The mean (arithmetic average) of 21 different observations is 40. If each observation that is greater than the median of the data is increased by 21, what will be the new mean of all 21 observations?

Introduction / Context:
This question checks understanding of how the arithmetic mean changes when a fixed amount is added to some of the observations. The twist is that the modification applies only to those observations that are greater than the median, which requires careful counting of how many such observations there are in a set of 21 values.

Given Data / Assumptions:

• Number of observations = 21.
• All observations are distinct.
• Original mean of all 21 observations = 40.
• Each observation greater than the median is increased by 21.
• Median of 21 observations is the 11th value when arranged in order.

Concept / Approach:
The key facts are:

• The sum of all observations is mean * number of observations.
• For 21 ordered observations, 10 values lie above the median, and 10 lie below it.
• Adding a constant k to n observations increases the total sum by n * k.
• New mean = new total sum / number of observations.

We simply compute the original total, adjust it by the net increase, and divide by 21 to get the new mean.

Step-by-Step Solution:
Step 1: Compute original total of the 21 observations. Original total = 40 * 21 = 840. Step 2: Determine how many observations are greater than the median. With 21 ordered observations, the median is the 11th value. There are 10 observations greater than this median. Step 3: Compute the increase in total when each of these 10 values is increased by 21. Increase in total = 10 * 21 = 210. Step 4: Compute the new total. New total = 840 + 210 = 1050. Step 5: Compute the new mean. New mean = 1050 / 21 = 50.

Verification / Alternative check:
A simple check is to note that only 10 out of 21 observations shifted upward by the same amount. If every observation had increased by 21, the mean would also increase by 21. Here the fraction of observations increased is 10 / 21, so the mean increases by 21 * (10 / 21) = 10. Starting from original mean 40, the new mean is 40 + 10 = 50, which matches our computed answer.

Why Other Options Are Wrong:
An answer of 30 suggests a decrease in mean, which is impossible when certain observations are increased. Values like 50.5 or 45 correspond to incorrect counts of how many observations were changed or incorrect arithmetic. Only 50 is consistent with both methods of reasoning and the given data.

Common Pitfalls:
Students sometimes mistakenly think that only one value (the median) is affected, or they miscount and take 11 observations instead of 10 above the median. Another frequent error is to try to change each observation individually rather than using total sums, which leads to messy calculations. The safe strategy is always to convert means into totals, apply the changes, and then convert back to the mean.

Final Answer:
The new mean of all 21 observations is 50.