared in smooth plastic?' or 'Are lobsters more likely to develop a left cutter when only one oyster chip is available?'. This visual representation is indispensable in studies that include different group sizes or when the focus is on the proportionate comparison of subgroups within the whole.

Statistical Data Analysis

Statistical data analysis is the backbone of interpreting results in studies like the lobster claw differentiation research. It involves collecting, summarizing, and drawing conclusions from data. In this study, researchers began by summarizing their results in a table, which served as the foundation for further analysis. They then moved on to visual tools, namely the stacked frequency and relative frequency bar charts, which aided in the visualization of complex data in a more digestible format.

Choosing the appropriate type of chart plays an important role in data analysis. While both charts provide valuable insights, the comparative usefulness depends on the research questions. For instance, if the study aims to compare raw counts of development outcomes across treatments, a stacked frequency bar chart is ideal. Conversely, if the goal is to compare proportions due to varying sample sizes or to focus on the prevalence of certain outcomes, a stacked relative frequency bar chart becomes more informative. Effective statistical data analysis can uncover trends, associations, or differences that mere observation may fail to reveal, thereby driving more accurate scientific conclusions.