To find the mean when class sizes (or class intervals) are different, you use a weighted average approach, focusing on the frequency and midpoint of each class interval. Here's how:
Understanding the Problem
When dealing with grouped data where the class intervals are not uniform (i.e., they have different widths), a simple average won't accurately represent the central tendency. This is because wider intervals may contain more data points, and their contribution to the overall mean needs to be appropriately weighted.
The Formula
The formula to calculate the mean in this scenario is:
Mean = ∑(fi * xi) / ∑fi
Where:
- fi = frequency of the i-th class interval
- xi = midpoint of the i-th class interval
Steps to Calculate the Mean
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Determine the Class Midpoints (xi): For each class interval, calculate the midpoint by averaging the upper and lower class limits.
xi = (Upper Class Limit + Lower Class Limit) / 2
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*Multiply Frequency by Midpoint (fi xi):** For each class interval, multiply the frequency (fi) by the corresponding midpoint (xi).
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*Sum the Products (∑(fi xi)):** Add up all the values obtained in step 2. This represents the sum of the weighted midpoints.
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Sum the Frequencies (∑fi): Add up the frequencies of all the class intervals. This represents the total number of observations.
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Calculate the Mean: Divide the sum of the products (from step 3) by the sum of the frequencies (from step 4). This gives you the mean of the grouped data.
Example
Let's say we have the following data on the weights of students:
| Class Interval (Weight in kg) | Frequency (Number of Students) |
|---|---|
| 40-50 | 10 |
| 50-60 | 15 |
| 60-75 | 20 |
| 75-90 | 5 |
-
Calculate Midpoints:
- 40-50: (40+50)/2 = 45
- 50-60: (50+60)/2 = 55
- 60-75: (60+75)/2 = 67.5
- 75-90: (75+90)/2 = 82.5
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Multiply Frequency by Midpoint:
- 40-50: 10 * 45 = 450
- 50-60: 15 * 55 = 825
- 60-75: 20 * 67.5 = 1350
- 75-90: 5 * 82.5 = 412.5
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Sum the Products: 450 + 825 + 1350 + 412.5 = 3037.5
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Sum the Frequencies: 10 + 15 + 20 + 5 = 50
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Calculate the Mean: 3037.5 / 50 = 60.75
Therefore, the mean weight of the students is 60.75 kg.
Importance of Class Midpoints
The class midpoint serves as the representative value for all data points within that interval. By using the midpoint, we're essentially assuming that the data is evenly distributed within each class interval. This is an approximation, but it's generally a reasonable one, especially when dealing with a large dataset.
Conclusion
Calculating the mean with varying class sizes requires a weighted approach using the class midpoints and their respective frequencies. This method provides a more accurate representation of the central tendency of the data compared to a simple average.
