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Chapter 2: Problem 5

Class Limits A data set with whole numbers has a low value of 20 and a high value of \(82 .\) Find the class width and class limits for a frequency table with 7 classes.

Short Answer

Expert verified
The class width is 9, and the class limits for 7 classes are 20-28, 29-37, 38-46, 47-55, 56-64, 65-73, and 74-82.

Step by step solution

01

Calculate Range of the Data

First, determine the range of the data by subtracting the low value from the high value. In this case, the low value is 20 and the high value is 82. Therefore, the range is calculated as follows:\[\text{Range} = 82 - 20 = 62\]
02

Determine the Class Width

Next, calculate the class width by dividing the range by the number of classes, and then round up to the nearest whole number. There are 7 classes:\[\text{Class Width} = \frac{\text{Range}}{\text{Number of Classes}} = \frac{62}{7} \approx 8.86\]Since we need to round up, the class width is 9.
03

Establish the Class Limits

Use the class width to determine the class limits. Start with the low value and add the class width to get the upper limit of the first class. - First class: 20 to 20 + 9 - 1 = 28 - Second class: 29 to 29 + 9 - 1 = 37 - Continue this pattern for the remaining classes. The full class limits are: 1. 20 - 28 2. 29 - 37 3. 38 - 46 4. 47 - 55 5. 56 - 64 6. 65 - 73 7. 74 - 82

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Class Width
The Class Width is an essential part of creating a frequency distribution. It helps organize data into tidy, manageable classes. To find the class width, you need to understand the data's range and how many classes you wish to divide it into. In our specific problem, we have a data range from 20 to 82, and we want 7 classes.

Here's how to calculate the class width:
  • First, calculate the range. Subtract the smallest number from the largest number. For this case, it's 82 - 20 = 62.
  • Next, divide the range by the number of classes. In our example, that's \( \frac{62}{7} = 8.86 \)
  • Finally, we round up to the nearest whole number because class width should be a whole number. So, 8.86 rounds up to 9.
Rounding up is crucial as it ensures all data is covered adequately in each class.
Class Limits
Class limits define the boundaries of each class in a frequency distribution. They're essential for organizing data efficiently and avoiding overlap or gaps in the classes. Each class has a lower and upper limit.

To establish class limits:
  • Start with the smallest value in your dataset, which is 20 in this situation.
  • Add the class width (9 in our example) to find the upper limit of the first class. Remember to subtract 1 to get the exact range. So, the calculation becomes \( 20 + 9 - 1 = 28 \).
  • The second class starts at one number higher: 29. Repeat the process to find subsequent class limits:
    • First class: 20 - 28
    • Second class: 29 - 37
    • Continue this pattern until you have all classes covered, up to the high value of 82.
Ensure the classes cover the entire data range without overlaps.
Data Range
Understanding the Data Range is a significant step when dealing with frequency distributions. It defines the span of the entire dataset. Knowing the range helps you figure out how to segment the data into classes properly.

Here's how you find and use the range:
  • Identify the smallest and largest numbers in your dataset.
  • Subtract the smallest from the largest to get your range. In this case, that's 82 - 20 = 62.
  • This range tells us about the overall spread of the data, helping to determine class width and ensuring the frequency distribution includes all data points.
The data range provides a clear view of how widely distributed your data is, crucial for accurate data analysis and presentation.

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Most popular questions from this chapter

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