Problem 10 - Basic Engineering Mathematics

Question

The amount of money earned weekly by 40 people working part-time in a factory, correct to the nearest \(£ 10\), is shown below. Form a frequency distribution having 6 classes for these data. \(\begin{array}{rrrrrrrr}80 & 90 & 70 & 110 & 90 & 160 & 110 & 80 \\ 140 & 30 & 90 & 50 & 100 & 110 & 60 & 100 \\ 80 & 90 & 110 & 80 & 100 & 90 & 120 & 70 \\\ 130 & 170 & 80 & 120 & 100 & 110 & 40 & 110 \\ 50 & 100 & 110 & 90 & 100 & 70 & 110 & 80\end{array}\)

Step-by-Step Solution

Answer

The frequency distribution table has been successfully created by classifying data into 6 classes based on calculated class intervals.

Step 1: Find the Range

First, identify the smallest and largest values in the data set. These are £30 and £170, respectively. The range is the difference between the largest and smallest values: \[ \text{Range} = 170 - 30 = 140 \]

Step 2: Determine Class Width

Divide the range by the number of classes (6) to approximate the class width. Then, round up if necessary to ensure coverage of all data:\[ \text{Class Width} = \frac{140}{6} \approx 23.33 \] Rounding up gives a class width of 24.

Step 3: Build Class Intervals

Using the class width, create 6 intervals that cover the entire range: 1. 30-53 2. 54-77 3. 78-101 4. 102-125 5. 126-149 6. 150-173 Each class interval covers 24 units, starting from the smallest data value.

Step 4: Tally Data into Classes

Go through the data and tally how many data points fall into each class interval: 1. 30-53: 4 values 2. 54-77: 3 values 3. 78-101: 14 values 4. 102-125: 11 values 5. 126-149: 3 values 6. 150-173: 5 values

Step 5: Create Frequency Distribution

Compile the tallies into a frequency distribution table:\[\begin{array}{|c|c|}\hline\text{Class Interval} & \text{Frequency} \\hline30-53 & 4 \54-77 & 3 \78-101 & 14 \102-125 & 11 \126-149 & 3 \150-173 & 5 \\hline\end{array}\]

Key Concepts

Class Interval

When creating a frequency distribution, a key step is to establish class intervals. Class intervals are specific ranges within your dataset that help to organize data into meaningful groups. For instance, in a dataset, if the lowest value is £30 and the highest is £170, you'd want classes to cover this entire range. In our example, we decided on 6 class intervals, each encapsulating a segment of this range.
It's crucial to ensure that each data point can fit into one of these intervals. This way, we avoid gaps between intervals which could lead to misrepresenting the dataset. A correctly formed set of class intervals makes it easier to understand, analyze, and visualize the data distribution.

Class Width

The class width is essentially the size of each class interval in your dataset. It tells you how broad each interval should be. To calculate the class width, start by determining the range of your dataset, which is the difference between the largest and smallest data values. Then, divide the range by the number of classes you want (in our case, 6 classes).
For example, with a range of 140 (from £30 to £170), dividing by 6 gives approximately 23.33. It's often useful to round up, ensuring that all data is included. Therefore, our class width became 24.
A well-chosen class width contributes to clear and actionable data insights because it ensures that all classes neatly cover the entire data spectrum. Avoid choosing very wide or very narrow classes to keep the data organized and easy to interpret.

Tallying Data

Tallying involves counting how many data points fall into each class interval. It's a crucial step to transform raw data into a structured frequency distribution. As you go through each data value, you mark a tally in the corresponding class interval.
For instance, with data between £30 and £53, you would tally every time a new data point fits into this range. After tallying all your data, you can easily see the concentration and spread of your dataset across different intervals.
This tallying technique helps you visualize data trends and patterns before you formally compile them into a frequency distribution table. Doing so ensures an organized, meaningful layout of the dataset for further analysis or presentation.

Range Calculation

The range is a basic yet powerful statistical measure of distribution spread. To calculate the range, subtract the smallest data value from the largest one. In our dataset, the smallest value is £30, and the largest is £170, resulting in a range of 140.
This range tells you how widely the data varies, giving you a sense of the spread and boundaries of your data. It's vital for determining an appropriate class width and setting up meaningful class intervals. A broader range could demand more class intervals or a wider class width, while a narrower range might result in fewer or smaller classes.
Understanding your data's range allows you to better organize and present the dataset, facilitating easier analysis and interpretation.