Problem 8 - Basic Engineering Mathematics

Question

The data given below refer to the gain of each of a batch of 40 transistors, expressed correct to the nearest whole number. Form a frequency distribution for these data having seven classes. \(\begin{array}{llllllll}81 & 83 & 87 & 74 & 76 & 89 & 82 & 84 \\ 86 & 76 & 77 & 71 & 86 & 85 & 87 & 88 \\ 84 & 81 & 80 & 81 & 73 & 89 & 82 & 79 \\ 81 & 79 & 78 & 80 & 85 & 77 & 84 & 78 \\ 83 & 79 & 80 & 83 & 82 & 79 & 80 & 77\end{array}\)

Step-by-Step Solution

Answer

The frequency distribution has intervals with frequencies: 2, 3, 7, 9, 8, 5, 1.

Step 1: Determine the Range

To begin classifying the data, identify the range by finding the difference between the highest and lowest data points. The maximum value is 89, and the minimum is 71, so the range is \(89 - 71 = 18\).

Step 2: Decide the Number of Classes

The problem states that we need to have seven classes. Hence, we will use this number to determine the class width. The classes will help categorize the data neatly into intervals.

Step 3: Calculate Class Width

Divide the range by the number of classes to determine the class width. \[ \text{Class width} = \frac{18}{7} \approx 2.57 \] Since class width needs to be a whole number, round up to 3 to ensure all data points are covered.

Step 4: Create Class Intervals

Start from the minimum value and add the class width to create seven class intervals. The intervals will be: 1. 71-73 2. 74-76 3. 77-79 4. 80-82 5. 83-85 6. 86-88 7. 89-91 (no data in this range)

Step 5: Tally the Frequencies

Count how many data points fall into each class interval:- 71-73: \(2\)- 74-76: \(3\)- 77-79: \(7\)- 80-82: \(9\)- 83-85: \(8\)- 86-88: \(5\)- 89-91: \(1\)

Step 6: Present the Frequency Distribution

Use the tallied frequencies to present the data as a frequency distribution: - 71-73: 2 - 74-76: 3 - 77-79: 7 - 80-82: 9 - 83-85: 8 - 86-88: 5 - 89-91: 1

Key Concepts

Understanding Class Intervals

When organizing data into a frequency distribution, defining class intervals is crucial. Class intervals allow us to group the data points into smaller categories, making analysis easier and more intuitive. In practical terms, a class interval is a range of values within which data points fall. These intervals should be mutually exclusive, meaning no data point can belong to more than one class interval. They should also be continuous without any gaps. For instance, we might have intervals like 71-73, 74-76, and 77-79, covering all possibilities between the smallest and largest numbers in a dataset.

When deciding on class intervals, consider the extent of the data. Begin by calculating the range (the difference between the maximum and minimum values), which helps determine how wide each interval should be. The key is to find a balance between having enough intervals to highlight interesting features in the data, without having too many which may complicate the analysis.

Remember, once intervals are chosen, they guide the classification of the rest of the data in your dataset.

Exploring Data Classification

Data classification is the process of organizing data into categories or classes. In our case, using class intervals helps to classify, or group, the data efficiently. When we set up a frequency distribution, we aim to summarize the data so that patterns and trends become obvious.

Classification can simplify complex data and make large datasets more comprehensible. There are several advantages to classifying data:

• It makes it easier to visualize data as categories condense the information.
• It facilitates comparison between different groups, assisting in spotting similarities and differences.
• It highlights the distribution and frequency of data within the dataset, helping to identify dominant categories or outliers.

It's vital to define appropriate classes using consistent criteria, like numerical ranges, to ensure the data is grouped logically and meaningfully. By doing so, frequency and patterns across the data become much clearer and make the results valuable for interpretation.

Creating and Interpreting a Histogram

A histogram is a visual representation of data distribution using bars of different heights. It is particularly useful in representing frequency distributions, helping us recognize patterns at a glance. Once class intervals and frequencies have been established, each interval can correspond to a bar in the histogram.

To create a histogram:

• Draw axes on your graph. The horizontal (x-axis) will represent class intervals, while the vertical axis (y-axis) represents frequency.
• For each class interval, draw a bar with height corresponding to the interval's frequency.
• Ensure no gaps between bars, as the data represented is continuous.

Histograms can reveal central trends, variability, shape of the data, and any unusual observations. They make it visually straightforward to see where data is most concentrated and which intervals have the highest or lowest occurrences.

By examining a histogram, one gains insights into the dataset's distribution, such as whether data clusters around certain values or spreads uniformly. This information can influence decision-making and hypothesis testing, making histograms a powerful tool in data analysis.