Problem 12 - Basic Engineering Mathematics

Question

The masses of 50 ingots in kilograms are measured correct to the nearest \(0.1 \mathrm{~kg}\) and the results are as shown below. Produce a frequency distribution having about 7 classes for these data and then present the grouped data as (a) a frequency polygon and (b) a histogram. \(\begin{array}{llllllllll}8.0 & 8.6 & 8.2 & 7.5 & 8.0 & 9.1 & 8.5 & 7.6 & 8.2 & 7.8 \\ 8.3 & 7.1 & 8.1 & 8.3 & 8.7 & 7.8 & 8.7 & 8.5 & 8.4 & 8.5 \\ 7.7 & 8.4 & 7.9 & 8.8 & 7.2 & 8.1 & 7.8 & 8.2 & 7.7 & 7.5 \\ 8.1 & 7.4 & 8.8 & 8.0 & 8.4 & 8.5 & 8.1 & 7.3 & 9.0 & 8.6 \\ 7.4 & 8.2 & 8.4 & 7.7 & 8.3 & 8.2 & 7.9 & 8.5 & 7.9 & 8.0\end{array}\)

Step-by-Step Solution

Answer

Create a frequency distribution with intervals of 0.3 kg, then plot a frequency polygon and histogram for visualization.

Step 1: Sort the Data

Sort the given data in ascending order. This helps in identifying the range and determining class intervals more easily. Sorted data: 7.1, 7.2, 7.3, 7.4, 7.4, 7.5, 7.5, 7.6, 7.7, 7.7, 7.7, 7.8, 7.8, 7.8, 7.9, 7.9, 7.9, 8.0, 8.0, 8.0, 8.0, 8.1, 8.1, 8.1, 8.1, 8.2, 8.2, 8.2, 8.2, 8.2, 8.3, 8.3, 8.3, 8.3, 8.4, 8.4, 8.4, 8.4, 8.5, 8.5, 8.5, 8.5, 8.6, 8.6, 8.7, 8.7, 8.8, 8.8, 9.0, 9.1.

Step 2: Determine the Range

Calculate the range of the data to determine the class intervals for the frequency distribution. The range is the difference between the maximum and minimum values. Range = 9.1 - 7.1 = 2 kg.

Step 3: Determine Class Intervals

Divide the range into approximately 7 equal intervals. A convenient choice might be class intervals of 0.3 kg each. Class intervals: 7.1-7.4, 7.4-7.7, 7.7-8.0, 8.0-8.3, 8.3-8.6, 8.6-8.9, 8.9-9.2.

Step 4: Create Frequency Distribution

Count the number of data entries falling within each class interval based on the sorted data. Frequency distribution: - 7.1-7.4: 5 - 7.4-7.7: 11 - 7.7-8.0: 9 - 8.0-8.3: 12 - 8.3-8.6: 8 - 8.6-8.9: 4 - 8.9-9.2: 1.

Step 5: Draw Frequency Polygon

To draw a frequency polygon, plot the midpoints of each class interval on the x-axis and their corresponding frequency on the y-axis. Connect the plotted points with straight lines. Midpoints: - 7.25 - 7.55 - 7.85 - 8.15 - 8.45 - 8.75 - 9.05

Step 6: Draw Histogram

Create a histogram by drawing adjacent rectangles where the base of each rectangle is the class interval, and the height is the frequency of the class. This visualizes the frequency distribution of the data.

Key Concepts

Histogram

A histogram is a type of bar graph that visually represents the frequency distribution of a dataset. In a histogram, the x-axis represents the class intervals, while the y-axis shows the frequency of data points within those intervals. Each bar in the histogram corresponds to a class interval, and the height of the bar represents the frequency of that interval.
Histograms are valuable tools for understanding the distribution of data. They allow us to see patterns, such as which ranges have the most data and how data is spread across various intervals. This can be incredibly helpful for tasks like identifying the skewness or modality of the data.
The key characteristics of a histogram include:

• Bars are adjacent, symbolizing continuous data.
• No gaps exist between bars except when a frequency is zero.
• The width of the bars is uniform if the class intervals are equal.

Use histograms to get a clear and immediate sense of the underlying frequency distribution and to summarize large data sets in a condensed visual form.

Frequency Polygon

A frequency polygon is another graphical representation of data which shows the overall shape of a frequency distribution. It is drawn by using a line graph that connects the midpoints of each class interval with straight lines. These midpoints are plotted against the frequency of each class.
Creating a frequency polygon involves:

• Identifying and plotting the midpoint of each class interval on the x-axis.
• Plotting the frequency of each class as the y-coordinate.
• Connecting these points with straight lines.

In many cases, frequency polygons are drawn on the same axes as histograms for comparison. It highlights trends over the range of the data and can be particularly useful for comparing two distributions on the same graph.

Class Interval

Class intervals are specific ranges that divide the whole data into separate blocks. They help us organize the data and prepare for creating frequency distributions. In a dataset, when data values are given like measurements (such as weights or time), these are sorted into class intervals to simplify analysis.
To create meaningful class intervals, consider the following:

• The number of class intervals (usually between 5 and 15).
• The range of the data divided by the number of intervals gives you an approximate width for each class.
• A uniform class width makes comparing frequencies easy.

Once defined, class intervals make it easier to see how the data set is distributed across these ranges and facilitate the calculation of the frequency distribution.

Data Sorting

Sorting data is a fundamental preliminary step when working with large datasets in statistical analysis. It involves arranging data values in a specific order, often in ascending or descending order, depending on the analysis objectives.
Sorting is particularly useful for:

• Identifying patterns and trends.
• Easily calculating the range for class intervals.
• Facilitating the process of creating other statistical graphs like histograms or frequency polygons.

Taking the time to sort your data correctly can greatly simplify subsequent analysis and ensure accurate results when calculating statistics like the median or when divided into class intervals.