Question

Trypanosomes are parasites that cause disease in humans and animals. In an early study of trypanosome morphology, researchers measured the lengths of 500 individual trypanosomes taken from the blood of a rat. The results are summarized in the accompanying frequency distribution. \({ }^{18}\) $$\begin{array}{|cccc|}\hline \begin{array}{c}\text { Length } \\\\(\mu \mathrm{m})\end{array} & \begin{array}{c}\text { Frequency } \\\\\text { (number of } \\\\\text { individuals) }\end{array} &\begin{array}{c}\text { Length } \\\\(\mu \mathrm{m})\end{array} & \begin{array}{c}\text { Frequency } \\\\\text { (number of } \\\\\text { individuals) }\end{array} \\\\\hline 15 & 1 & 27 & 36 \\\16 & 3&28 & 41 \\\17 & 21 & 29 & 48 \\\18 & 27 & 30 & 28 \\\19 & 23 & 31 & 43 \\\20 & 15 & 32 & 27 \\\21 & 10 & 33 & 23 \\\22 & 15 & 34 & 10 \\\23 & 19 & 35 & 4 \\\24 & 21 & 36 & 5 \\\25 & 34&37 & 1 \\\26 & 44 & 38 & 1 \\\\\hline\end{array}$$ (a) Construct a histogram of the data using 24 classes (i.e., one class for each integer length, from 15 to 38 ). (b) What feature of the histogram suggests the interpretation that the 500 individuals are a mixture of two distinct types? (c) Construct a histogram of the data using only 6 classes. Discuss how this histogram gives a qualitatively different impression than the histogram from part (a).

Short answer

The 24-class histogram will have narrow bars showing the frequency for each length from 15 to 38, which may reveal a bimodal distribution indicating two types of trypanosomes. The 6-class histogram will have wider bars and fewer details, sometimes hiding the bimodal distribution and providing a less specific impression of the data.

Step-by-step solution

Constructing a 24-Class Histogram

To construct a histogram with 24 classes, create 24 bins on the horizontal axis representing each integer length from 15 to 38. Then, plot a bar for each bin where the height represents the frequency of individuals with that length.

Analyzing the Histogram for Distinct Types

Observe the shape of the histogram. The presence of two or more peaks in the histogram suggests that the sample may contain two distinct types of trypanosomes. These peaks represent the most common lengths within each type.

Constructing a 6-Class Histogram

To create a histogram with 6 classes, combine the lengths into fewer, broader categories. For instance, you might have class intervals such as 15-19, 20-24, 25-29, 30-34, and 35-38. Calculate the frequency for each class by adding the individual frequencies within each range and drawing bars corresponding to these new frequencies.

Comparing Histograms

Analyze how the two histograms give different impressions. The histogram with fewer classes will have broader and fewer peaks and might smooth out the differences between two distinct types, masking the bimodal distribution seen in the 24-class histogram.

Key concepts

Frequency Distribution

When researchers study characteristics like the lengths of trypanosomes, they use a frequency distribution to summarize the data collected. A frequency distribution is a way to show how many times each value in a set of data occurs. More technically, it's a summary of how the counts (frequencies) are distributed across the possible values or intervals.

For instance, in the trypanosome study, the length of each parasite is measured and recorded. To create a frequency distribution, the lengths are grouped into intervals, and the number of trypanosomes within each length interval is counted. This count becomes the 'frequency.' From this distribution, researchers can infer whether certain lengths are more common than others and possibly raise questions about why those variations exist.

Creating a precise frequency distribution is essential for subsequent analyses, such as constructing histograms. It provides a clear picture of the data's structure at a glance, allowing for more informed interpretations of biological and statistical significance.

Histogram Construction

A histogram is a type of graph that represents frequency distribution data visually. It is constructed by plotting rectangular bars to show the frequency of data within successive numerical intervals, also known as bins or classes.

To build a histogram, the range of data (e.g., trypanosome lengths) must first be divided into bins, with each bin representing an interval of lengths. The number of intervals can vary, but in the trypanosome study, the researchers were directed to create an informative representation with 24 classes. Each bar's height reflects the frequency of trypanosomes within the specific length interval that the bin represents.

The process of constructing the histogram provides insights into the distribution's overall shape and spread, which are key to understanding patterns. For example, if most of the trypanosomes have similar lengths, the bars will be higher in the central area of the graph, indicating a concentration of values around a central value.

Bimodal Distribution

In the context of the trypanosome length data, one particular feature that can be observed from a well-constructed histogram is a bimodal distribution. A bimodal distribution occurs when two distinct peaks, or modes, are present in the histogram. This suggests that the dataset essentially contains two overlapping distributions, indicative of two different types or groups within the sample.

If a histogram of trypanosome lengths shows two peaks, it could be that there are two different populations of trypanosomes with characteristic lengths that differ. Such a finding could have important implications for biological understanding and medical treatment strategizing. However, the clarity of these peaks can change with the number of classes chosen for the histogram. A histogram with too few classes might blend the peaks together, making it difficult to discern the bimodal nature of the data. The selection of bin size and number is, therefore, crucial in preserving the fidelity of the data's story.