Directions: Consider the following data and answer questions: S. No. Class limit Frequency 1. 101-200 4 2. 201-300 12 3. 301-400 24 4. 401-500 40 5. 501-600 16 6. 601-700 12 7. 701-800 10 8. 801-900 5 9. 901-1000 2
Which one of the following is the relative frequency in percentage for class limit 301-400 from the given data set.
19.2
Calculating Relative Frequency Percentage
The question asks for the relative frequency in percentage for the class limit 301-400 from the provided data set. To find the relative frequency, we first need to determine the total frequency of all classes. Relative frequency for a specific class is calculated by dividing the frequency of that class by the total frequency. To express it as a percentage, we multiply the result by 100.
Understanding the Data Set
The given data set is a frequency distribution table showing the frequency for different class limits:
| S. No. | Class Limit | Frequency |
|---|---|---|
| 1. | 101-200 | 4 |
| 2. | 201-300 | 12 |
| 3. | 301-400 | 24 |
| 4. | 401-500 | 40 |
| 5. | 501-600 | 16 |
| 6. | 601-700 | 12 |
| 7. | 701-800 | 10 |
| 8. | 801-900 | 5 |
| 9. | 901-1000 | 2 |
Step-by-Step Calculation of Relative Frequency Percentage
Step 1: Calculate the Total Frequency
The total frequency is the sum of frequencies of all classes.
\( \text{Total Frequency} = 4 + 12 + 24 + 40 + 16 + 12 + 10 + 5 + 2 \)
\( \text{Total Frequency} = 125 \)
Step 2: Identify the Frequency for the Class Limit 301-400
From the table, the frequency for the class limit 301-400 is 24.
\( \text{Frequency of class 301-400} = 24 \)
Step 3: Calculate the Relative Frequency
Relative frequency is the ratio of the frequency of the specific class to the total frequency.
\( \text{Relative Frequency} = \frac{\text{Frequency of class 301-400}}{\text{Total Frequency}} \)
\( \text{Relative Frequency} = \frac{24}{125} \)
\( \text{Relative Frequency} = 0.192 \)
Step 4: Convert Relative Frequency to Percentage
To get the relative frequency percentage, multiply the relative frequency by 100.
\( \text{Relative Frequency Percentage} = \text{Relative Frequency} \times 100 \)
\( \text{Relative Frequency Percentage} = 0.192 \times 100 \)
\( \text{Relative Frequency Percentage} = 19.2\% \)
Therefore, the relative frequency in percentage for the class limit 301-400 is 19.2%.
Revision Table: Frequency Distribution Analysis
| Class Limit | Frequency (f) | Total Frequency (N) | Relative Frequency (f/N) | Relative Frequency Percentage (f/N * 100%) |
|---|---|---|---|---|
| 101-200 | 4 | 125 | \(4/125 = 0.032\) | \(0.032 \times 100\% = 3.2\%\) |
| 201-300 | 12 | 125 | \(12/125 = 0.096\) | \(0.096 \times 100\% = 9.6\%\) |
| 301-400 | 24 | 125 | \(24/125 = 0.192\) | \(0.192 \times 100\% = 19.2\%\) |
| 401-500 | 40 | 125 | \(40/125 = 0.320\) | \(0.320 \times 100\% = 32.0\%\) |
| 501-600 | 16 | 125 | \(16/125 = 0.128\) | \(0.128 \times 100\% = 12.8\%\) |
| 601-700 | 12 | 125 | \(12/125 = 0.096\) | \(0.096 \times 100\% = 9.6\%\) |
| 701-800 | 10 | 125 | \(10/125 = 0.080\) | \(0.080 \times 100\% = 8.0\%\) |
| 801-900 | 5 | 125 | \(5/125 = 0.040\) | \(0.040 \times 100\% = 4.0\%\) |
| 901-1000 | 2 | 125 | \(2/125 = 0.016\) | \(0.016 \times 100\% = 1.6\%\) |
| Total | 125 | \(125/125 = 1.000\) | \(1.000 \times 100\% = 100\%\) |
Additional Information on Frequency Distributions
A frequency distribution is a summary of how often different values appear in a data set. It is a fundamental tool in statistics for organizing and understanding data.
- Frequency: The number of times a particular value or range of values (class) appears in the data.
- Class Limit: The boundaries for each group or category in a grouped frequency distribution. For example, 301-400 is a class limit.
- Class Interval/Width: The difference between the upper and lower class boundaries. For the class 301-400, assuming continuous data or appropriate boundary adjustments, the width would be 100.
- Relative Frequency: The proportion of the total observations that fall into a specific class. It shows the fraction of data in that class.
- Relative Frequency Percentage: The relative frequency expressed as a percentage. It indicates the percentage of data in a specific class.
- Cumulative Frequency: The sum of frequencies for a class and all preceding classes.
Frequency distributions and relative frequencies are crucial for creating visualizations like histograms and frequency polygons, which help in understanding the distribution pattern of the data (e.g., symmetric, skewed).
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