Let's analyze the problem from the image. The set of numbers represents collectible card values for 10 students. The values are already ordered from least to greatest:

387, 430, 457, 471, 476, 538, 557, 594, 595, 685

The question asks what happens to the median and mean when the number 685 is changed to 585.

a) What happens to the median?

• The median is the middle value in a sorted dataset. Since there are 10 numbers, the median is the average of the 5th and 6th values in the ordered list: $\text{Median} = \frac{476 + 538}{2} = 507$ Changing 685 to 585 does not affect the middle values (476 and 538). Therefore, the median remains the same.

b) What happens to the mean?

• The mean is calculated by summing all the numbers and dividing by the number of values. Let's compute the original mean and the new mean after replacing 685 with 585.

Original mean:

$\text{Original sum} = 387 + 430 + 457 + 471 + 476 + 538 + 557 + 594 + 595 + 685 = 5190$ $\text{Original mean} = \frac{5190}{10} = 519$

New mean (after changing 685 to 585):

$\text{New sum} = 5190 - 685 + 585 = 5090$ $\text{New mean} = \frac{5090}{10} = 509$

So, the mean decreases from 519 to 509.

Final answers:

a) The median remains the same.
b) The mean decreases.

Would you like further details or have any other questions?

Here are 5 related questions:

1. How is the median affected if the change were in one of the middle values?
2. How does the standard deviation change with this modification?
3. What happens to the range of the dataset with this change?
4. How would removing a value instead of changing it affect the median and mean?
5. How does the mode change when you modify values in a dataset?

Tip: The median is less sensitive to extreme values compared to the mean, making it a better measure of central tendency in skewed distributions.