Consider the following data for the next two (02) items that follow : Class 40-50 50-60 60-70 70-80 Frequency 4 3 1 2
What is the mean of the distribution?
56
Calculating the Mean of a Frequency Distribution
The question asks us to find the mean of the given frequency distribution. We are provided with class intervals and their corresponding frequencies. To find the mean of a grouped frequency distribution, we typically use the formula:
\( \bar{x} = \frac{\sum f_i x_i}{\sum f_i} \)
Where:
- \( \bar{x} \) is the mean
- \( f_i \) is the frequency of the i-th class
- \( x_i \) is the class mark (midpoint) of the i-th class
- \( \sum f_i \) is the sum of all frequencies (total number of observations)
- \( \sum f_i x_i \) is the sum of the products of each frequency and its corresponding class mark
First, let's find the class mark (\( x_i \)) for each class interval. The class mark is the midpoint of the class interval. It is calculated as:
\( \text{Class Mark} = \frac{\text{Lower Limit} + \text{Upper Limit}}{2} \)
Now, let's calculate the class mark for each given class interval:
- For class 40-50: \( x_1 = \frac{40 + 50}{2} = \frac{90}{2} = 45 \)
- For class 50-60: \( x_2 = \frac{50 + 60}{2} = \frac{110}{2} = 55 \)
- For class 60-70: \( x_3 = \frac{60 + 70}{2} = \frac{130}{2} = 65 \)
- For class 70-80: \( x_4 = \frac{70 + 80}{2} = \frac{150}{2} = 75 \)
Next, we multiply each frequency (\( f_i \)) by its corresponding class mark (\( x_i \)).
- For class 40-50: \( f_1 x_1 = 4 \times 45 = 180 \)
- For class 50-60: \( f_2 x_2 = 3 \times 55 = 165 \)
- For class 60-70: \( f_3 x_3 = 1 \times 65 = 65 \)
- For class 70-80: \( f_4 x_4 = 2 \times 75 = 150 \)
Now, we can organize this information and calculate the sums in a table:
| Class | Frequency (\( f_i \)) | Class Mark (\( x_i \)) | \( f_i x_i \) |
| 40-50 | 4 | 45 | \( 4 \times 45 = 180 \) |
| 50-60 | 3 | 55 | \( 3 \times 55 = 165 \) |
| 60-70 | 1 | 65 | \( 1 \times 65 = 65 \) |
| 70-80 | 2 | 75 | \( 2 \times 75 = 150 \) |
| Total | \( \sum f_i = 10 \) | \( \sum f_i x_i = 180 + 165 + 65 + 150 = 560 \) |
Finally, we apply the formula for the mean:
\( \bar{x} = \frac{\sum f_i x_i}{\sum f_i} = \frac{560}{10} = 56 \)
So, the mean of the given frequency distribution is 56.
Revision Table: Key Concepts for Mean Calculation
| Term | Definition | How to Calculate for Grouped Data |
| Mean | The average of a dataset. | \( \frac{\sum f_i x_i}{\sum f_i} \) |
| Frequency (\( f_i \)) | The number of times a value or observation occurs. | Given in the frequency distribution table. |
| Class Mark (\( x_i \)) | The midpoint of a class interval. | \( \frac{\text{Lower Limit} + \text{Upper Limit}}{2} \) |
| \( \sum f_i \) | Sum of all frequencies. | Add up all values in the frequency column. |
| \( \sum f_i x_i \) | Sum of (frequency × class mark) for all classes. | Calculate \( f_i x_i \) for each class and add them up. |
Additional Information: Measures of Central Tendency
Besides the mean, other important measures of central tendency include the median and the mode. For grouped data like the frequency distribution provided:
- Median: The middle value when the data is arranged in order. For grouped data, the median is calculated using a specific formula involving the median class (the class where the cumulative frequency is greater than or equal to \( \frac{N}{2} \), where \( N = \sum f_i \)). The formula is \( \text{Median} = L + \frac{(\frac{N}{2} - CF)}{f} \times h \), where L is the lower boundary of the median class, CF is the cumulative frequency of the class preceding the median class, f is the frequency of the median class, and h is the class width.
- Mode: The value that appears most frequently. For grouped data, the mode is found using the modal class (the class with the highest frequency). The formula is \( \text{Mode} = L + \frac{(f_1 - f_0)}{(2f_1 - f_0 - f_2)} \times h \), where L is the lower boundary of the modal class, \( f_1 \) is the frequency of the modal class, \( f_0 \) is the frequency of the class preceding the modal class, \( f_2 \) is the frequency of the class succeeding the modal class, and h is the class width.
Understanding these measures helps in analyzing and summarizing characteristics of a dataset.
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