Points to Remember:
- Median is the middle value in an ordered dataset.
- For grouped data (frequency distribution), the median needs to be calculated using the cumulative frequency.
- The formula for calculating the median for grouped data involves identifying the median class and applying a specific formula.
Introduction:
The question requires calculating the median for a given frequency distribution. A frequency distribution shows the number of times each value (x) occurs (f). The median represents the central tendency of the data, meaning the value that separates the higher half from the lower half of the data when arranged in ascending order. In this case, we have the values (x) 1 through 9 and their corresponding frequencies (f). We will use the cumulative frequency to determine the median.
Body:
1. Calculating Cumulative Frequency:
First, we need to calculate the cumulative frequency (cf). The cumulative frequency is the running total of frequencies.
| x | f | cf |
|—|—|—|
| 1 | 8 | 8 |
| 2 | 10 | 18 |
| 3 | 11 | 29 |
| 4 | 16 | 45 |
| 5 | 20 | 65 |
| 6 | 25 | 90 |
| 7 | 15 | 105 |
| 8 | 9 | 114 |
| 9 | 6 | 120 |
The total number of observations (N) is 120.
2. Identifying the Median Class:
The median is the (N/2)th observation. In this case, (120/2) = 60th observation. The median class is the class interval containing the 60th observation. Looking at the cumulative frequency column, we see that the 60th observation falls within the class interval with x = 5 (cf = 65).
3. Calculating the Median:
The formula for calculating the median for grouped data is:
Median = L + [(N/2 – cf)/f] * h
Where:
- L = Lower limit of the median class (5)
- N = Total number of observations (120)
- cf = Cumulative frequency of the class preceding the median class (45)
- f = Frequency of the median class (20)
- h = Class width (1, as the intervals are consecutive integers)
Substituting the values:
Median = 5 + [(60 – 45)/20] * 1 = 5 + (15/20) = 5 + 0.75 = 5.75
Conclusion:
The median for the given frequency distribution is 5.75. This indicates that approximately half of the observations are below 5.75 and half are above. This calculation provides a robust measure of central tendency, even with the presence of skewed data. Understanding and applying the concept of cumulative frequency is crucial for accurately determining the median in grouped data. Further analysis could involve comparing the median to other measures of central tendency like the mean and mode to gain a comprehensive understanding of the data’s distribution. This approach ensures a balanced and accurate representation of the data’s central tendency.