The mean of 21 observations, all of them distinct, is 40. If each observation greater than the median is increased by 21, what will be the new mean of all observations?

Introduction / Context:
This statistics question links the mean and median of a data set and describes a transformation applied only to the values greater than the median. You are told the mean of 21 distinct observations. Then each observation greater than the median is increased by 21, and you must find the new mean. This problem is useful for understanding how selective changes in a data set affect the overall mean.

Given Data / Assumptions:
- There are 21 observations, all different from one another.
- The mean (average) of these 21 observations is 40.
- This means the total of all 21 observations is 21 * 40.
- Afterward, each observation greater than the median is increased by 21.
- For 21 ordered observations, the median is the 11th observation.
- Therefore, there are 10 observations greater than the median.
- We must find the new mean after each of those 10 observations is increased by 21.

Concept / Approach:
The mean is total sum divided by the number of observations. If we know how much the total sum changes, we can easily calculate the new mean. The key observation here is that only the 10 largest values change, and each of them increases by 21. This means the total increase in the sum is 10 * 21. Since the number of observations remains 21, we divide the new total by 21 to get the new mean. We do not need to know the individual values or even the median itself.

Step-by-Step Solution:
Step 1: Original mean = 40.Step 2: Number of observations = 21.Step 3: Original total sum of all observations = 21 * 40 = 840.Step 4: For 21 ordered observations, the median is the 11th observation.Step 5: Observations greater than the median are the top 10 observations (positions 12 to 21).Step 6: Each of these 10 observations is increased by 21, so the total increase in the sum = 10 * 21 = 210.Step 7: New total sum after the change = original total + increase = 840 + 210 = 1050.Step 8: The number of observations remains 21.Step 9: New mean = new total sum / number of observations = 1050 / 21.Step 10: Compute 1050 / 21 = 50.Step 11: Therefore, the new mean of the observations is 50.

Verification / Alternative check:
You can imagine a simple illustrative set of 21 values arranged in increasing order. Increasing each of the 10 largest values by 21 clearly adds 210 to the total sum. Since the number of observations stays the same, the mean must increase by 210 / 21 = 10. Starting from an original mean of 40, the new mean becomes 40 + 10 = 50. This reasoning perfectly matches the calculated result.

Why Other Options Are Wrong:
A new mean of 30 or 45 would correspond to a decrease or a smaller increase in the total, which contradicts the fact that several values have been raised by a positive amount. A mean of 40 would imply no change in total, which is not true because 10 numbers are increased by 21 each. The value 50.5 might appear close, but it corresponds to a total sum that is not an integer multiple of the increased increments. Only 50 is consistent with adding 210 to the total and dividing by 21.

Common Pitfalls:
Some learners mistakenly think that only the median increases or that the number of modified observations is 11 instead of 10. Others attempt to find the actual median value, which is unnecessary. The key is to understand that there are exactly 10 values greater than the median in a set of 21 distinct ordered observations and that each of those 10 values increases by 21, so the effect on the mean is straightforward to compute.

Final Answer:
The new mean of the observations will be 50.