Points to Remember:
- Median is the middle value in an ordered dataset.
- For grouped data (class intervals and frequencies), the median needs to be calculated using interpolation.
- The cumulative frequency is crucial for finding the median class.
Introduction:
The question requires calculating the median from grouped data, which is data presented in class intervals with corresponding frequencies. Unlike ungrouped data where the median is simply the middle value after sorting, grouped data necessitates a slightly more complex calculation involving interpolation. The median represents the central tendency of the data, indicating the value that separates the lower 50% from the upper 50%. We will use the cumulative frequency to identify the median class and then apply the interpolation formula to determine the precise median value.
Body:
1. Understanding the Data:
To calculate the median, we need the class intervals and their corresponding frequencies. Let’s assume the following data (replace with the actual data provided in the question):
| Class Interval | Frequency (f) | Cumulative Frequency (cf) |
|—|—|—|
| 10-20 | 5 | 5 |
| 20-30 | 8 | 13 |
| 30-40 | 12 | 25 |
| 40-50 | 7 | 32 |
| 50-60 | 3 | 35 |
2. Finding the Median Class:
The total number of observations (N) is 35. The median is the (N+1)/2 = (35+1)/2 = 18th observation. We look at the cumulative frequency column to find the class interval containing the 18th observation. This is the median class (20-30) because its cumulative frequency (13) is the smallest cumulative frequency greater than or equal to 18.
3. Calculating the Median using Interpolation:
The formula for calculating the median for grouped data is:
Median = L + [(N/2 – cf) / f] Ã h
Where:
- L = Lower limit of the median class (20)
- N = Total number of observations (35)
- cf = Cumulative frequency of the class preceding the median class (5)
- f = Frequency of the median class (8)
- h = Class width (10)
Substituting the values:
Median = 20 + [(18 – 5) / 8] Ã 10 = 20 + (13/8) Ã 10 = 20 + 16.25 = 36.25
Conclusion:
The median of the given grouped data (assuming the data provided above) is 36.25. This value represents the central point of the dataset, dividing the data into two equal halves. The calculation involved identifying the median class using cumulative frequency and then applying the interpolation formula to obtain a more precise estimate of the median. This method is essential when dealing with grouped data where individual data points are not available. For future analysis, ensuring accurate data collection and appropriate class interval selection is crucial for obtaining a reliable median value. Further analysis could involve comparing the median with other measures of central tendency like the mean and mode to gain a comprehensive understanding of the data distribution. This would contribute to more informed decision-making based on a holistic understanding of the data.