Frequency distributions are essential tools for organizing data into classes, allowing for easier analysis of the frequency of occurrences within specified ranges. When constructing a frequency distribution, particularly when only the number of classes is provided, a systematic approach is necessary to determine the class limits and widths.

The first step in this process is to calculate the class width. The class width is defined as the difference between two consecutive lower or upper class limits. To find the class width, you can use the formula:

Class Width = \(\frac{\text{Maximum} - \text{Minimum}}{\text{Number of Classes}}\)

For example, if the maximum value in your dataset is 115 and the minimum is 5, and you need to create 8 classes, the calculation would be:

Class Width = \(\frac{115 - 5}{8} = \frac{110}{8} = 13.75\)

Since class widths are typically rounded to whole numbers or convenient values, you might choose to round 13.75 up to 14 or 15, depending on the context. In this case, using 15 as the class width simplifies calculations.

Next, you will determine the lower class limits. The first lower class limit is often set at or below the minimum value of the dataset. In this example, you can use 5 as the first lower limit. Subsequent lower class limits are found by adding the class width to the previous lower limit:

Lower Class Limits: 5, 20, 35, 50, 65, 80, 95, 110

After establishing the lower class limits, the upper class limits can be calculated. The first upper class limit is found by subtracting 1 from the second lower class limit to avoid overlap. Continuing this pattern, the upper class limits would be:

Upper Class Limits: 19, 34, 49, 64, 79, 94, 109, 124

With both lower and upper class limits defined, you can now tally the frequency of data points that fall within each class. This involves counting how many data values lie between each pair of limits. For instance, if you have data values that fall between 5 and 19, you would count those occurrences and record them accordingly.

After completing the frequency counts for all classes, you will have a complete frequency distribution that summarizes the dataset effectively. This structured approach not only aids in organizing data but also enhances the clarity of analysis, making it easier to interpret results and draw conclusions.

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