To find the mean for grouped data, you can follow these steps:
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Create a Frequency Distribution Table: Start with your grouped data organized in classes or intervals, along with their corresponding frequencies.
Example:
| Class Interval | Frequency (f) | |----------------|----------------| | 1 - 10 | 5 | | 11 - 20 | 12 | | 21 - 30 | 8 |
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Calculate the Midpoint for Each Class: The midpoint (often denoted as \(x_i\)) of each class interval is found by averaging the lower and upper bounds of the interval.
\[ x_i = \frac{\text{Lower limit} + \text{Upper limit}}{2} \]
For the table above:
| Class Interval | Frequency (f) | Midpoint (x_i) | |----------------|----------------|-----------------| | 1 - 10 | 5 | 5.5 | | 11 - 20 | 12 | 15.5 | | 21 - 30 | 8 | 25.5 |
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Calculate \(f_i \cdot x_i\): Multiply the frequency of each class by the corresponding midpoint.
| Class Interval | Frequency (f) | Midpoint (x_i) | \(f_i \cdot x_i\) | |----------------|----------------|-----------------|--------------------| | 1 - 10 | 5 | 5.5 | 27.5 | | 11 - 20 | 12 | 15.5 | 186 | | 21 - 30 | 8 | 25.5 | 204 |
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Sum \(f_i\) and \(f_i \cdot x_i\): Calculate the total frequency (\(N\)) and the total of \(f_i \cdot x_i\) values.
- Total Frequency \(N\) = \(5 + 12 + 8 = 25\)
- Sum of \(f_i \cdot x_i\) = \(27.5 + 186 + 204 = 417.5\)
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Calculate the Mean: Use the formula for the mean of grouped data:
\[ \text{Mean} = \frac{\sum (f_i \cdot x_i)}{N} \]
\[ \text{Mean} = \frac{417.5}{25} = 16.7 \]
So, the mean of the grouped data would be 16.7.