A data Collection has 20 distinct observations with median 22. One more number is included in the data, but the median does not change. Find this number

Answer

To solve the problem step by step, we need to analyze the given information about the data collection and the median.

### Step 1: Understand the initial data collection
We have a data collection with 20 distinct observations, and the median of this data is 22.

### Step 2: Determine the position of the median
For a dataset with an even number of observations (in this case, 20), the median is calculated as the average of the two middle numbers. Since there are 20 observations, the two middle numbers would be the 10th and 11th terms when the data is arranged in ascending order.

### Step 3: Identify the values of the 10th and 11th terms
Given that the median is 22, we can conclude that the average of the 10th and 11th terms is 22. Therefore, we can express this mathematically:
\[ \frac{T_{10} + T_{11}}{2} = 22 \]
This implies:
\[ T_{10} + T_{11} = 44 \]

### Step 4: Adding one more number to the dataset
When we add one more number to the dataset, the total number of observations becomes 21. Since 21 is an odd number, the median will now be the middle term, which is the 11th term in the ordered dataset.

### Step 5: Analyzing the effect of the new number on the median
To maintain the median at 22 after adding a new number, this new number must be placed such that it does not affect the position of the 11th term.

### Step 6: Determine the new number
To keep the median at 22, the new number must be equal to or less than the 11th term (which is 22) and greater than or equal to the 10th term (which is also less than or equal to 22). Therefore, the new number must be 22.

### Conclusion
Thus, the number that can be added to the data collection while keeping the median unchanged is **22**.

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