Question

For each of 31 healthy dogs, a veterinarian measured the glucose concentration in the anterior chamber of the right eye and also in the blood serum. The following data are the anterior chamber glucose measurements, expressed as a percentage of the blood glucose. \(\begin{array}{rrrrrrrr}81 & 85 & 93 & 93 & 99 & 76 & 75 & 84 \\ 78 & 84 & 81 & 82 & 89 & 81 & 96 & 82 \\ 74 & 70 & 84 & 86 & 80 & 70 & 131 & 75 \\ 88 & 102 & 115 & 89 & 82 & 79 & 106 & \end{array}\) Construct a frequency distribution and display it as a table and as a histogram.

Short answer

Organize data in ascending order, determine range and class intervals, create a frequency distribution table, and then construct a histogram based on the frequency table.

Step-by-step solution

Organize the Data

First, arrange the glucose percentage measurements in ascending order to make it easier to determine the range and create intervals for the frequency distribution.

Determine the Range and Class Intervals

Calculate the range of the data by subtracting the smallest measurement from the largest measurement. Then, choose an appropriate number of class intervals based on the range and round them to convenient numbers for easy interpretation.

Create Frequency Distribution Table

Using the determined class intervals, tally the number of measurements that fall into each interval to create a frequency distribution table.

Construct the Histogram

Use the frequency distribution table to construct a histogram. Represent each class interval on the horizontal axis and the frequency of each interval on the vertical axis. Draw bars for each class interval with heights corresponding to the frequencies.

Key concepts

Understanding Histograms

A histogram is a graphical representation of the frequency distribution of numerical data. It is used to depict the distribution patterns of a dataset and is an essential tool for data analysis. By condensing large datasets into manageable visual forms, histograms help identify the shape and spread of the data, such as whether it is skewed or symmetrical, and where it tends to concentrate.

In our exercise, after organizing the dog glucose measurements from the lowest to the highest values, we can visualize this data using a histogram. Constructing a histogram involves plotting class intervals on the horizontal axis and the frequency of each interval on the vertical axis. Each measurement falls into a range—named a 'class interval' or 'bin'. The number of measurements within a given bin is represented by the height of a bar that spans the bin's range on the x-axis. In the veterinarian's dataset for dogs' glucose levels, a well-constructed histogram would allow us to see at a glance the most common ranges of glucose levels and any outliers such as the measurement of 131%.

Data Organization Strategies

Effective data organization is crucial for clear data analysis and interpretation. The step-by-step solution illustrates the importance of organizing data before delving into more complex representations like histograms. Starting by putting the glucose measurements in ascending order is more than just a preparatory step; it's a fundamental part of data organization that allows for easier identification of ranges, outliers, and overall patterns within the dataset.

Organizing data involves several techniques depending on the nature of the data and the desired outcome. For numerical data, as in our exercise, this could include sorting, removing duplicates, and identifying the range. These preliminary steps pave the way for constructing accurate and meaningful histograms. Additionally, proper organization aids in detecting any errors or anomalies in the data, which could significantly affect the results and conclusions drawn from the analysis.

Determining Class Intervals

Class intervals are the foundation of creating a histogram and organizing continuous data. A class interval, or class width, is a span of values within which a data point is categorized in a frequency distribution table. The choice of class intervals impacts how the histogram looks and the insights we can gain from it. Too wide an interval may hide important data variations, while too narrow may lead to a complex, cluttered histogram that's difficult to interpret.

In our worked-out exercise, the step of determining class intervals is critical. The largest glucose measurement (131%) and the smallest (70%) define the range, which is the difference between these two figures. Based on this range, the educator would decide on the number of intervals that make sense for representing the data clearly without losing meaningful insights. An essential tip for students is that class intervals are usually of equal size and should be chosen to have round numbers for ease of understanding. This step ensures that when the histogram is constructed, it can be easily read and interpreted by its intended audience.