Understanding Class Intervals and Frequency
To find the median from grouped data, we need to follow a systematic approach. In the given data, the class intervals along with their corresponding frequencies need to be organized:
- Class Intervals and Frequencies:
- 100-150: 415
- 150-200: 520
- 200-250: 1225
- 250-300: 2300
- 300-350: 2
Step 1: Calculate Cumulative Frequency
The first step is to calculate the cumulative frequency (CF):
- Cumulative Frequency Calculation:
- CF for 100-150: 415
- CF for 150-200: 415 + 520 = 935
- CF for 200-250: 935 + 1225 = 2160
- CF for 250-300: 2160 + 2300 = 4460
- CF for 300-350: 4460 + 2 = 4462
Step 2: Find Total Frequency (N)
- Total Frequency (N):
- N = 4462
Step 3: Determine Median Class
To find the median, we need to locate the median class:
- Median Position:
- Median Position = N/2 = 4462/2 = 2231
- Since 2231 falls in the cumulative frequency of 250-300, this is the median class.
Step 4: Calculate Median
Using the median formula for grouped data:
- Formula:
- Median = L + [(N/2 - CF)/f] × c
Where:
- L = lower boundary of the median class = 250
- N = total frequency = 4462
- CF = cumulative frequency of the class before the median class = 2160
- f = frequency of the median class = 2300
- c = width of the median class = 50
Step 5: Substitute Values
Now substituting the values into the formula:
Median = 250 + [(2231 - 2160) / 2300] × 50
Calculating gives:
Median ≈ 250 + [71/2300] × 50 ≈ 250 + 1.54 ≈ 251.54
Conclusion
Thus, the median of the given data is approximately 251.54.