Points to Remember:
- Median: The middle value in a dataset when arranged in ascending order. For an even number of data points, it’s the average of the two middle values.
- Cumulative Frequency: The sum of frequencies up to a given point in a frequency distribution.
- Frequency Distribution: A table showing the number of times each value (or range of values) occurs in a dataset.
Introduction:
This question requires a factual and analytical approach. We are given a frequency distribution table showing the life length (in hours) of 100 electric bulbs. The median life is stated as 20 hours. Our task is to determine the missing frequency, ‘x’, using the properties of the median and cumulative frequency. The median is a crucial measure of central tendency, providing the middle value of a dataset. Understanding cumulative frequency is key to solving this problem.
Body:
1. Understanding the Data:
The provided data (assumed, as the table is missing from the prompt) would likely be in the form of a table with columns for “Life Length (hours)”, “Frequency”, and “Cumulative Frequency”. The “Life Length” column would show ranges of bulb lifespans (e.g., 0-5 hours, 5-10 hours, etc.). The “Frequency” column would show how many bulbs fell into each lifespan range. The “Cumulative Frequency” column would be the running total of frequencies. One of the frequency values would be represented by ‘x’.
2. Calculating the Median:
Since we have 100 bulbs (an even number), the median is the average of the 50th and 51st values when the data is arranged in ascending order. This means the cumulative frequency up to the median class (the class containing the median value) must be at least 50.
3. Finding the Missing Frequency (x):
To find ‘x’, we need to work with the cumulative frequency. Let’s assume (for illustrative purposes) a simplified example of the frequency distribution:
| Life Length (hours) | Frequency | Cumulative Frequency |
|—|—|—|
| 0-10 | 10 | 10 |
| 10-20 | 20 | 30 |
| 20-30 | x | 30 + x |
| 30-40 | 30 | 30 + x + 30 |
| 40-50 | 20 | 80 + x |
| 50-60 | 20 | 100 |
In this example, the median (20 hours) falls within the 20-30 hour range. Since the median is the average of the 50th and 51st values, the cumulative frequency up to the upper limit of the median class (30 hours) must be greater than or equal to 50. Therefore:
30 + x ⥠50
Solving for x:
x ⥠20
However, the total frequency is 100. We need to consider the cumulative frequency after the median class. If x were significantly larger, the median would shift. A more precise solution requires knowing the exact class boundaries and frequencies of the other classes. Further analysis would involve interpolation within the median class to pinpoint the exact value of x.
Conclusion:
Determining the missing frequency ‘x’ requires a systematic approach using the properties of the median and cumulative frequency. The solution involves analyzing the cumulative frequency distribution to ensure the median falls within the expected range. Without the complete frequency distribution table, a precise value for ‘x’ cannot be calculated. However, the methodology outlined above demonstrates how to solve this type of problem. Further information is needed to complete the calculation. The process highlights the importance of accurate data collection and analysis in statistical studies. A complete and accurate frequency distribution is crucial for reliable statistical inferences.