Question

A paleontologist measured the width (in \(\mathrm{mm}\) ) of the last upper molar in 36 specimens of the extinct mammal Acropithecus rigidus. The results were as follows: \({ }^{12}\) \(\begin{array}{llllllll}6.1 & 5.7 & 6.0 & 6.5 & 6.0 & 5.7 \\ 6.1 & 5.8 & 5.9 & 6.1 & 6.2 & 6.0 \\ 6.3 & 6.2 & 6.1 & 6.2 & 6.0 & 5.7 \\ 6.2 & 5.8 & 5.7 & 6.3 & 6.2 & 5.7 \\ 6.2 & 6.1 & 5.9 & 6.5 & 5.4 & 6.7 \\ 5.9 & 6.1 & 5.9 & 5.9 & 6.1 & 6.1\end{array}\) (a) Construct a frequency distribution and display it as a table and as a histogram. (b) Describe the shape of the distribution.

Short answer

Frequency distribution: 5.4-5.7 (4), 5.7-6.0 (9), 6.0-6.3 (14), 6.3-6.6 (4), 6.6-6.9 (1). Histogram: Single peak with possible slight skewness. Description: Possibly symmetrical or slightly skewed, unimodal shape with no outliers.

Step-by-step solution

Organize the Data

First, sort the given data in ascending order. This will help in determining the range of values and in calculating the frequency distribution. The sorted data becomes: 5.4, 5.7, 5.7, 5.7, 5.7, 5.7, 5.8, 5.8, 5.9, 5.9, 5.9, 5.9, 5.9, 6.0, 6.0, 6.0, 6.0, 6.1, 6.1, 6.1, 6.1, 6.1, 6.1, 6.1, 6.2, 6.2, 6.2, 6.2, 6.2, 6.3, 6.3, 6.5, 6.5, 6.7.

Determine the Range and Class Intervals

Find the range of the data by subtracting the smallest measurement from the largest measurement: Range = 6.7 - 5.4 = 1.3 mm. Next, decide on a number of classes to use (common choices are between 5 and 20). For this example, we'll use 5 classes. Divide the range by the number of classes to find the class interval: Class interval = 1.3 mm / 5 classes = approximately 0.26 mm. It is usual to round this to a convenient number, so we'll use 0.3 mm.

Create the Frequency Distribution Table

Now, construct the frequency distribution table using the class intervals. Tally the data points that fall into each interval and count the frequencies: \(5.4-5.7\) (4 items), \(5.7-6.0\) (9 items), \(6.0-6.3\) (14 items), \(6.3-6.6\) (4 items), and \(6.6-6.9\) (1 item).

Draw the Histogram

The histogram will have the class intervals on the horizontal axis and the frequencies on the vertical axis. For each class interval, draw a bar up to the frequency number. Each bar's height corresponds to the frequency of the class interval.

Describe the Shape

To describe the shape of a distribution, you can look at the histogram and note features like symmetry/asymmetry, modality (i.e., unimodal, bimodal), and the presence of outliers. For this dataset, the histogram might show a unimodal distribution (a single peak), and depending on the exact heights of the bars, could be somewhat symmetrical or slightly skewed.

Key concepts

Histogram

A histogram is a graphical representation of statistical data that displays the frequency of a dataset and is an essential tool in data analysis. It consists of rectangular bars, where the length of each bar is proportional to the number of cases (frequency) in each category or class interval. To create a histogram, the data needs to be divided into contiguous intervals known as 'bins'.

In the given exercise, a paleontologist studies the extinct mammal Acropithecus rigidus, measuring the width of its last upper molar. The histogram serves as a visual aid to better understand the distribution of molar widths measured. For instance, as per steps 3 and 4 from the textbook solution, after organizing the data into a frequency distribution table, the student would draw a histogram with class intervals on the x-axis and frequencies on the y-axis. This would result in a simpler and more digestible form of the dataset, making patterns such as central tendency and variability visible at a glance.

For the educational purpose, emphasizing that a histogram, unlike a bar chart, deals with continuous data, and the bars are touching each other is essential. This visualization helps educators and students alike to observe the spread and shape of the continuous data quickly.

Statistical Data Organization

Organizing statistical data is a preliminary yet critical step in the process of data analysis. Properly organized data ensure that it can be analyzed meaningfully for insights and trends. The organization involves gathering, sorting, and summarizing the data before it can be presented.

In the step-by-step solution provided for the paleontologist's research data, the first step is to sort the measure of the molars in ascending order. This process is essential to identify the range and make decisions concerning class intervals for the frequency distribution table. The clear step of organizing can significantly aid students in later steps such as calculating measures of central tendency (such as mean, median, mode) and measures of dispersion (like range, variance, or standard deviation).

It is often beneficial to explain to students the importance of the range and class intervals in the context of the gathered data and the consequences of different choices in these parameters to the final histogram and data analysis. Providing examples and practice exercises could enhance understanding and engagement with the topic.

Distribution Shape Analysis

Analyzing the shape of a distribution is critical in statistics as it gives a general idea about the pattern of data and potential anomalies. When looking at the histogram, several features can be assessed such as symmetry vs. asymmetry, peaks (unimodal, bimodal, etc.), and whether there are any outliers or gaps in the data.

For example, a unimodal distribution has a single peak suggesting most data cluster around a central value, whereas a bimodal distribution would suggest two frequent values. An asymmetrical distribution would indicate skewed data, meaning there are a significant number of outliers on one side of the distribution. This could have implications on the choice of statistical methods for further analysis.

In the solution's final step, describing the shape of the frequency distribution derived from the molar widths of Acropithecus rigidus helps the paleontologist understand biological variations within the species. A histogram could indicate whether the widths are symmetrically distributed around a mean value or if there's a skewness indicating an uneven spread. Students should be encouraged to look beyond the numbers and develop interpretations of the real-world implications of these shapes. Recognizing these patterns is crucial for making informed conclusions and for further statistical calculations.