Finding the median of grouped data involves identifying the median class and then using a specific formula to pinpoint the median value within that class.
Here's a step-by-step guide:
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Organize the Data: Ensure your data is presented in a grouped frequency distribution table, with class intervals and corresponding frequencies.
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Calculate Cumulative Frequencies: Create a "cumulative frequency" column. For each class, add the frequency of that class to the cumulative frequency of the preceding class. The cumulative frequency of the last class represents the total number of observations (n).
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Identify the Median Class: The median class is the class interval where the cumulative frequency is greater than or equal to n/2 (where n is the total number of observations).
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Apply the Median Formula: Use the following formula to calculate the median:
Median = l + [ (n/2 - c) / f ] * h
Where:
- l = Lower limit of the median class
- n = Total number of observations (sum of frequencies)
- c = Cumulative frequency of the class preceding the median class
- f = Frequency of the median class
- h = Class width (the difference between the upper and lower limits of any class interval, assuming equal class widths)
Example:
Let's say we have the following grouped data:
| Class Interval | Frequency (f) | Cumulative Frequency (c) |
|---|---|---|
| 0-10 | 5 | 5 |
| 10-20 | 8 | 13 |
| 20-30 | 12 | 25 |
| 30-40 | 7 | 32 |
| 40-50 | 3 | 35 |
Here, n = 35. Therefore, n/2 = 17.5
The median class is 20-30 (because its cumulative frequency, 25, is the first one greater than or equal to 17.5).
Applying the formula:
- l = 20 (lower limit of the median class)
- n = 35
- c = 13 (cumulative frequency of the class preceding the median class)
- f = 12 (frequency of the median class)
- h = 10 (class width)
Median = 20 + [(35/2 - 13) / 12] 10
Median = 20 + [(17.5 - 13) / 12] 10
Median = 20 + [4.5 / 12] 10
Median = 20 + 0.375 10
Median = 20 + 3.75
Median = 23.75
Therefore, the median of this grouped data is 23.75.
Key Considerations:
- Equal Class Widths: The formula assumes equal class widths. If class widths are unequal, adjustments may be necessary, or alternative methods used.
- Continuous Data: This method is typically used for continuous data that has been grouped.
- Interpretation: The median represents the middle value of the dataset. Half of the observations fall below the median, and half fall above it.
