The problem states:

"A set of 23 numbers is given. The smallest number in the set is reduced by 40.
a) How does the arithmetic mean change?
b) How does the range change?
c) How does the median change?"

Solution:

1. Change in the arithmetic mean:
   The arithmetic mean is calculated as the sum of all numbers divided by the total number of numbers.
   If the smallest number decreases by 40, the total sum decreases by 40, while the count of numbers (23) remains the same.
   The change in mean is:
   $\Delta \text{Mean} = \frac{-40}{23} \approx -1.74$
   The arithmetic mean decreases by approximately 1.74.

2. Change in the range:
   The range is defined as the difference between the largest and smallest numbers in the set.
   Since the smallest number is reduced by 40, the range increases by 40.

3. Change in the median:
   The median is the middle number when all numbers are arranged in order.
   Reducing the smallest number does not affect the order of the middle numbers, so the median remains unchanged.

Would you like further details or clarifications?

Related Questions:

1. How is the median affected if the largest number is increased instead of the smallest number?
2. What happens to the range if both the smallest and largest numbers are decreased by the same value?
3. How would the mean change if a number other than the smallest or largest is altered?
4. If the dataset had an even number of elements, how would the median calculation differ?
5. Why does the mean depend on every value in the set, but the median often does not?

Tip:

Remember, the mean is sensitive to every value in the dataset, while the median depends only on the central value(s).